<<

Fundamental Optics Optical Specifications Material Properties Optical Coatings 1.1 1.2 1.3 1.6 1.8 1.46 1.49 1.52 1.11 1.17 1.18 1.20 1.23 1.26 1.27 1.29 1.32 1.36 1.37 1.41 1 www.cvimellesgriot.com Fundamental Optics Fundamental Etalons Ultrafast Theory Introduction Paraxial Formulas Properties of Systems Lens Combination Formulas Performance Factors Lens Shape Lens Combinations Effects Lens Selection Spot Size Aberration Balancing Definition of Terms Paraxial Lens Formulas Principal-Point Locations Fundamental Optics Fundamental 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.1 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:28PMPage1.2

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics covered in systems withuniformillumination;opticalforGaussianbeamsare Finally, itshouldbenotedthatthediscussioninthischapterrelatesonlyto 1.2 readers, aglossaryoftermsisprovidedin Because someofthetermsusedinthischaptermaynotbefamiliartoall may thereforerequiremodification. such ascomponentsize, cost,orproductavailability. Systemparameters In practice, thesecondstepmayrevealconflictswithdesignconstraints, process is simple generalizationscanbeused,especiallywhenthelensselection simplest opticalsystemsgenerallyrequirescomputerraytracing,but effects ofaberrations. Atrulyrigorousperformanceanalysisforallbutthe their actualperformanceisevaluated withspecialattentionpaidtothe Second, actualcomponentsarechosenbasedontheseparaxialvalues, and paraxial calculationsarecoveredinthenextsectionofthischapter. length(s), clearaperture(),andobjectimageposition.These are madetodeterminecriticalparameterssuchasmagnification,focal broken downintotwomainsteps. First, paraxialcalculations(firstorder) The processofsolvingvirtuallyanyopticalengineeringproblemcanbe Introduction detailed specificationsonCVIMellesGriotproducts. contact usforhelpinproductselectionortoobtainmore neers optical engineeringsupportisrequired,ourapplicationsengi- commonly to selectthemostappropriatecataloglensesfor information giveninthischapterissufficienttoenabletheuser applications engineersateachofourfacilitiesworldwide. The CVI MellesGriotmaintainsastaffofknowledgeable, experienced Support Fundamental Optics are available toprovideassistance. Donothesitateto confined toalimitedrangeofcomponentshapes. Gaussian Beam Optics Beam Gaussian encountered applications. However, whenadditional . Definition of Terms of Definition . NIERN PROCESS ENGINEERING OPTICAL THE c Determine ifperformanc solve forremaining omponent valuesc c and Using Determine o Estimate performanc harac Pic orig Determine ifc c b parameters, suc b system mag harac jec ased knownparameters, k lensc with anyb d inal paraxialformulas t/imag erived teristic nific onparaxially teristic d c onstraints esig omponents b ation and values e asic s ofsystem d n s meet asic istanc hosen system g h as oals values onflic

Fundamental Optics

es e e t www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.3 ″ h e in alled c www.cvimellesgriot.com imag e (whether real Fundamental Optics Fundamental tively t (whether real t and jec ipal point H″ Fundamental Optics ollec ipal point H jec are c ″ s

ht of princ es, with ob and s e ( e, positive for ob istanc ht ht d ″ s istanc istanc ate ate planes), positive for imag d t heig t e heig e d jec jec b b onjug onjug for image to right of H″ for image to left of H″ for an inverted image for an upright image c or virtual) to the left of princ c or virtual) to the rig for object to right of H for object to left of H for convex (diverging) mirrors for concave (converging) 4 = = 4 =o = imag =o = imag 4 = = 4 s s is is i i al points. ″ ″ is is is is ″ ″ h h s s s s m m s For mirrors: f f s rear focal point F″ f ium e positive oth FH and y med b

H″ d h may b e H e infinite if principal points represents b f ation or f to b ally 90% of f) th (EFL) whic ative. nific s CA t and image relative to front and rear foc lens is surround (the second principal point) ) al leng is infinite s s F = mag h v or / A/2 ″ ″ C ate ratio, said s ex 1.0 iameter h tive foc d , assuming = c ″ sin ( F c /s ″ ation of objec ″ onjug lear (typic (as shown) or neg H of ind s c either c front focal point = =effe =ar = lens h object Sign conventions A = A = f v m f C Note loc for image to left of H″ for image to right of H″ for an inverted image for an upright image f for object to left of H (the first principal point) for object to left of H (the for object to right of H 4 = = 4 = 4 is is is is is is ″ ″ m m s When using the thin-lens approximation, simply refer to the left and right of the lens. When using the thin-lens approximation, simply s s s Sign Conventions sign conventions: formulas is dependent on adherence to the following of the paraxial lens validity The 1.1) to figure For : (refer Figure 1.1 Figure Paraxial Formulas Paraxial 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.3 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:28PMPage1.4

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics or realimageis0.33mmhighandinverted. formed, andwhatisthemagnification?(Seefigure1.2.) principal pointofaLDX-25.0-51.0-C( A 1-mm-highobjectisplacedontheopticalaxis, 200mmleftofthe Example 1:ObjectoutsideFocalPoint when dealingwithsimpleopticalsystems. plification, calledthethin-lensapproximation,canspeedupcalculation 1.4 known asthehiatus, center ofthelens. Byneglectingthedistancebetweenlens’principalpoints, focal lengthareallreferencedtotheprincipalpoints, nottothephysical With areallensoffinitethickness, theimagedistance, objectdistance, and where (s forms: This relationshipcanbeusedtorecastthefirstformulaintofollowing By definition,magnificationistheratioofimagesizetoobjector in This formulaisreferencedtofigure1.1andthesignconventionsgiven position, andimagepositionisgivenby (object andimagedistance).The relationshipamongfocallength,object length basedonconstraintssuchasmagnificationorconjugatedistances Typically, thefirststepinopticalproblemsolvingistoselectasystemfocal Sign Conventions Sign s m mss s sm m s sf f fm f 11 11 111 f () ″ ″ ″

= = = =

= = =− =− =+ = +=+ s 67 66 m m s Fundamental Optics 50 s s ″ sm ″ 1 s () s ss ++ + () . = ″ m = ss + 1 ) istheapproximateobject-to-imagedistance. 2 mm + 66 s 200 h + 200 h s 1 ″ ″ 1 1 ″ . ″ 77 . m 1 2

= s . ″ = 033 . s ″ becomes theobject-to-imagedistance. This sim- f = 50 mm).Whereistheimage (1.6) (1.5) (1.4) (1.3) (1.2) (1.1) viewed onlybackthroughthelens. In thiscase, thelensisbeingusedasamagnifier, andtheimagecanbe or virtualimageis2.5mmhighandupright. (See figure1.3.) same lens. Whereistheimageformed,andwhatmagnification? The sameobjectisplaced30mmleftoftheprincipalpoint Example 2:ObjectinsideFocalPoint Figure 1.2 principal pointofanLDK-50.0-52.2-C( A 1-mm-highobjectisplacedontheopticalaxis, 50mmleft of thefirst Example 3:ObjectatFocalPoint Figure 1.3 or virtualimageis0.5mmhighandupright. formed, andwhatisthemagnification?(Seefigure1.4.) object s s m m s s 11 11 ″ ″ ″ ″ = = =− =− = =− s s 50 s s − ″ ″ 25 75 50 = = image mm mm − − − Example 1 30 Example 2 50 30 1 25 75 50 1 =− =− F 05 25 1 . . 200 object ( ( f f = = 50 mm,s 50 mm,s F 1 f = 4 = = 50 mm).Whereistheimage 200 mm,s Fundamental Optics 30 mm,s www.cvimellesgriot.com 66.7 F 2 ″= ″=66.7 mm) F 4 2 image 75 mm) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.5 v www.cvimellesgriot.com Fundamental Optics Fundamental f Fundamental Optics NOTE principal surface F-number and F-number and numerical 2 CA Because the sign convention given previously is not used universally Because the sign convention given previously is not the reader may notice differences in the paraxial in all optics texts, will be correct as long as a consistent set results However, formulas. of formulas and sign conventions is used. Figure 1.5 Figure f-numbers can also be defined for any arbitrary ray if its conjugate Ray f-numbers can also be defined for any arbitrary the principal surface of distance and the diameter at which it intersects the optical system are known. (1.8) (1.9) (1.7) 1 25 mm) F 4 ″= 50 mm, s = 50 mm, s 4 = f ( . . f 2 f CA 1 CA image = Example 3 v based on the concepts of focal ratio (f-number or f/#) and based on the concepts of focal ratio (f-number projecting a high-power image. f-number () uniformly with collimated . The f-number defines the angle f-number defines uniformly with collimated light. The 2 sin 2 = == F object NA NA f-number and of the cone of light leaving the lens which ultimately forms the image. This of the cone of light leaving the lens which ultimately forms the image. is an important concept when the throughput or light-gathering power of an optical system is critical, such as when focusing light into a mono- chromator or this cone angle is numerical other term used commonly in defining The NA is the sine of the angle made by the marginal ray with The aperture. using simple , By referring to figure 1.5 and the optical axis. it can be seen that To visualize the f-number, consider a lens with a positive visualize the f-number, To illuminated F-NUMBER AND NUMERICAL APERTURE paraxial calculations used to determine the necessary element The diameter are f-number is the ratio of the focal length of the numerical aperture (NA). The the clear aperture (CA). lens to its “effective” diameter, A simple graphical method can also be used to determine paraxial image A simple graphical method can also be used to graphical approach relies on two simple location and . This a ray that enters the system parallel properties of an optical system. First, focal point. Second, a ray to the optical axis crosses the optical axis at the exits the system from the that enters the first principal point of the system its exit angle (i.e., second principal point parallel to its original direction method has angle). This with the optical axis is the same as its entrance illustrated in figures 1.2 been applied to the three previous examples this second through 1.4. Note that by using the thin-lens approximation, through the center property reduces to the statement that a ray passing of the lens is undeviated. Figure 1.4 Figure 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.5 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 7/6/20091:42PMPage1.6

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.6 consider thefollowingexample. To understandhowtousethisrelationshipbetweenmagnification andNA, SAMPLE CALCULATION as theopticalinvariant. throughput orimagebrightness).This conceptissometimesreferred to notimproveperformance(i.e., will but will increasesystemsizeandcost the objectorimageside, increasingthelensdiameterbeyondthisvalue oneither constrained is if oneisdealingwithasysteminwhichtheNA fication havebeenselected,thevalue ofNAsetsthevalue ofCA.Thus, Since theNAofarayisgivenbyCA/2 diameter hasnoeffectonlight-collectionefficiencyorimagebrightness. cation andNA,therecanbeatheoreticallimitbeyondwhichincreasingthe a brighterimage. However, becauseoftherelationshipbetweenmagnifi- increasing itsCA,wewillbeabletocollectmorelightandtherebyproduce natural toassumethat,byincreasingthediameter( When alensoropticalsystemisusedtocreateanimageofsource, itis involving apertureconstraints. often beusedtodeterminetheoptimumlensdiameterinsituations is completelyindependentofthespecificsopticalsystem,anditcan on theobjectandimagesidesofsystem.This powerfulandusefulresult The magnificationofthesystemisthereforeequaltoratioNAs issimplyt Since Referring tofigure1.6, To understandtheimportanceofNA,consideritsrelationtomagnification. THE OPTICALINVARIANT Imaging Properties ofLensSystems and we arriveat leading to wwhich canberearrangedtoshow CA CA NA″ NA (objectside) m s s ″

= s =

s = = ″ Fundamental Optics (imageside) sin sin 2 2 NA NA s s v ″″ v sin ″ sin ″ = . v v NA NA hhe magnificationofthesyst == ″ = . sin sin v v

= CA 2 s CA 2 s neafcllnt n magni- s, onceafocallengthand ″ f ) ofthelens, thereby em, (1.15) (1.14) (1.13) (1.12) (1.11) (1.10) three. The followingchapter, Making somesimplecalculationshasreducedourchoiceoflensestojust LDX-6.0-7.7-C andLDX-5.0-9.9-C,whicharebiconvex. that meetthesecriteriaareLPX-5.0-5.2-C,whichisplano-convex,and the limitedinputNAofopticalfiber. The singletlensesinthiscatalog diameter lenswillnotresultinanygreatersystemthroughputbecause of optic witha9.1-mmfocallengthand5-mmdiameter. Usingalarger Accomplishing thisimagingtaskwithasinglelensthereforerequiresan and With animageNAof0.25anddistance( CA, theoptimumclearaperture(effectivediameter)oflens. We cannowusetherelationship NA and findthats distances, to determinethatthefocallengthis9.1mm.To determinetheconjugate (using thethin-lensapproximation),wecanuseequation1.3, By definition,themagnificationmustbe0.1.Letting appropriate? distance fromthesourcetofiberbe110mm.Whichlensesare 100 as showninfigure1.7.Assumethatthefiberhasacorediameterof incandescent bulbwithafilament1mmindiameterintoanopticalfiber Suppose itisnecessary, usingasingletlens, tocoupletheoutputofan entrance (image)size. in termsofthef-number. InadditiontothefixedNA,theybothhavea — theyhaveafixedNA.For , thislimitisusuallyexpressed problems appeartobequitedifferent,theybothhavethesamelimitation an opticalfiberorintotheentranceslitofamonochromator. Althoughthese Two verycommonapplicationsofsimpleopticsinvolvecouplinglightinto Example: SystemwithFixedInputNA make afinalchoiceoflensesbasedonvarious performancecriteria. 025 s s sm Amm CA fm m ), () . m andanNAof0.25,thatthedesignrequirestotal =

= +=+ = s 1 5 CA ( ( 20 and m ss + + = s 1 . 100 mmands ″ ″ ) , weutilizeequation1.6, ) 2 ″ , Gaussian Beam Optics Beam Gaussian ″=10 mm. = CA/2s Fundamental Optics or NA″=CA/2s www.cvimellesgriot.com s ″ s ) of10mm, , discusseshowto = s ″ total 110mm (see eq.1.6) (see eq.1.3) ″ to derive Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.7 www.cvimellesgriot.com Fundamental Optics Fundamental ″ s image side 2 Fundamental Optics CA fiber core h″ = 0.1 mm = 10 mm = = 0.25 ″ ″ s v″ NA ″ s f = 9.1 mm CA = 5 mm optical system ! 2s CA s = 100 mm s NA = = 0.025 0.1 1.0 v = 110 mm ″ ″ h h s + s 2 Optical system for focusing the output of an incandescent bulb into an Optical system geometry for focusing the output of an incandescent bulb into Numerical aperture and magnification Numerical aperture CA filament h = 1 mm object side CA magnification = = = 0.1 Figure 1.7 Figure Figure 1.6 Figure 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.7 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.8

Fundamental Optics www.cvimellesgriot.com Lens Combination Formulas

Fundamental Optics Many optical tasks require several lenses in order to achieve an acceptable level of performance. One possible approach to lens combinations is to con- sider each image formed by each lens as the object for the next lens and so Symbols on. This is a valid approach, but it is time consuming and unnecessary. It is much simpler to calculate the effective (combined) focal length and f = combination focal length (EFL), positive if combination principal-point locations and then use these results in any subsequent c final focal point falls to the right of the combination secondary paraxial calculations (see figure 1.8). They can even be used in the optical principal point, negative otherwise (see figure 1.8c). invariant calculations described in the preceding section. = f1 focal length of the first element (see figure 1.8a). EFFECTIVE FOCAL LENGTH = f2 focal length of the second element. The following formulas show how to calculate the effective focal length and principal-point locations for a combination of any two arbitrary com- d = distance from the secondary principal point of the first

Gaussian Beam Optics ponents. The approach for more than two lenses is very simple: Calculate element to the primary principal point of the second element, the values for the first two elements, then perform the same calculation for positive if the primary principal point is to the right of the this combination with the next lens. This is continued until all lenses in the secondary principal point, negative otherwise (see figure 1.8b). system are accounted for. ″= s1 distance from the primary principal point of the first The expression for the combination focal length is the same whether lens element to the final combination focal point (location of the separation distances are large or small and whether f1 and f2 are positive final image for an object at infinity to the right of both lenses), or negative: positive if the focal point is to left of the first element’s primary principal point (see figure 1.8d). ff f = 12 . (1.16) ffd+− ″= 12 s2 distance from the secondary principal point of the second element to the final combination focal point (location This may be more familiar in the form of the final image for an object at infinity to the left of both 111 d lenses), positive if the focal point is to the right of the second Optical Specifications =+ − . (1.17) element’s secondary principal point (see figure 1.8b). fff12ff 12 z = Notice that the formula is symmetric with respect to the interchange of the H distance to the combination primary principal point lenses (end-for-end rotation of the combination) at constant d. The next measured from the primary principal point of the first element, two formulas are not. positive if the combination secondary principal point is to the right of secondary principal point of second element (see figure 1.8d). COMBINATION FOCAL-POINT LOCATION z ″=distance to the combination secondary principal point For all values of f , f , and d, the location of the focal point of the combined H 1 2 measured from the secondary principal point of the second system (s ″), measured from the secondary principal point of the second 2 element, positive if the combination secondary principal point lens (H ″), is given by 2 is to the right of the secondary principal point of the second element (see figure 1.8c). −

Material Properties ff() d s ″ = 21 . 2 +− (1.18) ffd12 Note: These paraxial formulas apply to coaxial combinations of both thick and thin lenses immersed in air or any other = 4 fluid with independent of position. They This can be shown by setting s1 d f1 (see figure 1.8a), and solving assume that light propagates from left to right through an optical system. 111=+ ″ fss212

″ for s2 . Optical Coatings 1.8 Fundamental Optics 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.9

Fundamental Optics

www.cvimellesgriot.com Fundamental Optics

COMBINATION SECONDARY PRINCIPAL-POINT LOCATION Because the thin-lens approximation is obviously highly invalid for most combinations, the ability to determine the location of the secondary princi- 12 34 pal point is vital for accurate determination of d when another element is added. The simplest formula for this calculates the distance from the secondary principal point of the final (second) element to the secondary principal point of the combination (see figure 1.8b): asinBeam Optics Gaussian

= ″ − zs2 f. (1.19) d>0 COMBINATION EXAMPLES 12 It is possible for a lens combination or system to exhibit principal planes 34 that are far removed from the system. When such systems are themselves combined, negative values of d may occur. Probably the simplest example of a negative d-value situation is shown in figure 1.9. Meniscus lenses with steep surfaces have external principal planes. When two of these lenses are brought into contact, a negative value of d can occur. Other combined-lens examples are shown in figures 1.10 through 1.13. d<0 Optical Specifications

Figure 1.9 “Extreme” meniscus-form lenses with external principal planes (drawing not to scale)

H H ″ 1 1 ″ Hc

lens lens 1 combination Material Properties ″ zH f1 (a) (c) fc = 4 s1 d f 1 H H H ″ H H ” c 1 1 2 2

lens 1 lens and combination lens 2

zH

d (d) Optical Coatings (b) ″ s2 fc

Figure 1.8 Lens combination focal length and principal planes

Fundamental Optics 1.9 1ch_FundamentalOptics_Final_a.qxd 6/15/20094:09PMPage1.10

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics symmetry isnotrequired. 1.10 The signsoff This approximation isadequateformostthick-lenssituations. This isthebasisforHuygensandRamsdeneyepieces. achromatism requires that,inthethin-lensapproximation, both elementsare madefrom thesamematerial.Achieving combinations canbemadenearlyachromatic, eventhough Figure 1.11 than Figure 1.10 other aberrations. shapes are unrestricted andcanbechosentocompensatefor value thatguaranteestheexistenceofanairspace.Element H d 1 ″ = f 1 () Fundamental Optics = ff 12 f + 2 2 : 1 , Achromatic combinations: Positive lensesseparatedbydistancegreater f f f is negativeandboths 2

1 . , andd d d are unrestricted, butd focal plane f 1 f 2 H 2 s 2 H ″ 2 2 ″ ″ and principal plane combination Air-spaced lens z z secondary are positive.Lens f must havea 2 f<0 lens shapesshowninthefigure). Forexample,f positive lensfollowedbyanegative(butnotnecessarilythe the firstlenssurfacetoimagewouldsuggestbyusinga the imagesize,canbemademuchlargerthandistancefrom characteristic ofthetelephotolensisthatEFL,andhence f d catalog lensesLDX-50.8-130.4-CandLDK-42.0-52.2-C,with Figure 1.12 tion conjugatedistancesmustbe measured from thesepoints. combination issimilarlylocatedinthefirstelement.Combina- the element.Bysymmetry, theprimaryprincipalpointof is theelementcenterthicknessandn d lensescouldalsobeplanoaspheres.) Because (The contact. on ofapairidenticalplano-convexlensesare vertices Figure 1.13 d the vertexofplanosurfacesecondelement,where t d point ofthecombinationcoincideatH″ pal pointofthesecondelementandsecondaryprincipal 2 = = = larger thanf = 78.2 mm,willyields 0, f 4 1 2(Galileantelescopeorbeamexpander),andpositivefor /2 f f principal plane secondary combination 1 = /2. Thenf /2. f 1 /2 1 = Telephoto combination: Condenser configuration: /2. To maketheexampleevenmore specific, /2. f 2 /2, is negativeford z<0 n t f c 1 /2 2 ″=2.0 m,f = s s 2 ″ , andz H f H″ less thanf d = is therefractive indexof s = 5.2 m,andz , atdeptht ″ 0. Thesecondaryprinci- Fundamental Optics The mostimportant n t The convex c www.cvimellesgriot.com 1 /2, infinitefor /2, 1 is positiveand combination c / = n 43.2 m. beneath s 2 ″ c Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.11 2 v www.cvimellesgriot.com 1 v Fundamental Optics Fundamental Fundamental Optics

l of light at a boundary 1 2 n n Melles Griot applications engineers are able to provide a Melles Griot applications engineers are able to material 1 index material 2 index APPLICATION NOTE APPLICATION Technical Assistance Technical is Detailed performance analysis of an optical system software. accomplished by using computerized ray-tracing CVI If you systems. ray-tracing analysis of simple catalog-component of your optical need assistance in determining the performance your particular system, or in selecting optimum components for application, please contact your nearest CVI Melles Griot office. Figure 1.14 Figure , (1.20) is the angle of refraction, and both angles 2 v 2 sinv 2 n alignment, impact the performance of an optical system. alignment, impact the = 1 is the angle of incidence, is the angle of incidence, 1 v sinv 1 n are measured from the surface as shown in figure 1.14. After paraxial formulas have been used to select values for component focal to select values formulas have been used After paraxial As in any is to select actual lenses. diameter(s), the final step length(s) and selection process involves a number of tradeoffs engineering problem, this where ABERRATIONS determine the precise performance of a lens system, we can trace the To at each optical interface law path of light rays through it, using Snell’s called , process, to determine the subsequent ray direction. This When this process is completed, is usually accomplished on a computer. it is typically found that not all the rays pass through the points or posi- deviations from ideal imaging These tions predicted by paraxial theory. are called lens aberrations. at the interface between direction of a light ray after refraction The index of refraction is given isotropic media of differing two homogeneous, law: by Snell’s DIFFRACTION poses nature, its Diffraction, a natural property of light arising from Diffraction is always a fundamental limitation on any optical system. if the system has significant present, although its effects may be masked When an optical system is essentially free from aberrations, aberrations. and it is referred to as its performance is limited solely by diffraction, diffraction limited. the focal length(s) and In calculating diffraction, we simply need to know factors such as aperture diameter(s); we do not consider other lens-related shape or index of refraction. and aberrations decrease Since diffraction increases with increasing f-number, determining optimum system performance with increasing f-number, of these factors often involves finding a point where the combination has a minimum effect. Performance Factors Performance cost, weight, and environmental factors. including performance, real optical systems is limited by several factors, performance of The magnitude of these and light diffraction. The including lens aberrations with relative ease. effects can be calculated such as lens manufacturing tolerances and Numerous other factors, component considered explicitly in the following discussion, Although these are not that if calculations indicate that a lens system in mind it should be kept in practice it may fall short only just meets the desired performance criteria, In critical applications, of this performance as a result of other factors. performance is it is generally better to select a lens whose calculated significantly better than needed. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.11 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:28PMPage1.12

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics v by developingamethodofcalculatingaberrationsresultingfromthe excessively tediousandtimeconsuming.Seidel*addressedthisissue real lenssurfaces. Beforetheadventofelectroniccomputers, thiswas As alreadystated,exactraytracingistheonlyrigorousway toanalyze the paraxialprediction. theory, aberrationsarereallyameasureofhowtheimagediffersfrom would formitsimageatthepointandtosizeindicatedbyparaxial rations. Becauseaperfectopticalsystem(onewithoutanyaberrations) real performancebecausesinv marginal rays),paraxialtheoryyieldsincreasinglylargedeviationsfrom high f-numberlenses).Withmorehighlycurvedsurfaces(andparticularly The assumptionthatsinv this approximationusingtheparaxialformulas. expansions areused.Designofanyopticalsystemgenerallystartswith called first-orderorparaxialtheorybecauseonlythefirsttermsofsine 1.12 von* Ludwig Seidel,1857. with theirarguments(i.e., replacesinv The firstapproximationwecanmake istoreplaceallthesinefunctions functions inSnell’s lawcanbeexpandedinaninfiniteTaylor series: The firststepindevelopingtheseroughguidelinesistorealizethatthesine better startingpointforanyfurthercomputeroptimization. in theinitialstagesofsystemspecification,butcanalsohelpachievea method forquicklyestimatinglensperformance. This notonlysavestime ing easiertouseandarereadilyavailable, itisstillquiteusefultohavea Even thoughtoolsforthepreciseanalysisofanopticalsystemarebecom- widely used. powerful ray-tracingsoftware, Seidel’s formulaforsphericalaberrationisstill good descriptionofopticalsystemimagequality. Infact, evenintheeraof system ofclassifyingthem,whichmakes analysismuchsimpler, givesa In actualpractice, aberrationsoccurincombinationsratherthan alone. This terms inthesineexpansions. aberrations withoutactuallytracinglargenumbersofraysusingallthe eral . Seideldevelopedmethodstoapproximateeachofthese tion. Inpolychromaticlighttherearealsochromaticaberrationandlat- are sphericalaberration,,fieldcurvature, ,anddistor- system intoseveraldifferentclassifications. Inmonochromaticlightthey To simplifythesecalculations, Seidelputtheaberrationsofanoptical called Seidelaberrations. 1 3 /3! term.The resultantthird-orderlensaberrationsaretherefore i ... ! / ! / ! / ! / sin v v v v vvv 111 Fundamental Optics −+−+− + − + =− 3 3579 = 1 5 v is reasonablyvalid forv ≠ v. These deviationsareknownasaber- 1 7 1 with 1 9 v 1 itself andsoon).This is

close tozero(i.e., (1.21) (TSA).These quantitiesarerelatedby which theseraysintercepttheparaxialfocalplaneiscalledtransverse rays) iscalledlongitudinalsphericalaberration(LSA).The heightat rays)andtheraysthatgothroughedgeoflens(marginal (paraxial between theinterceptofraysthatarenearlyonopticalaxis it focuses(crossestheopticalaxis).The distancealongtheopticalaxis ther fromtheopticalaxisrayenterslens, thenearertolens shows thesituationmoretypicallyencounteredinsinglelenses. The far- collimated light.AllrayspassthroughthefocalpointF Figure 1.15illustrateshowanaberration-freelensfocusesincoming SPHERICAL ABERRATION Figure 1.15 for agiventaskarepresentedlaterinthischapter. However, thethird- present. Parameters forchoosingthebestlensshapeandorientation conjugate Spherical aberrationisdependentonlensshape, orientation,and TSA = LSA# longitudinal sphericalaberration ratio, aswellontheindexofrefractionmaterials tan(u Spherical aberrationofaplano-convexlens ″ ). aberration-free lens u ″ transverse sphericalaberration paraxial focalplane Fundamental Optics LSA www.cvimellesgriot.com ″ . The lowerfigure F F TSA ″ ″ (1.22) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.13 www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics sagittal image (focal line) paraxial meridional, rays. Rays not in this plane are referred to as skew Rays meridional, rays. focal plane ASTIGMATISM the natural asym- is focused by a spherical lens, When an off-axis object system appears to have two different The metry leads to astigmatism. focal lengths. the plane containing both optical axis and object As shown in figure 1.16, Rays that lie in this plane are called plane. point is called the tangential tangential, or ray goes from the object point through the or principal, chief, The rays. the plane perpendicular to the lens system. The center of the aperture of or the principal ray is called the sagittal tangential plane that contains radial plane. figure illustrates that tangential rays from the object come to a focus closer The When the image is evaluated to the lens than do rays in the sagittal plane. we see a line in the sagittal direction. A line in at the tangential conjugate, Between these conjugate. the tangential direction is formed at the sagittal Astigmatism the image is either an elliptical or a circular blur. conjugates, is defined as the separation of these conjugates. amount of astigmatism in a lens depends on lens shape only when The with the lens itself. there is an aperture in the system that is not in contact although in many cases (In all optical systems there is an aperture or stop, strongly Astigmatism it is simply the clear aperture of the lens element itself.) depends on the conjugate ratio. (focal line) v tangential image ≠ (1.23) sagittal plane principal ray optical system . f

3 f/# . 0 067 = tangential plane

Astigmatism represented by sectional views Astigmatism represented

al axis al c opti produce spherical or cylindrical surfaces. The manufacture of The surfaces. produce spherical or cylindrical object point spot size due to spherical aberration spot size due to spherical Figure 1.16 Figure Theoretically, the simplest way to eliminate or reduce spherical aberration simplest way the Theoretically, an (i.e., radius of curvature with a varying the lens surface(s) is to make to exactly compensate for the fact that sin v aspheric surface) designed with high surface most lenses however, practice, In at larger angles. by grinding and polishing techniques that accuracy are manufactured naturally chromatic, spherical aberration of a plano-convex lens used aberration of a monochromatic, spherical order, by ratio can be estimated at infinite conjugate of complex, and it is difficult to produce a lens aspheric surfaces is more aberration completely. sufficient surface accuracy to eliminate spherical these aberrations can be virtually eliminated, for a chosen Fortunately, by combining the effects of two or more spherical (or set of conditions, cylindrical) surfaces. spherical aberration, In general, simple positive lenses have undercorrected spherical aberration. By and negative lenses usually have overcorrected with a negative lens combining a positive lens made from low-index it is possible to produce a combination in which made from high-index glass, powers do not. The the spherical aberrations cancel but the focusing series such as the LAO simplest examples of this are cemented doublets, properly used. which produce minimal spherical aberration when 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.13 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:28PMPage1.14

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.14 Figure 1.17 some extentbycombiningpositiveandnegativelenselements. lenses haveoutward curvingfields. Field curvature canthusbecorrectedto Positive lenselementsusuallyhaveinward curvingfields, andnegative the blurfromfieldcurvature to avalue of0.25itsoriginalsize. height. Therefore, byreducingthefieldangleone-half, itispossibletoreduce Field curvature varies withthesquareoffieldangle orthesquareofimage correspond tothetangentialandsagittalconjugates. problem iscompoundedbecausetwoseparateastigmaticfocalsurfaces field curvature (seefigure1.19). Inthepresenceofastigmatism,this to imagebetteroncurvedsurfacesthanflatplanes. This effectiscalled Even intheabsenceofastigmatism,thereisatendencyopticalsystems FIELD CURVATURE marginal rays. placing anaperture, orstop, inanopticalsystemtoeliminatethemore surfaces. Alternatively, asharperimagemaybeproducedbyjudiciously As withsphericalaberration,correctioncanbeachievedbyusingmultiple still exhibitcomaoffaxis. Seefigure1.18. corrected andthelensbringsallraystoasharpfocusonaxis, alensmay appears asacharacteristiccomet-like flare. Evenifsphericalaberrationis object points. Anoff-axisobjectpointisnotasharpimagepoint,butit a comaticcircle. This causesblurringintheimageplane(surface)ofoff-axis in figure1.17,eachconcentriczoneofalensformsring-shapedimagecalled of magnification.This givesrisetoanaberrationknownascoma.Asshown In sphericallenses, differentpartsofthelenssurfaceexhibitdegrees COMA Fundamental Optics Imaging anoff-axis pointsource byalenswithpositivetransversecoma S 1 1 ′ 1 ′ 1 S P, O 1′ 1 33 points onlens 42 2 3′ 4′ 2′ Figure 1.18 Figure 1.19 1′ 1 1 1′ 0 2′ 4′ 3′ 4 spherical fosurfae Positive transversecoma Field curvature positive transversecoma 4 corresponding points on 4′ 60∞ 1 3′ 3 1 focal plane ′ 2′ S 2 Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.15 ) 2 2 3 y y y y y — longitudinal www.cvimellesgriot.com )( 2 2 3 red focal point v v v Fundamental Optics Fundamental —— —— — Fundamental Optics blue focal point )( red light ray 3 2 2 fv fv f f f — — (f Aperture Angle Field Image Height blue light ray . From Snell’s law (see equation 1.20), it can be law (see Snell’s . From Longitudinal chromatic aberration Longitudinal chromatic white light ray Material Properties Lateral Spherical Astigmatism Longitudinal Spherical Coma Curvature Field Chromatic Aberration seen that light rays of different or will be refracted wavelengths seen that light rays of different 1.21 shows the index is not a constant. Figure at different angles since collimated light is incident on a positive the result when polychromatic index of refraction is higher for shorter wave- lens element. Because the closer to the lens than the longer wavelengths. these are focused lengths, the axial distance from Longitudinal chromatic aberration is defined as of spherical aberration, the nearest to the farthest focal point. As in the case of chromatic aberration. positive and negative elements have opposite signs aberration to form Once again, by combining elements of nearly opposite corrected. It is neces- a , chromatic aberration can be partially so that characteristics, sary to use two with different negative element can balance the aberration of the stronger, the weaker positive element. Figure 1.21 Figure of Aberrations with Aperture, Variations Field Angle, and Image Height CHROMATIC ABERRATION CHROMATIC described are purely a function of the shape aberrations previously The light. they can be observed with monochromatic and of the lens surfaces, are used to trans- arise when these optics however, Other aberrations, index of refraction of a The wavelengths. form light containing multiple Known as dispersion, this is discussed material is a function of wavelength. in BARREL DISTORTION DISTORTION PINCUSHION Pincushion and barrel distortion Pincushion and barrel Furthermore, the amount of distortion usually increases with the amount Furthermore, OBJECT Figure 1.20 Figure DISTORTION but may also be distorted. may have curvature image field not only The point may be formed at a location on this surface image of an off-axis The distor- This by the simple paraxial equations. other than that predicted to (where rays from an off-axis point fail tion is different from coma plane). Distortion means that even if a perfect meet perfectly in the image formed, its location on the image plane is not off-axis point image is correct. effect of this can be seen as two different The increasing image height. barrel (see figure 1.20). Distortion does kinds of distortion: pincushion and it simply means that the image shape does not lower system resolution; to the shape of the object. Distortion is a sepa- not correspond exactly predicted location ration of the actual image point from the paraxially or as an absolute value on the image plane and can be expressed either as a percentage of the paraxial image height. has opposite types of It should be apparent that a lens or lens system This or backward. distortion depending on whether it is used forward a photograph, and then used means that if a lens were used to make in the final screen in reverse to project it, there would be no distortion systems at 1:1 magnification perfectly symmetrical optical Also, image. have no distortion or coma. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.15 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.16

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.16 Figure 1.22 is inadequate. Inthesecases, exactraytracingisabsolutelyessential. having largeaperturesoraangularfieldofview, third-ordertheory to quantifyaberrations. However, inhighlycorrectedsystemsorthose For manyopticalsystems, thethird-ordertermisallthatmaybeneeded stop location. magnification dependsoncolor. Lateralcolorisverydependentonsystem is whyraysintercepttheimageplaneatdifferentheights. Statedsimply, wavelength, bluelightisrefractedmorestronglythanredlight,which positive lensandaseparateaperture. Becauseofthechangeinindexwith Figure 1.22showsthechiefrayofanopticalsystemconsistingasimple Lateral coloristhedifferenceinimageheightbetweenblueandredrays. LATERAL COLOR aperture Fundamental Optics Lateral Color blue lightray red lightray focal plane lateral color estimate thespotsizeofadoublet,tablesin Although thereisnosimpleformulathatcanbeusedto purely monochromaticlight. for focusingcollimatedlightorcollimatingpointsources, evenin chromatic aberration,theyareoftensuperiortosimplelenses Because achromaticdoubletscorrectforsphericalaswell Simple Lenses Achromatic Doublets Are Superiorto catalog achromaticdoublets. sample values thatcanbeusedtoestimatetheperformanceof APPLICATION NOTE Fundamental Optics www.cvimellesgriot.com Spot Size Spot give Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.17 2 www.cvimellesgriot.com Fundamental Optics Fundamental 1.5 Fundamental Optics s″=2f), a similar analysis would = s 1 0.5 1), with convex side toward the infinite conjugate, performs nearly the infinite conjugate, side toward 1), with convex exact longitudinal spherical aberration (LSA) = At infinite conjugate with a typical glass singlet, the plano-convex shape singlet, the plano-convex with a typical glass At infinite conjugate (q show that a symmetric biconvex lens is the best shape. Not only is Not only biconvex lens is the best shape. show that a symmetric but coma, distortion, and lateral chro- spherical aberration minimized, true results are cancel each other out. These matic aberration exactly which explains the utility index or wavelength, regardless of material as well as symmetrical optical systems in of symmetric convex lenses, if a remote stop is present, these aberrations may not general. However, cancel each other quite as well. is definitely not the optimum the best-form shape wide-field applications, For since it yields especially at the infinite conjugate ratio, singlet shape, ideal shape is determined by the situation The maximum field curvature. It is possible to achieve much and may require rigorous ray-tracing analysis. more than one element. better correction in an optical system by using cases of an infinite conjugate ratio system and a unit conjugate ratio The system are discussed in the following section. as well as the best-form lens. Because a plano-convex lens costs much lens. as well as the best-form an asymmetric biconvex singlet, these lenses less to manufacture than exhibits near-minimum this lens shape Furthermore, are quite popular. coma when used off axis, and near-zero total transverse aberration thus enhancing its utility. ( imaging at unit magnification For CTOR (q) SHAPE FA (1.24) v. It is also 40.5 0 ≠ v , more exactly, its , more exactly, . In this particular 41 exact transverse spherical aberration (TSA) 41.5

5 4 3 2 1

. use of the Coddington shape factor, q, defined as factor, use of the Coddington shape 42 Aberrations of positive singlets at infinite conjugate ratio as a function of shape

S MILLIMETER IN S ABERRATION − +

21 21 rr rr () () = q Aberrations described in the preceding section are highly dependent on section are highly dependent described in the preceding Aberrations of the lens (or and material lens shape, application, Figure 1.23 Figure index of refraction). The singlet shape that minimizes spherical aberration index of refraction). The criterion for best-form is called best-form. The at a given conjugate ratio at the marginal rays are equally refracted at each any conjugate ratio is that minimizes the effect of sin This of the lens/air interfaces. Lens Shape Lens the criterion for minimum surface-reflectance loss. Another benefit is Another benefit surface-reflectance loss. the criterion for minimum at both infinite minimized for best-form shape, that absolute coma is nearly and unit conjugate ratios. it is the dependence of aberrations on lens shape, further explore To make helpful to 1.23 shows the transverse and longitudinal spherical aberrations Figure q of a singlet lens as a function of the shape factor, the lens has a focal length of 100 mm, operates at f/5, has an instance, 546.1 nm), green line, index of refraction of 1.518722 (BK7 at the mercury It is also assumed ratio. and is being operated at the infinite conjugate An asymmetric shape that corre- that the lens itself is the aperture stop. is 0.7426 for this material and wavelength of about sponds to a q-value to note that the the best singlet shape for on-axis imaging. It is important with a example, For best-form shape is dependent on refractive index. the infinite high-index material, such as silicon, the best-form lens for conjugate ratio is a meniscus shape. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.17 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.18

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.18 but withabettercorrectionofthesphericalaberration. the otherhand,iftheselensesarereversed,wehavesystemjustdescribed orientation isoppositetothatshownbeoptimumforthisshapelens. On an infiniteconjugateratio, butwiththeconvexsurfacetoward thefocus. This middle, we so dramatically, considerthatifthebiconvexlenswere splitdownthe biconvex lens. To of thesetwolensesyieldsalmostexactlythesamefocallengthas Those showninfigure1.25arebothLPX-20.0-20.7-C.The combination plano-convex lenseswithconvexsurfacesfacingandnearlyincontact. A dramaticimprovementinperformanceisgainedbyusingtwoidentical f/13.3 doestheraycloselyapproachparaxialfocus. significant sphericalaberrationispresentinthislensatf/2.7.Notuntil lens (LDX-21.0-19.2-C),thebest-formsingletinthisapplication.Clearly, light wavelength of546.1nm. The firstsystemisasymmetricbiconvex All areshowntothesamescaleandusingrayf-numberswitha Figure 1.25showsthreepossiblesystemsforuseattheunitconjugateratio. UNIT CONJUGATE RATIO affects resultsslightly. estimate be scaledtofitaplano-convexlensofanyfocallength,canused focal point,andthef/3.8rayisfairlyclose. This usefuldrawing,whichcan unacceptable. The raywithf-number7.5practicallyinterceptstheparaxial Figure 1.24alsoshowsthef-numberatwhichsingletperformancebecomes the doublet’s performanceissuperior. doublets, itisimportanttorememberthatevenwithmonochromaticlight corrected inthedoublet.Eventhoughtheselensesareknownasachromatic convex lens. Additionally, chromaticaberration(notshown)ismuchbetter in micrometers, ratherthaninmillimeters, asinthecaseofplano- paraxial focalpoint;however, inthiscase, thedepartureismeasured cerned inthelens. Ofcourse, notalloftherayspassexactlythrough same scaleastheplano-convexlens. Nosphericalaberrationcanbedis- negative meniscushigh-index(flintglass)element.This isdrawntothe consists ofapositivelow-index(crownglass)elementcementedto part ofthefigureshowsaprecisionachromat(LAO-21.0-14.0), which This situationcanbeimprovedbyusingatwo-elementsystem.The second strikes theparaxialfocalplanesignificantlyoffopticalaxis. through it,isshowntoexactscale. The marginalray(rayf-number1.5) light atawavelength of546.1nm.This drawing,includingtheraystraced shows aplano-convexlens(LPX-15.0-10.9-C)withincomingcollimated at infiniteconjugateratiosisgenerallynearlyplano-convex.Figure 1.24 As showninthepreviousdiscussion,best-formsingletlensforuse INFINITE CONJUGATE RATIO Lens Combinations Fundamental Optics the magnitudeofitssphericalaberration,althoughlensthickness would havetwoidenticalplano-convexlenses, eachworkingat understand whythisconfigurationimprovesperformance lenses. Onceagain,sphericalaberrationisnotevident,eveninthe f/2.7ray. third partoffigure1.25showsasystemcomposedtwoLAO-40.0-18.0 plano-convex singletswithachromats, yieldingafour-element system.The working attheinfiniteconjugateratio, thenextstepistoreplace the Since theunitconjugatecasecanbethoughtofastwolenses, each to asingletwhenusedattheinfiniteconjugateratioandlowf-numbers. Previous examplesindicatethatanachromatissuperiorinperformance with atwo-elementachromat Figure 1.24 7.5 3.8 2.5 1.9 1.5 7.5 3.8 2.5 1.9 1.5 ray f-numbers ACHROMAT PLANO-CONVEX LEN Single-element plano-convexlenscompared paraxial imageplane LAO-21.0-14.0 LPX-15.0-10.9-C S Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.19 www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics paraxial image plane S LPX-20.0-20.7-C LDX-21.0-19.2-C LAO-40.0-18.0 2.7 3.3 4.4 6.7 13.3 S IDENTICAL ACHROMAT SYMMETRIC BICONVEX LEN 2.7 3.3 4.4 6.7 13.3 IDENTICAL PLANO-CONVEX LENSES 2.7 3.3 4.4 6.7 13.3 ray f-numbers Three possible systems for use at the unit conjugate ratio Three Figure 1.25 Figure 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.19 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.20

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.20 Figure 1.26 image plane. ratio, far-field anglesaretransformedintospatialdisplacementsinthe limiting performanceofopticalsystems. Moregenerally, atanyconjugate the focalplane;therefore, itisFraunhofer diffractionthatdeterminesthe plane-wave inputmapsthefar-field diffractionpatternofitsapertureonto gate ratios. Notso. Alensorsystemoffinitepositivefocallengthwith whereas Fresnel diffractionequationsshouldbeconsideredatfiniteconju- diffraction From these overlysimpledefinitions, onemightassumethatFraunhofer and thelightissensedataninfinitedistance(far-field) fromthisaperture. (or object)isilluminatedwithaninfinitesource(plane-wave illumination) systems. T ,however, isoftenimportanteveninsimpleoptical such applicationsasdigitaloptics, fiberoptics, andnear-field microscopy. important inmostclassicalopticalsetups, butitbecomesveryimportantin or object,knownasthenearfield.Consequently, Fresnel diffractionisrarely is thusonlyofconcernwhentheilluminationsourceclosetothisaperture light intheimmediateneighborhoodofadiffractingobjectoraperture. It hofer types. Fresnel diffractionisprimarilyconcernedwithwhathappensto Diffraction effectsaretraditionallyclassifiedintoeitherFresnel orFraun- of wave theory. describes diffraction,butrigorousexplanationdemandsadetailedstudy from theinitialdirectionofpropagation.Huygens’principlenicely fringe patternthatrapidlydecreasesinintensitywithincreasingangle . Interferencebetweenthesecondarywavelets givesrisetoa wavelets. The propagatingwave isthentheenvelopeoftheseexpanding states thateachpointonapropagatingwavefront isanemitterofsecondary considering thewave natureoflight.Huygens’principle(figure1.26) and inescapablephysicalphenomenon.Diffractioncanbeunderstoodby geometric propagation.This effect,knownasdiffraction,isafundamental In alllightbeams, someenergyisspreadoutsidetheregionpredictedby Diffraction Effects Fundamental Optics his isthelight-spreadingeffectofanaperturewhen is importantonlyinopticalsystemswithinfiniteconjugate, aperture wavelets secondary wavefront Huygens’ principle into thisregion some lightdiffracted wavefront the spotsize, causedbydiffraction,ofacircularlensis limits ofperformanceforcircularlenses. Itisimportanttorememberthat Fraunhofer diffractionatacircularaperturedictatesthefundamental CIRCULAR APERTURE where distribution inthispatterncanbedescribedby rings. Eachringisseparatedbyacircleofzerointensity. The irradiance disc (seefigure1.27),whichissurroundedbyanumberofmuchfainter aperture actuallyconsistsofacentralbrightregion,knownastheAiry The diffractionpatternresultingfromauniformlyilluminatedcircular this limitingspotsize. it isthef-numberoflens, notitsabsolutediameter, thatdetermines illumination andl where d circular aperture Figure 1.27 = Jx I II 2.44l 0 x 1 () = = d peak irradiance intheimage peak irradiance is thediameteroffocusedspotproducedfromplane-wave =− = 0 (f/#) ⎩ ⎪ ⎧ x Bessel functionofthee firstkindoforderunity 2 ∑ n Jx ∞ = 1 1 x () () Center ofatypicaldiffractionfor pattern AIRY DISCDIAMETER=2.44 is thewavelength oflightbeingfocused.Noticethat 1 ⎭ ⎪ ⎫ n 2 + 1 () nn − 12 x 22 !! n − 2 n −− 1 Fundamental Optics www.cvimellesgriot.com l f/# (1.26) (1.25) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings are 1.21 . This (1.28) (1.29) x -values and -values w, and l x (see eq. 1.25) is the secondary D, ″ s . Furthermore, when . Furthermore, www.cvimellesgriot.com where Fundamental Optics Fundamental mode has a smooth Gaussian tanv Fundamental Optics ″ . 00 s = r Gaussian Beam Optics has its usual meaning, and p Gaussian Beam Optics () l x x f D w l l p p is the aperture diameter. For a slit aperture, this relationship is this a slit aperture, For is the aperture diameter. ./# is the slit width, = = 244 v v w D = d sin sin irradiance profile. Formulas used to determine the focused spot size from Formulas irradiance profile. such a beam are discussed in given by where Linear instead of angular all in the same units (preferably millimeters). field positions are simply found from GAUSSIAN BEAMS alters diffraction Apodization, or nonuniformity of aperture irradiance, formulas and results given If pupil irradiance is nonuniform, the patterns. is important to remember because most This previously do not apply. out- The pupil irradiance. optical systems do not have uniform -based put beam of a laser operating in the TEM ENERGY DISTRIBUTION TABLE ENERGY DISTRIBUTION shows the major features of pure (unaberrated) accompanying table The table The of circular and slit . diffraction patterns Fraunhofer pattern and percentage of total , shows the position, relative to each ring or band. It is especially convenient energy corresponding to either pattern with the same variable characterize positions in the circular aperture case by is related to field angle in variable where dealing with Gaussian beams, the location of the focused spot also departs the location of the dealing with Gaussian beams, This from that predicted by the paraxial equations given in this chapter. is also detailed in conjugate distance. This last result is often seen in a different form, namely This conjugate distance. for a circular lens is the diffraction-limited spot-size equation, which, achieved by an represents the smallest spot size that can be value This and it is the f-number, optical system with a circular aperture of a given has dropped to zero. diameter of the first dark ring, where the intensity graph in figure 1.28 shows the form of both circular and slit aperture The Aperture scale. diffraction patterns when plotted on the same normalized between diameter is equal to slit width so that patterns are the same. angular deviations in the far-field (1.27) , is . D () l

2 iidth 1.22 f/# ⎫ ⎪ ⎭ x l sin v x w l peak irradiance in image sin slit w pv wavelength sin angular deviation from pattern maximum. ⎧ ⎪ ⎩ = = = = 0 = NA D 0 l 0.61 aperture diameter p I x v l w wavelength angular radius from the pattern maximum. angular radius from the ======x APPLICATION NOTE APPLICATION d Rayleigh Criterion ultimately limited spatial resolution is In imaging applications, by diffraction. Calculating the maximum possible spatial resolution of an optical system requires an arbitrary definition In the Rayleigh of what is meant by resolving two features. criterion, it is assumed that two separate point sources can be resolved when the center of the Airy disc from one overlaps the first dark ring in the diffraction pattern of the second. In this d distance, the smallest resolvable case, II x l D v where SLIT APERTURE is useful in relation to which is mathematically simpler, A slit aperture, distribution in the diffraction irradiance The cylindrical optical elements. is described by pattern of a uniformly illuminated slit aperture where irradiance distribution from a uniformly the far-field useful formula shows This of diameter illuminated circular aperture and where 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.21 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.22

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.22 of acircular aperture Figure 1.28 Note: Positionvariable( Energy DistributionintheDiffractionofaCircular Pattern orSlitAperture Fifth Dark Fourth Dark Third Dark Second Dark First Dark Central Maximum Ring orBand Fourth Bright Third Bright Second Bright First Bright Fundamental Optics Fraunhofer diffraction ofasingletslitsuperimposedontheFraunhoferdiffraction pattern pattern x) isdefinedinthetext. Position 5.24p 4.71p 4.24p 3.70p 3.24p 2.68p 2.23p 1.64p 1.22p 0.0 (x )(

NORMALIZED PATTERN IRRADIANCE (y) SLIT APERTURE 0.0 1.0 CIRULAR APERTURE .1 .2 .3 .4 .5 .6 .7 .8 .9 4 Circular Aperture 8 4 Intensity 7 Relative 0.0 0.0008 0.0 0.0016 0.0 0.0042 0.0 0.0175 0.0 1.0 I 4 x / aperture 6 I 0 4 ) 5 POSITION INIMAGEPLANE(x) 4 91.0% withinfirstbrightrin adjoining subsiiarymaxima slit 4 4 95.0% withinthetwo 3 central maximum 83.9% inAirydisc 4 2 4 90.3% in 1 in Ring Energy Relative (%) 83.8 012345678 1.0 1.5 2.8 7.2 aperture cirular Position 5.00p 4.48p 4.00p 3.47p 3.00p 2.46p 2.00p 1.43p 1.00p 0.0 (x )( Slit Aperture Intensity 0.0 0.0050 0.0 0.0083 0.0 0.0165 0.0 0.0472 0.0 1.0 I x / I Fundamental Optics 0 ) www.cvimellesgriot.com in Band Energy 90.3 (%) 0.5 0.8 1.7 4.7 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.23

Fundamental Optics

www.cvimellesgriot.com Fundamental Optics Lens Selection

Having discussed the most important factors that affect the performance limited, it is pointless to strive for better resolution. This level of resolution of a lens or a lens system, we will now address the practical of can be achieved easily with a plano-convex lens. selecting the optimum catalog components for a particular task. The fol- While angular divergence decreases with increasing focal length, spherical lowing useful relationships are important to keep in mind throughout aberration of a plano-convex lens increases with increasing focal length. the selection process: To determine the appropriate focal length, set the spherical aberration formula for a plano-convex lens equal to the source (spot) size:

$ Diffraction-limited spot size = 2.44 l f/# Beam Optics Gaussian 0. 067 f = $ 3 1 mm. Approximate on-axis spot size f /# of a plano-convex lens at the infinite = 0. 067 f This ensures a lens that meets the minimum performance needed. To conjugate resulting from spherical aberration 3 f /# select a focal length, make an arbitrary f-number choice. As can be seen NA from the relationship, as we lower the f-number (increase collection $ Optical invariant = m = . NA″ efficiency), we decrease the focal length, which will worsen the resultant divergence angle (minimum divergence = 1 mm/f). In this example, we will accept f/2 collection efficiency, which gives us Example 1: Collimating an Incandescent Source a focal length of about 120 mm. For f/2 operation we would need a min- Produce a collimated beam from a halogen bulb having a 1-mm- imum diameter of 60 mm. The LPX-60.0-62.2-C fits this specification square filament. Collect the maximum amount of light possible and produce exactly. would be about 8 mrad.

a beam with the lowest possible divergence angle. Finally, we need to verify that we are not operating below the theoretical Optical Specifications This problem, illustrated in figure 1.29, involves the typical tradeoff diffraction limit. In this example, the numbers (1-mm spot size) indicate between light-collection efficiency and resolution (where a beam is being that we are not, since collimated rather than focused, resolution is defined by beam diver- gence). To collect more light, it is necessary to work at a low f-number, diffraction-limited spot size = 2.44#0.5 mm#2 = 2.44 mm. but because of aberrations, higher resolution (lower divergence angle) will be achieved by working at a higher f-number. Example 2: Coupling an Incandescent Source into a Fiber In terms of resolution, the first thing to realize is that the minimum divergence angle (in radians) that can be achieved using any lens system In Imaging Properties of Lens Systems we considered a system in which the is the source size divided by system focal length. An off-axis ray (from output of an incandescent bulb with a filament of 1 mm in diameter was the edge of the source) entering the first principal point of the system to be coupled into an optical fiber with a core diameter of 100 µm and a exits the second principal point at the same angle. Therefore, increasing numerical aperture of 0.25. From the optical invariant and other constraints = = = the system focal length improves this limiting divergence because the given in the problem, we determined that f 9.1 mm, CA 5 mm, s 100

″= ″= = Material Properties source appears smaller. mm, s 10 mm, NA 0.25, and NA 0.025 (or f/2 and f/20). The singlet lenses that match these specifications are the plano-convex LPX-5.0-5.2-C An optic that can produce a spot size of 1 mm when focusing a perfectly or biconvex lenses LDX-6.0-7.7-C and LDX-5.0-9.9-C. The closest achromat collimated beam is therefore required. Since source size is inherently would be the LAO-10.0-6.0.

v min

source size

v min = Optical Coatings f f

Figure 1.29 Collimating an incandescent source

Fundamental Optics 1.23 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.24

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.24 Figure 1.30 combination discussionin f-number. Basedonthelens we needalenswithan8-µmspotsizeatthis We want tocollectandfocusatanumericalapertureof0.15orf/3.3, This problem,illustratedinfigure1.30,isessentiallya1:1imagingsituation. 0.5 µm. into anotherfiberwiththesamecharacteristics. Assumeawavelength of Couple anopticalfiberwith8-µmcoreanda0.15numericalaperture Example 3:SymmetricFiber-to-Fiber Coupling certain toprovideadequateperformance. trace wouldberequiredtodetermineitsexactperformance, itisvirtually aberration. Abetterchoiceistheachromat.Althoughacomputerray likely tobemarginalinthissituation,especiallyifweconsiderchromatic be largerthanthatgivenbyoursimplecalculation.This lensistherefore However, sincewearenotworkingatinfiniteconjugate, thespotsizewill This isslightlysmallerthanthe100-µmspotsizewearetryingtoachieve. conjugate. We willignore, forthemoment,thatwearenotworkingatinfinite on thefocusingsidebyusingoursphericalaberrationformula: spherical We canimmediatelyrejectthebiconvexlensesbecauseof setup iseitherapairofidenticalplano-convexlensesorachromats, faced spot size Fundamental Optics aberration. We canestimatetheperformanceofLPX-5.0-5.2-C == 6 10 067 0 Symmetric fiber-to-fiber coupling .() 2 3

Lens Combination Formulas Combination Lens 84 m. m s = f , ourmostlikely gradient-index lenses(LGTseries). be touseapairofsphericalballlenses(LMS-LSFNseries)oronethe An entirelydifferentapproachtoafiber-coupling tasksuchasthiswould assumed thatdiffractionwillnotplayasignificantrolehere. Since thisishalfthespotsizecausedbyaberrations, itcanbesafely limit: to make surethatthesystemisnotbeingasked toworkbelowthediffraction Because fairlysmallspotsizesarebeingconsideredhere, itisimportant 8-m spotdiameter. given in length, largerachromats, suchastheLAO-10.0-6.0. The performancedata, mance, thenextstepmightbetouseapairofslightlylongerfocal pair oflongerfocallengthsingletswouldresultinunacceptableperfor- erations ofhandling,mounting,andpositioningthem.Becauseusinga with utilizingthesetiny, shortfocallengthlensesisthepracticalconsid- 1.3 mm.The LPX-4.2-2.3-BAK1 meetsthesecriteria.The biggestproblem This formulayieldsafocallengthof4.3mmandminimumdiameter lens, weagainusethesphericalaberrationestimateformula: front tofront.To determinethenecessaryfocallengthforaplano-convex 4 53 4 33 05 244 067 0 .. . 33 . Spot Size Spot × 3

f = mm 008 0 .. m s , showthatthiscombinationdoesprovidetherequired ″ = ×

mm f .. = Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.25 optical fiber Applied www.cvimellesgriot.com b f Fundamental Optics Fundamental LMS-LSFN coupling sphere Fundamental Optics collimated light section uncoated narrow band b f , vol. 20, no. 18, pp 3136–45, 1981. Off-axis aberrations , vol. 20, no. fiber LMS-LSFN coupling sphere optical APPLICATION NOTE APPLICATION are absent since the fiber are so much smaller than are absent since the fiber diameters are so much the coupler focal length. Spherical Ball Lenses for Fiber Coupling Spherical Ball Lenses at the focal Spheres are arranged so that the fiber end is located output from the first sphere is then collimated. If two point. The the beam will be spheres are aligned axially to each other, Translational transferred from one focal point to the other. the beam. alignment sensitivity can be reduced by enlarging the spherical Slight negative defocusing of the ball can reduce all coupling aberration third-order contribution common to Additional information can be found in “Lens Coupling systems. by A. Nicia, Optic Devices: Efficiency Limits,” in Fiber Optics 100 mm, = f

f 3 × /# f 10 mm. . = 0 067 f m wavelength light becomes diffraction lim- light becomes diffraction m wavelength /

14 f m mf 54 9

××= =× 50 mm, and f/4.8 at = /# ( . ) . .. /# 244 05 f or When working with these focal lengths (and under the conditions previously When working with these focal lengths (and under performance above stated), we can assume essentially diffraction-limited that this treatment does not take in mind, however, Keep these f-numbers. aberration, which will into account manufacturing tolerances or chromatic be present in polychromatic applications. f/7.2 at f ited (i.e., the effects of diffraction exceed those caused by aberration). the effects ited (i.e., size set the equations for diffraction-limited spot solve this problem, To result The aberration equal to each other. and third-order spherical since aberrations scale with focal length, depends upon focal length, By substituting dependent upon f-number. while diffraction is solely into this formula, we get f/8.6 at some common focal lengths Example 4: Diffraction-Limited Performance Diffraction-Limited Example 4: a plano-convex lens being used at an infinite Determine at what f-number m conjugate ratio with 0.5- 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.25 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.26

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.26 *Diffraction-limited performanceisindicatedbyDL. Focal Length= *Diffraction-limited performanceisindicatedbyDL. Focal Length= increase linearlyasalensisscaledup. focal lengthlensesproducesmallerspotsizesbecauseaberrations limited spotsize, thenotation“DL” appearsnexttotheentry. The shorter the spotsizecausedbyaberrationsissmallerorequaltodiffraction- lens isfacinginthedirectionthatproducesaminimumspotsize. When illuminated, several differentf-numbers. Allthetablesareforon-axis, uniformly The tablesgivethediameterofspotforavariety oflensesusedat lens intheproductlistings. the tolerancesandspecificationsclearlydescribedforeachCVIMellesGriot When actuallychoosingalensorsystem,itisimportanttonote provides ahigherdegreeofcorrectionbutmakes alignmentmoredifficult. sidered. Furthermore, rememberthatusingmorethanoneelement geometric optics. Effectsofmanufacturingtoleranceshavenotbeencon- obtained fromcomputerraytracingconsideronlytheeffectsofideal In interpretingthesetables, rememberthatthesetheoreticalvalues constructed fromstandardcatalogoptics. results foravariety ofsimpleandcompoundlenssystems, whichcanbe already beenpresented.The followingtablesgivesomequantitative mum workingconditionsforsomeofthelensesinthiscataloghave however, somesimpleguidelinescanbeusedforlensselection.The opti- and systemsofcatalogcomponentsonrequest.Incertainsituations, applications engineerscansupplyray-tracingdataforparticularlenses should bedeterminedbyanexacttrigonometricraytrace. CVIMellesGriot In general,theperformanceofalensorsysteminspecificcircumstance Spot Size f/10 f/10 f/5 f/2 f/# f/5 f/3 f/2 f/# f/3 Fundamental Optics collimated inputlightat546.1nm.They assumethatthe 60 mm 10 mm D-.-.- P-.-.- LAO-10.0-6.0 LPX-8.0-5.2-C LDX-5.0-9.9-C 33(L 33(L 13.3(DL) 13.3(DL) 13.3 (DL) — 36 8 LDX-50.0-60.0-C 13.3 (DL) 816 217 45 Spot Size(m 6.7 (DL) 25 94 m)* LPX-30.0-31.1-C 13.3 (DL) 600 160 33 6.7 (DL) 11 7 rapidly offaxis. natic meniscuslenscombinationsbecausetheirperformancedegrades the lensesareusedoffaxis. This isparticularlytrueoftheachromat/apla- Unfortunately, theseresultscannotbegeneralizedtosituationsinwhich combine toformaminimum. some optimumperformancepointatwhichbothaberrationsanddiffraction first decreasesandthenincreaseswithf-number, meaningthatthereis increases linearlywithf-number. Thus, forsomelenstypes, spotsizeat be evenhigher. Ontheotherhand,spotsizecausedbydiffraction dependent onthecubeoff-number. For doublets, thisrelationshipcan on f-number. For aplano-convexsinglet,sphericalaberrationisinversely The effectonspotsizecausedbysphericalaberrationisstronglydependent *Diffraction-limited performanceisindicatedbyDL. Focal Length= / 974(DL) 6.9(DL) 3 13.8(DL) 6.7(DL) 7 13.3(DL) — 17 13.3(DL) 79 f/10 MENP-18.0-4.0-73.5-NSF8 295 f/5 LAO-30.0-12.5 f/3 LPX-18.5-15.6-C f/2 f/# LAO-60.0-30.0 13.3 (DL) Spot Size(m — 10 34 m)* 30 mm LAO-100.0-31.5 &MENP-31.5-6.0-146.4-NSF8 Spot Size(m 13.3 (DL) Fundamental Optics 6.7 (DL) 4 (DL) m)* — LAO-50.0-18.0 & www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.27 oriented in its reverse oriented in its www.cvimellesgriot.com 2 f 0.272 Fundamental Optics Fundamental Fundamental Optics oriented in the normal direction is oriented in the 1 f plano-convex (normal) plano-concave (normal) . 2

2 2 2.65. Figure 1.32(b) shows a system of catalog f/# = kf .. 393 1 ff

2 f/# .. . . 0 272 1 069 0 272 1 069 =+ would be needed and a complex diverging lens as well. objective would be needed and a complex diverging =− =− 2 1 f f LSA 0.403 combined with a plano-concave lens of focal length a plano-concave lens combined with If a plano-convex lens of focal length lens of focal length If a plano-convex direction, the total spherical aberration of the system is direction, the total spherical we obtain to zero, After setting this equation elements of aberration contributions of the two the magnitude make To system, select the focal equal so they will cancel out, and thus correct the that of the negative length of the positive element to be 3.93 times element. 1.32(a) shows a beam-expander system made up of catalog elements, Figure is corrected to simple system inthe focal length ratio is 4:1. This which at 632.8 nm, even though the objective is operating about 1/6 wavelength is remarkably good wavefront This at f/4 with a 20-mm aperture diameter. normally assume that a correction for such a simple system; one would doublet tolerances. into account manufacturing analysis does not take This also be derived from this A of lower magnification can symmetric-convex objective is used a If information. together] with a the aberration coefficients are in reversed plano-concave diverging lens, the ratio of 1.069/0.403 closest possible given the lenses that provides a magnification of 2.7 (the error in this case is only maximum wavefront lengths). The focal available even though the objective is working at f/3.3. a quarter-wave, symmetric-convex symmetric-concave longitudinal spherical aberration (3rd order) = f/# 1.069 plano-convex (reversed) plano-concave (reversed) Third-order longitudinal spherical aberration of typical lens shapes longitudinal Third-order for six of the most common positive and negative lens shapes for six of the most common positive and negative . Normally, such a process requires computerized analysis and such a . Normally, positive lenses aberration cient coeffi (k) negative lenses To improve system performance, optical designers make sure that the make optical designers system performance, improve To sums to together all surfaces taken contribution from total aberration nearly zero Figure 1.31 Figure optimization. However, there are some simple guidelines that can be there are some simple guidelines optimization. However, approach in this catalog. This lenses available used to achieve this with at a much lower f-number than can usually can yield systems that operate lenses. be achieved with simple from two examine how to null the spherical aberration we will Specifically, thus technique will monochromatic light. This or more lenses in collimated, beam focusing and expanding. be most useful for laser third-order longitudinal spherical aberration 1.31 shows the Figure coefficients when used with parallel, monochromatic incident light. The plano-convex light. The when used with parallel, monochromatic incident spherical aberration when and plano-concave lenses both show minimum parallel beam. All oriented with their curved surface facing the incident aberration. With other configurations exhibit larger amounts of spherical systems can be it is now possible to show how various these lens types, corrected for spherical aberration. In this case, example. A two-element laser beam expander is a good starting sum of their focal lengths, two lenses are separated by a distance that is the system will not This so that the overall system focal length is infinite. By the beam diameter. focus incoming collimated light, but it will change same f-number. definition, each of the lenses is operating at the equation for longitudinal spherical aberration shows that, for two lenses The directly with the focal lengths of aberration varies with the same f-number, it should as focal length. Thus, sign of the aberration is the same The the lenses. this Galilean-type beam be possible to correct the spherical aberration of which consists of a positive focal length objective and a negative expander, diverging lens. Aberration Balancing Aberration 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.27 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.28

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.28 balancing Figure 1.32 c) SPHERICALLY CORRETED25-mmEFLf/2.0OBJETIVE Fundamental Optics plano-conave 10-mm diameter f plano-conave 25-mm diameter f plano-conave 10-mm diameter f = = = b) CORRETED2.7!BEAMEXPANDER 420 mm 425 mm 420 mm a) CORRETED4 Combining cataloglensesforaberration !BEAM EXPANDER plano-convex 22.4-mm diameter f f symmetric-onvex 32-mm diameter plano-convex 27-mm diameter f = 80mm = 54mm = 50mm(2) or In thiscase, f negative elementisjustabouthalfthatofeachthepositivelenses. Therefore, acorrectedsystemshouldresultifthefocallengthof aberration fromthisconfiguration,wemustsatisfy are proportionaltotheirfocallengths. To obtainzerototalspherical operate atthesamef-number, sothattheiraberrationcontributions two identicalplano-convexpositiveelements. Again,alloftheelements shows anobjectiveconsistingofinitialnegativeelement,followedby objectives thatmightbeusedaslaserfocusinglenses. Figure 1.32(c) These sameprinciplescanbeutilizedtocreatehighnumericalaperture output beamdiameters. larly usefulwithNd:YAG andargon-ionlasers, whichtendtohavelarge minimizing thelengthofthesebeamexpanders. They wouldbeparticu- The relativelylowf working distance. corrected to1/6wave, hastheadditionaladvantage ofaverylong of about25mmandanf-numberapproximatelyf/2.This objective, 6 7 7 0 272 0 272 0 069 1 f f ... 1 2 =−

fff 122 051 .. 1 ++= = 425 mmandf numbers oftheseobjectivesisagreatadvantage in 2 = 50 mmyieldatotalsystemfocallength Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.29 ond enter of enter of ht) www.cvimellesgriot.com ative if c urvature of first urvature of sec F″ Fundamental Optics Fundamental back focal point e (positive if c e (neg Fundamental Optics ius of c ius of c d d urvature is to left) urvature is to rig c surfac c surfac =ra =ra ) 1 2 ″″ r r b f f B kness kness secondary principal surface secondary principal point called the principal surface, at which this can happen. called the principal surface, reversed ray locates front focal secondary vertex dge thic enter thic point or primary principal surface 2 c 2 r A =e = FOCAL POINT (FFOCAL POINT OR F the on or originate at either focal point must be, Rays that pass through the basis for fact is This parallel to the optical axis. opposite side of the lens, locating both focal points. SURFACE PRINCIPAL PRIMARY originating at the front focal point F (and therefore Let us imagine that rays after emergence from the opposite side of the lens) parallel to the optical axis instead of twice refracted (once imaginary surface, are singly refracted at some is a unique imaginary There actually happens. at each lens surface) as surface, consider a single ray traced from the air on locate this unique surface, To through the lens and into the air on the other side. one side of the lens, of these are external Two lens. into three segments by the ray is broken The external segments The (in the air), and the third is internal (in the glass). and (certainly near, can be extended to a common point of intersection principal surface is the locus of all such usually within, the lens). The e c t t H″ e c t t H is not 1 f th; 1 r th th al leng al leng al leng ative e positive (as shown) f tive foc b f f c A k foc c primary vertex A a or neg may b is measured with reference to is measured with reference =effe = front foc = f b f f f in figure 1.33) and the back focal length primary principal point f primary principal surface F e e ) us to front f istanc istanc d d ray from object at infinity ray from object at infinity Focal length and focal points appears frequently in lens formulas and in the tables of appears frequently in lens us ). b dge reversed ray) is that light travels from left to right. apparent when a lens is visually inspected. apparent when a lens is front focal point e foc in figure 1.33) determines magnification and hence the image in figure 1.33) determines optical axis f = front foc = rear edge to rear B A The convention in all of the figures (with the exception of a single The deliberately Figure 1.33 Figure FOCAL LENGTH ( FOCAL LENGTH Definition of Terms Definition lengths associated with every lens or distinct terms describe the focal Two focal length focal length (EFL) or equivalent effective lens system. The (denoted term f The size. because f Unfortunately, standard lenses. the meaning of usually inside the lens, principal points which are immediately length relates the focal plane positions directly second type of focal The surfaces (namely the vertices) which are imme- to landmarks on the lens size but is especially It is not simply related to image diately recognizable. correct lens positioning or convenient for use when one is concerned about Examples of this second type of focal length are the mechanical clearances. front focal length (FFL, denoted f (BFL, denoted f 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.29 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.30

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics the opticalaxis. The secondaryvertexistheintersectionoflens surfacewith optical axis. The primaryvertexistheintersectionoflenssurfacewith 1.30 EFFECTIVE FOCALLENGTH( SECONDARY VERTEX (A PRIMARY VERTEX (A used tocollimateasource(therefore, either situation inwhichalensiseitherfocusingincomingcollimatedlightorbeing from H″ CONJUGATE DISTANCES ( optical axis. This pointistheintersectionofsecondaryprincipalsurfacewith OR SECONDARY NODALPOINT SECONDARY PRINCIPAL POINT(H optical axis. This pointistheintersectionofprimaryprincipalsurfacewith PRIMARY PRINCIPAL POINT(H)ORFIRSTNODAL being refractedbybothlenssurfaces. thought ofasoncerefractedatthesecondaryprincipalsurface, insteadof point F″ for acollimatedbeamincidentfromtheleftandfocusedtobackfocal This termisdefinedanalogouslytotheprimaryprincipalsurface, butitisused SECONDARY PRINCIPAL SURFACE it issometimesreferredtoastheprincipalplane. Near theopticalaxis, theprincipalsurfaceisnearlyflat,andforthisreason, focal point. surface ofaperfectlycorrectedopticalsystemisspherecenteredonthe points ofintersectionextendedexternalraysegments. The principal the designwavelength (l to therearfocalpoint(F″ principal point(H)andthedistancefromsecondary(H 1.0), thisisboththedistancefromfrontfocalpoint(F)toprimary Assuming thatthelensissurroundedbyairorvacuum (refractiveindex Specifically, The conjugatedistancesaretheobjectdistance, to theimagelocation.The terminfiniteconjugateratioreferstothe on theright.Raysinthatpartofbeamnearestaxiscanbe Fundamental Optics s is thedistancefromobjecttoH,ands ). Laterweusef 0 1 ). ) 2 s ) AND EFL, s to designatetheparaxialEFLfor f ″ ″ ) ) ) s or s s ″ , andimagedistance, s is infinity). ″ is thedistance ″ ″ ) . vertex (A This lengthisthedistancefromfrontfocalpoint(F)toprimary point (F″ with theopticalaxismultipliedbyindexofrefraction( marginal ray(thethatexitsthelenssystematitsouteredge)makes system, suchasinthecaseofamagnifyingglass. A virtualimagecanbeviewedonlybylookingbackthroughtheoptical A virtualimagedoesnotrepresentanactualconvergenceoflightrays. were placedatthepointoffocus, animagewouldbeformedonit. A realimageisoneinwhichthelightraysactuallyconverge;ifascreen EDGE-TO-FOCUS DISTANCES ( BACK FOCALLENGTH( FRONT FOCALLENGTH( The NAofalenssystemisdefinedtobethesineangle, NUMERICAL APERTURE (NA) by theheightatwhichitinterceptsprincipalsurface. clear aperture. Rayf-numberistheconjugatedistanceforthatraydivided a lenssystemisdefinedtobetheeffectivefocallengthdividedby The f-number(alsoknownasthefocalratio, relativeaperture, orspeed)of F-NUMBER (f/#) This lengthisthedistancefromsecondaryvertex(A with theopticalaxismultipliedbyindexofrefraction: The NAcanbedefinedforanyrayasthesineofanglemade bythatray point. Bothdistancesarepresumedalways tobepositive. B A NA is thedistancefromfrontfocalpointtoprimaryvertexoflens. is thedistancefromsecondaryvertexoflenstorearfocal f/# = n sinv = ). 1 CA ). . f . f b f f ) ) A AND Fundamental Optics B ) www.cvimellesgriot.com 2 n ) totherearfocal ) ofthemedium. (see eq.1.7) v 1 , thatthe (1.30) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.31 www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics (Addison Wesley) Lens Design Fundamentals Elements of Modern Optical Design Optics Modern , , Max Born, Emil Wolf, (Academic Press) System Design Rudolph Kingslake, (Academic Press) Smith, Warren (McGraw Hill). Donald C. O’Shea, (John Wiley & Sons) Eugene Hecht, (Cambridge University Press) Rudolph Kingslake, Rudolph Kingslake, APPLICATION NOTE APPLICATION If you need help with the use of definitions and formulas If you need help with the use of definitions and our applications engineers will be presented in this guide, pleased to assist you. Technical Reference Technical the definitions and formulas presented further reading about For publications: refer to the following here, (1.31) (1.32) magnifier. # f () . To create a virtual image for viewing create a virtual . To in mm . # f f mm (). 254 = f 1000 = diopters in mm magnification AND distance in object space In an imaging system, depth of field refers to the criteria for The image. over which the system delivers an acceptably sharp the user; depth of field what is acceptably sharp is arbitrarily chosen by increases with increasing f-number. an imaging system, depth of focus is the range in image space over For In other words, image. which the system delivers an acceptably sharp this is the amount that the image surface (such as a screen or piece of ) could be moved while maintaining acceptable focus. Again, criteria for acceptability are defined arbitrarily. depth of focus refers to such as laser focusing, In nonimaging applications, the range in image space over which the focused spot diameter remains below an arbitrary limit. Thus, the smaller the focal length is, the larger the power in diopters will be. the smaller the focal length is, Thus, Thus, a 25.4-mm focal length positive lens would be a 10 a 25.4-mm focal length positive lens Thus, DIOPTERS term diopter is used to define the reciprocal of the focal length, which The inverse focal length of a lens The is commonly used for ophthalmic lenses. expressed in diopters is with the human , in principle, any positive lens can be used at an infinite any positive lens can be used at an principle, in with the , is usually a narrow range there However, number of possible . when Typically, be comfortable for the viewer. of magnifications that will object distance so that the image appears to be the viewer adjusts the is a comfortable viewing distance for most essentially at infinity (which by the relationship individuals), magnification is given MAGNIFICATION POWER MAGNIFICATION for use as simple magnifiers are rated with Often, positive lenses intended as 4 a single magnification, such 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:29 PM Page 1.31 2:29 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:30PMPage1.32

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics t 1.32 Paraxial LensFormulas where Focal Length defined infigure1.33. in tothespeedoflightinlensglass. Allothervariables are The refractiveindexofthelensglass, n,istheratioofspeedlight formulas arevalid forboththickandthinlensesunlessotherwisenoted. values whichcorrespondtotheparaxialformulas. The followingparaxial edge ormargin.AllEFLvalues (f)tabulatedinthiscatalogareparaxial experienced by a The paraxialformulasdonotincludeeffectsofsphericalaberration be approximatedbytheanglesthemselves(measuredinradians). angles ofincidenceandrefractionaresmall(asusedinSnell’s law)andcan angles ofincidenceandrefractionaresmall.Asaresult,thesines surfaces arealways verynearlynormaltotheopticalaxis, andhenceall always verycloseandnearlyparalleltotheopticalaxis. Inthisregion,lens The followingformulasarebasedonthebehaviorofparaxialrays, whichare An oftenusefulapproximationistoneglects (refer tofigure1.34) Surface SagittaandRadiusofCurvature familiar lensmaker’s formula: second termoftheaboveequationvanishes, andweareleftwiththe convention previouslygivenfortheradii PARAXIAL FORMULASFORLENSESINAIR c ≅ r rrs rr sr 1 1 f f 0, andforplanolenseseither 22 =+ −−> − =− −− =− + − =− −+ =− n 28 sd ). () () () Fundamental Optics is therefractiveindex,t n n 1 1 s 2 2 . rr 11 12 marginal ray—apassingthroughthelensnearits rr 11 11 12 d 2 d 2 2 2 0 () n − n c is thecenterthickness, andthesign r 2 1 or rr 12 t r c r 1 2 and is infinite. Ineithercasethe /2. r 2 applies. For thinlenses, (1.33) (1.37) (1.36) (1.35) (1.34) Symmetric LensRadii( Figure 1.34 For symmetriclenses(r where theabovesignconventionapplies. Principal-Point Locations(signeddistancesfrom vertices) Since Plano LensRadius the positive, butthe4 where, inthefirstform,= With centerthicknessconstrained, nf rn f f rn HAH AH AH AH = 1 1 1 12 −+− =− =− − ± =− 2 r 2 sign mustbeusedregardlessofthef () () () nf = ″ is infinite, =− = =

21 111 1 1 rrtn t r nr rrtn t r nr rtn t nr ()() ()()

. ⎩ ⎪ ⎪ ⎧ 1 21 21 Surface sagittaandradiusof curvature

⎪ ⎩ ⎪ ⎧

−− +− −+ +− −+ ″ rt 1 sign mustbeusedif c − c − () rt rt 2 1 2 = 2 c c c c 4

( r r

r >0 nf . 2 r t 4 c 1 ft = n ), 1 sign ischosenforthesquarerootif 1 s c 4 ) ⎭ ⎪ ⎪ ⎫ r ⎭ ⎪ ⎪ ⎫ 1 ) f is negative. Inthesecondform, Fundamental Optics www.cvimellesgriot.com . s >0 d 2 (1.38) (1.42) (1.41) (1.40) (1.39) f is Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.33 (1.50) (1.51) (1.52) (1.53) (1.54) ). In this H and to the 1 in object space f www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics are again equal, so that the lens ″ f 1 ). This case is of considerable practical ). This ″ n and = c c 1 rt −+ − ), and image space medium (refractive index n 1

″ 21 . are the sagittas of the first and second surfaces. Bevel are the sagittas of the first and second surfaces.

. ()() + 2 c nr r t n n s f t AH b2

fs

fs f − f − =+ ″ ″

f is infinite, l’Hôpital’s rule from calculus must be used, whereby l’Hôpital’s is infinite, and

s sf sf s 1 1 A B = =+ s =− =− = = in image space. It is also necessary to distinguish the principal points It is also necessary to distinguish in image space. = f f1 ″ ff ff f and m and a lens and as a window lens serves both as The the nodal points. from separating the object space and image space media. fluid (figure situation of a lens immersed in a homogenous The 1.35) is included as a special case (n situation, the EFL assumes two distinct values, namely situation, the EFL assumes two distinct values, f two values The importance. Front Focal Length Front presented above applies to A where the sign convention PARAXIAL FORMULAS FOR LENSES IN ARBITRARY MEDIA IN ARBITRARY FORMULAS FOR LENSES PARAXIAL formulas allow for the possibility of distinct and completely arbitrary These (refractive index n′), lens refractive indices for the object space medium (refractive index n combination formulas are applicable to systems immersed in a common is more difficult, and it must general case (two different fluids) fluid. The be approached by ray tracing on a surface-by-surface basis. is neglected. Magnification or Conjugate Ratio Edge-to-Focus Distances positive lenses, For where radii. If r = H 1 (1.49) (1.46) (1.47) (1.48) (1.43) (1.44) (1.45) and to the radii. ″ . If the Abbé sine H 2 by 4p Q . To convert from steradians . To ″ ⎪ ⎬ ⎪ ⎭ ⎫ ⎫ ⎪ ⎭ , for plano-convex lenses in the , for plano-convex c t

12 rr 2 1

n − instead of s

n c may be calculated using the arc sine function in a symmetric lens, we obtain A →∞in a symmetric lens, () c 1 2 rt 11 . ⎧ ⎪ ⎩ . −+ − ″

c f nf n n 1 t 21 Aberration Balancing s . v 2 ()() is infinite, l’Hôpital’s rule from calculus must be used. rule from l’Hôpital’s is infinite, 2 CA c n nr r t n t 0 1 2 AH 1 =−

2 r ⎪ ⎨ ⎩ ⎧ − = + − ″ c c . These results are useful in connection with the following results are useful . These t t n or ″ ″ ″ (cos) sin f =− − =− 2 /2 1 1

p c 21 ″ ″ 4 AH AH arctan = = = =− = = b2 b is infinite, l’Hôpital’s rule from calculus must be used, whereby l’Hôpital’s is infinite, ff ff v Qp v HH HH and 2 H″=t r 2 where the sign convention presented above applies to A condition is known to apply, v condition is known to apply, instead of the arctangent. Back Focal Length is the apparent angular radius of the lens clear aperture. For an observer For is the apparent angular radius of the lens clear aperture. at an on-axis object point, use s where this result is in steradians, and where where this result is in steradians, simply divide to the more intuitive sphere units, Solid Angle observer situated at an on-axis for an solid angle subtended by a lens, The image point, is which, in the thin-lens approximation (exact for plano lenses), becomes which, in the thin-lens approximation (exact for paraxial lens combination formulas. paraxial lens combination Hiatus or Interstitium (principal-point separation) Thus, referring to referring Thus, If either r For flat plates, by letting r by flat plates, For correct orientation, A If 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:30 PM Page 1.33 2:30 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:30PMPage1.34

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.34 Figure 1.35 explicit functionofthecompletelensprescription(bothradii,t This numberappearsfrequentlyinthefollowingformulas. Itisan Lens Constant( that and bothmediaindices(n Principal-Point Locations Lens Formula(Newtonianform) Lens Formula(Gaussianform) Effective FocalLengths AH AH f f xx k n s f k 1 = 2 + = appears. ″ n nn s k = Fundamental Optics n ″ = ″ ″ ′ . r = − 21212 . nt k = ″ −

c nt + f k k ″ Symmetric lenswithdisparate objectandimagespaceindexes = nn ″ nn c nr k ″ F = nn ″ k − ′ ) r − 2 n 2 nn k ″ nr ″ ′ ′ . − ′ ′ 1 − and nn n nn tn c . ()( ) n index n=1(airorvacuum) ″ ′ ). This dependenceisimplicitanywhere − nrr ′ ″ f − f f ′ . A 1 c and (1.58) (1.57) (1.56) (1.55) (1.60) (1.59) n H ′ ) H ″ index n from Corresponding PrincipalPoint Separation ofNodalPoint Nodal-Point Locations Lens Maker’s Formula Magnification Second Principal-Point-to-ImageDistance Object-to-First-Principal-Point Distance NA HN AH AN HN AH AN m s s HN and N″ n f ″ = 11 22 = = = = sn ks N index n″=1.333(water)

= 1.51872(BK7) ns ns sn ks ″ A n f =+ H ″ N″ ns ns ″ ″ ″ 2 ″ = ″ to rightofH″ ″ N − − = . ″ ″=( k ″ . . . n ″ ″ + 4 f n ″ )/k ″″ . , positiveforNtorightofH f b . Fundamental Optics www.cvimellesgriot.com F ″ (1.66) (1.65) (1.64) (1.63) (1.62) (1.61) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.35 . f f = b f is the = f ″ becomes the www.cvimellesgriot.com ″ s without change of Fundamental Optics Fundamental appears to emerge ″ ″ Fundamental Optics in these formulas by assuming that in these formulas by assuming f , and f b f , f ). This makes the nodal slide method the most precise the nodal slide method the makes ). This becomes the lens-to-object distance; becomes the lens-to-object becomes the object-to-image distance. This is known as This distance. becomes the object-to-image ″ s n ″ s = = APPLICATION NOTE APPLICATION NOTE APPLICATION n lens-to-image distance; and the sum of conjugate distances lens-to-image distance; s the thin-lens approximation. For Quick Approximations be saved by ignoring the differences Much time and effort can among Points Physical Significance of the Nodal lens appears to A ray directed at the primary nodal point N of a emerge from the secondary nodal point N a ray directed at N direction. Conversely, during the center of rotation), the image remains stationary fact is the basis for the nodal slide method for rotation. This nodal The a lens. measuring nodal-point location and the EFL of points when points coincide with their corresponding principal are equal the image space and object space refractive indices ( method of principal-point location. from N without change of direction. At the infinite conjugate from N without change of direction. At the infinite if a lens is rotated about a rotational axis orthogonal to ratio, the optical axis at the secondary nodal point (i.e., if N Then Then (see eq. 1.46) (see eq. 1.48) (see eq. 1.50) . p . /CA, where CA is the diameter of the clear /CA, where CA is the diameter ″ ″ ″ s 2s CA s 22 CA 2s 2 CA CA ″

″ /CA and f v 2 v 2 arcsin arctan .. 1 arcsin 2 AH. = 2 = 4 AH. = ″ + = ″ (cos) sin v v sin ″ v v (1 cos (1 ) v p v ppv p 4 21 2 4 = sin =− = sin =−

= = ″″ b2 f ff ff n n where where where arctan where Q p v Q and Focal Ratios focal ratios are f The Front Focal Length Front Back Focal Length Back Focal To convert from steradians to spheres, simply divide by 4 convert from steradians to spheres, To Solid Angles (in steradians) aperture of the lens. Numerical Apertures 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:30 PM Page 1.35 2:30 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:30PMPage1.36

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.36 Figure 1.36 possiblethatbothprincipal shapes (shortradiiorsteepcurves),itis extrememeniscuslens the lensradii,andcanbefoundbyformula.in positions dependontheindexofrefractionlensmaterial,and shapes. The exact relation tothelenssurfacesforvarious standardlens fallin Figure 1.36indicatesapproximatelywheretheprincipalpoints Principal-Point Locations Fundamental Optics Principal pointsofcommonlenses H″ H″ H″ H″ H″ F″ F″ F″ F″ F″ the way totheplanevertex. point isatthecurvedvertex,andotherapproximatelyone-thirdof into threeapproximatelyequalsegments. For planolenses, oneprincipal principal pointsdividethatpartoftheopticalaxisbetweenvertices points willfalloutsidethelensboundaries. For symmetriclenses, the F″ F″ F″ F″ F″ H″ H″ H″ H″ H″ Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.37 hypotenuse face www.cvimellesgriot.com Fundamental Optics Fundamental hypotenuse face virtual image of object: visible only to observer Fundamental Optics entrance face exit face entrance face exit face Virtual imaging using a Real imaging using a prism right-angle prism can lead to TIR failure. An aluminum- or silver-coated hypotenuse An aluminum- or silver-coated can lead to TIR failure. object right-angle prism object is recommended for applications where the right-angle prism is frequently is a slight handled, or where convergent or divergent beams are used. There loss of reflectance at all internal angles with the coating, and no critical angle exists. Figure 1.38 Figure FOR PRISMS ABERRATIONS Prisms will introduce aberrations when they are used with convergent or diver- gent beams of light. Using prisms with collimated or nearly collimated light that include prisms Conjugate distances will help minimize aberrations. should be long. Prisms 1.37 Figure (1.67) 1.414, = 2 _ l 1 n has the same orientation as the real image shown, but it can has the same orientation as the real image shown, = ( ) arcsin c vl the critical angle will exceed 45 degrees, and total internal (TIR) and the critical angle will exceed 45 degrees, will fail for a collimated beam internally incident at 45 degrees on the hypotenuse face of a right-angle prism. Reflectance decreases rapidly at angles of incidence smaller than the critical angle. the TIR of a collimated beam index of BK7 is sufficiently high to guarantee The region. at 45 degrees internal incidence over the visible and near- or divergent beams possibility of significant TIR failure with convergent The TIR can also fail if the in mind if polarization is important. should be kept Even an almost invisible extremely clean. hypotenuse face is not kept and depends on the refractive index, which is a function of wavelength. If, and depends on the refractive index, which is a function of wavelength. the refractive index should fall to less than √ at some wavelength, TOTAL INTERNAL REFLECTION TOTAL an internal reflection) at angles Rays incident upon a glass/air boundary (i.e., 100-percent efficiency that exceed the critical angle are reflected with critical angle is given by The regardless of their initial polarization state. be viewed by the observer only by looking back through the prism system. be viewed by the observer only by looking back PRISM ORIENTATION orientation of a prism determines its effect on a beam of light or The an image. and sees a virtual image A viewer looks through a prism at an object from the original object, image may be displaced 1.37). This (see Figure if a dove prism is used, it may coincide with the object. Furthermore, or, the case of a right-angle image orientation may differ from the object; in prism, the image is reversed. 1.38) can be formed only if imaging optics are A real image (see Figure the image is virtual. A present in the system. Without imaging optics, virtual image Prisms are blocks of optical material whose flat, polished sides are Prisms are blocks of optical Prisms may be used angles to each other. arranged at precisely controlled can invert or deviate a beam of light. They in an optical system to deflect and disperse light into its component wavelengths, or rotate an image, be used to separate states of polarization. 1ch_FundamentalOptics_Final_a.qxd 7/6/2009 1:54 PM Page 1.37 1:54 PM 7/6/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:30PMPage1.38

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics the firstsurface, b the surfaceofadispersingprismapexangleA.The angleofrefractionat angle ofrefractionexitingtheprism, 1.38 Figure 1.39 As showninFigure 1.39,abeamofwidthW metric beamprofiles. beam expansionorcompression,andmaybeusedtocorrectcreateasym- exit anglesforabeamdiffer. This isusefulinanamorphic(one-dimensional) A prismexhibitsmagnificationintheplaneofdispersionifentranceand permitted totravelfarenoughsothebeamsseparatespatially. lengths followingthesamebeampath.Typically, thedispersedbeamsare . Inlaserwork,dispersingprismsareusedtoseparatetwowave- by theprism.Aspectrumisthenformedatfocalplaneofalensorcurved component colors. Dispersing prismsareusedtoseparateabeamofwhitelightintoits DISPERSING PRISMS e beam makes withitsoriginaldirection. The beamdeviation,e The magnificationW = a M d g b W = = = = = 1 sin A sin Fundamental Optics d o cos cos o cos cos 4 4 4 4 A b ag db 1 1 ( ((sina h Diagram ofdispersingprism sing , theangleofincidenceatsecondsurface, Generally, thelightisfirstcollimatedandthendispersed a 2 /W , isofgreatestimportance. Itistheangleexit ) ) /h 1 is givenby: ) b A d , areeasilycalculated: e g 1 is incidentatananglea d g W , andthe 2 (1.69) (1.68) on Another useisillustratednext. an isoscelesBrewsterprismresults. If, inaddition,thebaseanglesofprismarechosenasBrewster’s angle, eliminating lossesforp-polarizedbeams. The apexangletochooseis: angle, theequalincidentandexitanglesmaybemadeBrewster’s angle, refraction ofaprismcanbedetermined.Also, byproperchoiceofapex of By measuringtheangleofincidenceforminimumdeviation,index the internalraysareperpendiculartobisectorofapexangle. the incidentandexitanglesareequal,prismmagnificationisone, and of incidence: At agivenwavelength, thebeamdeviation where RPistheresolvingpowerofprism. equal totheminimumangularresolution,weobtain: the spectralresolvingpowerofaprism.Settingexpressionford limited angularresolutionatagivenbeamdiametersetsthelimiton angular resolutiond The resolvingpowerofaprismspectrometerangle f If thespectrumisformedbyadiffractionlimitedfocalsystemoflength of theprismisgivenby: Figure 1.40 where , theminimumspotsizeisdx~f RP A =p4 d a d ldb min dev h == = is theprismindexofrefractionatthatwavelength. Atthisangle, d ⎝ ⎜ ⎛ l ldb oos co = 2 sin sc

sinA v B ⎝ ⎜ ⎛ Translation ofaprismatminimumdeviation 4 o c cos wA 1 [ h 2 sin( A/2)] sin ~ ⎠ ⎟ ⎞ os ⎝ ⎜ ⎛ l d d /w forabeamofdiameterw. The diffraction l h ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ ⎝ ⎜ ⎛ d d l h l ⎠ ⎟ ⎞ / W. This correspondstoaminimum e Fundamental Optics is aminimumatanangle a , theangulardispersion www.cvimellesgriot.com (1.72) (1.71) (1.70) (1.73) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.39 www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics 2 l 1 l are superimposed in a collimated beam, 2 l and 1 2 2 , l > l 1 1 Longer wavelength is deviated more than Longer wavelength is deviated more One of the wavelengths deviates at exactly l l spersing prisms. This means that it will enter the right angle prism means that it will enter This spersing prisms. Figure 1.43 Figure the shorter wavelength Figure 1.42 Figure 90° to its initial direction as at the output of a harmonic generating , the diagram in Figure as at the output of a harmonic generating crystal, possible to find a rotation of the prism in 1.42 suggests that it is always will operate at its plane that ensures that one of the two wavelengths face of the first of the minimum deviation when refracting at the input half-di be turned exactly 90°, be presented to the normal to one of its faces, and hence exit the second half-dispersing prism in minimum deviation, Broca prism deviated at exactly 90° to its initial direction. Pellin less deviates the longer wavelength A simple dispersing prism always the longer Broca prism, whether In a Pellin than the shorter wavelength. is deviated more or less depends on the orientation of the wavelength is an important consideration when designing a high power prism. This 1.43 and 1.44. as shown in Figures Broca beam separator, Pellin Suppose wavelengths l Suppose wavelengths b, the b, (1.74) = , is 2135. 2 W /dl 0, and b = 1 for F2 glass at 590 nm. 41 -polarized, the prism will be mm 1 2 b b b 40.0854 is h/dl h l d d ) Ray path lengths of a prism at minimum 21 bb ( =− RP If the 25-mm prism is completely filled, the resolving power, l If the 25-mm prism is completely filled, the resolving power, is sufficient to resolve the Sodium D lines. This base of the prism. So, we have the classical result that the resolving base of the prism. So, of the prism times the power of a prism spectrometer is equal to the base dispersion of the prism material. consider CVI Melles Griot EDP-25-F2 prism, operating in As an example, and emergence are angle of incidence minimum deviation at 590 nm. The both then 54.09° and d PELLIN BROCA PRISMS prism is split in half along Broca prism, an ordinary dispersing In a Pellin prism, the two halves are Using a right angle the bisector of the apex angle. joined to create a dispersing prism with an internal right angle bend obtained 1.42. by total internal reflection, as shown in Figure prism to create a Pellin one can split any type of dispersing In principle, Broca prism is based on an Isosceles the Pellin Broca prism. Typically Brewster prism. Provided the light is p deviation essentially lossless. Figure 1.41 Figure b If the beam is made to fill the prism completely, where the relevant quantities are defined in Figure 1.41. in Figure quantities are defined where the relevant At minimum deviation, translating a prism along the bisector of the apex angle along the bisector of deviation, translating a prism At minimum is 1.40. This See Figure output rays. the direction of the does not disturb to laser design where intracavity prisms are used important in femtosecond dispersion. By aligning a prism for minimum compensate for length the it along its apex bisector, deviation and translating the contribution thus varying with no misalignment, in material may be varied possible to it is group velocity dispersion. Finally, of the material to overall show that at minimum deviation 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.39 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.40

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.40 image isunchanged. direction, withtheimagerotated180°.Asbefore, thehandedness ofthe of theprismsystemisabeamparalleltobutdisplacedfromitsoriginal first, isplacedsuchthatthebeamwilltraversebothprisms. The neteffect as showninFigure 1.46.Asecondprism,rotated90°withrespect tothe Porro prismsaremostoftenusedinpairs, formingadouble Porro prism, Figure 1.45 prisms haveroundededgestominimizebreakageandfacilitateassembly. image isreflectedtwice, thehandednessofimageisunchanged.Porro direction offsetfromitsentrancepoint,asshowninFigure 1.45.Sincethe traveling throughaPorro prismisrotatedby180°andexitsintheopposite the 45°slopedfaces, andexitsagainthroughthelargeface. Animage the largefaceofprism,undergoestotalinternalreflectiontwicefrom prism usedtoaltertheorientationofanimage. Inoperation,lightenters A Porro prism,namedforitsinventorIgnazioPorro, isatypeofreflection PORRO PRISMS fluence levelsof50mJ/cm specifically 180-nm to240-nmregion.Crystal-quartzPellin Brocaprismsare used from240nmto2000nm.Excimer-grade prismsareusedinthe and istheleastexpensive. UV-grade fusedsilicaPellin Brocaprismsare of sizesandmaterials. BK7prismsareusedinthevisibleandnearIR, CVI MellesGriotoffersBrewsteranglePellin Broccaprismsinanumber shorter wavelength Figure 1.44 catastrophic damage)abovethisfluence, probablyduetoself-focusing. l l Fundamental Optics 1 1 >l , l designed forhigh-powerQ-switched266-nmlaserpulsesat 2 2 Porro prismsretroreflect andinverttheimage Longer wavelengthisdeviatedlessthanthe 2 . Fusedsilicaprismstrack(i.e., sufferinternal l 2 l 1 with theimagerotated 180° parallel tobutdisplacedfrom itsoriginaldirection, Figure 1.46 . provide alonger, foldeddistancebetweentheobjectivelensesand an invertedimageandinmanybinocularstobothre-orientthe Double Porro prismsystemsareusedinsmallopticaltelescopestoreorient Double porro prismresults inabeam Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.41 are out are are y x y E x y E and z y www.cvimellesgriot.com and x x E E z Fundamental Optics Fundamental Fundamental Optics qq x x E E E E y y E E Circularly polarized light. Circularly Linearly polarized light. is the unit vector for right circularly polarized light; for negative helicity is the unit vector for right 4 Figure 1.48 Figure Figure 1.47 Figure Polarization e light; for light that rotates clockwise in a fixed plane as viewed facing into light; for light that rotates and for light whose electric field rotation disobeys the right the light wave; propagation. hand rule with thumb pointing in the direction of in of phase by angular frequency of phase by angular frequency (1.76) (1.77) (1.78) (1.75) . If the y and q − () ikz t ) y f i direction. These can be thought of as the can be direction. These y z and

are real numbers defining the magnitude and the x f y i x − + () () , and f x y =+ f f , traveling in the = y / / is the unit vector for left circularly polarized light; for positive 4 E 12 12 x , () () x = f = = e E (, ,) ( = − + APPLICATION NOTE APPLICATION Exyz xEe yEe e exiy exiy f Polarization Convention the orientation of a polarized electromagnetic wave Historically, has been defined in the optical regime by the orientation of the is the convention used by CVI Melles Griot. This electric vector. amplitude and phase shift of the field along two orthogonal directions. amplitude and phase shift of the field along two phase of the field components in the orthogonal unit vectors x phase of the field components in the orthogonal origin of time is irrelevant, only the relative phase shift origin of time is irrelevant, where need be specified. 2. CIRCULAR REPRESENTATION into circularly polarized In the circular representation, we resolve the field by the complex unit vectors basic states are represented The components. where 1. CARTESIAN REPRESENTATION 1. CARTESIAN the propagation equation for an electric field is given In Cartesian coordinates, by the formula Polarization States Polarization Fourier to describe a single plane wave numbers are required Four component helicity light; for light that rotates counterclockwise in a fixed plane as and for light whose electric field viewed facing into the light wave; rotation obeys the right hand rule with thumb pointing in the direction of propagation. 1ch_FundamentalOptics_Final_a.qxd 7/21/2009 9:41 AM Page 1.41 9:41 AM 7/21/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.42

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics angle to thex the parameterw 1.42 Figure 1.49 Let per opticalcycle. that thetipofelectricfieldvectorwilldescribeanellipse, rotatingonce An arbitrarypolarizationstateisgenerallyellipticallypolarized.This means 3. ELLIPTICALREPRESENTATION where E As inthecaseofCartesianrepresentation,wewrite . Letw Let Note thatthephaseshiftd The ellipticalrepresentationis: ellipse asshowninFigure 1.49. polarized components. Notethat nitudes andphasesofthefieldcomponentsleftrightcircularly = y E E E =e a ( e _ + =e and be thesemimajorandb =+ = = E () E and = Fundamental Optics w abee h + _ = ξη ^ , with respecttothex be theaxesofaright-handedcoordinatesystemrotatedbyan • • e E E E y be theanglethatsemimajoraxismakes withthe i f 4 axes. = ^ , The polarizationellipse is implicitintherotationofy f = = i δ e 0 , andf 4 h k t ikz () E o 4 − b above isrequiredtoadjustthetimeorigin,and ω e 4 i f be thesemiminoraxisofpolarization are fourrealnumbersdescribingthemag- 4 Y ) axis andalignedwiththepolarization e i(kz-q E t) W a and y h axes withrespect X x (1.81) (1.80) (1.79) (1.82) axis. the signoftheirquotient;forexample, ifg returned bythefunctioniscontrolledsignsofboth that atan(x 5 In theabove, atan( We definethefollowingquantities: elliptical transformations. The inversetransformationsarestraightforward. For brevity, wewillprovideonlytheCartesiantocircularand CONVERSIONS BETWEENREPRESENTATIONS B. CARTESIAN TOELLIPTICAL TRANSFORMATION A. CARTESIAN TOCIRCULAR TRANSFORMATION p /4 or4 φ φ φφ ψφφ φφ ugg vgg δφφ bvu Eu Ev avu EE gE E gE E gE gE 12 34 + − 1 01234 3 4 2 + − =+ =+ =− =+ =− =− =+ =+ = = = =+ = = = () = () () () ()/ ()/ tn(, ( atan tn(,) , ( atan 3 1 2 3 2 x x y y y x , x x y y y x x x y y y x x x x y y y xx x / 12 / 234 12 34 y p o sin cos o sin cos i cos sin i cos sin ) /4, notp = 2 2 / / 4 atan(y 2 gg 2 φφ φφ φφ gg 2 φφ 2 2 22 1 4 3 x 12 12 / / , − y ) isthefourquadrantarctangentfunction.This means 2 E 2 /4. ) / x ) withtheprovisionthatquadrantofangle 2 = g Fundamental Optics 1 = 4 www.cvimellesgriot.com 1, thenf x and 12 y , notjust above is (1.83) (1.94) (1.93) (1.92) (1.91) (1.90) (1.89) (1.88) (1.87) (1.86) (1.85) (1.84) (1.98) (1.97) (1.96) (1.95) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.43 beam p-polarized transmitted www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics reflected beam s-polarization At CVI Melles Griot, the DOT marks preferred incident beam Figure 1.50 Figure for transmitted input face. This is the tested direction is also higher for this Damage threshold wavefront. orientation as well. . p is the (1.99) 0, and (1.101) (1.100) = ⊥ , is given by p, the above 1 E . The extinction . The ⊥ T has a broad maxi- ⊥ v = 0 point, or do a null f i 2 = 1 , less than 1. The transmission , less than 1. The /2, and T L is a unit vector along the rejected p T r = 1/2 v ⊥ () ( ) + || f , in terms of the incident field , in terms of the incident i 2 and the pass direction 1 /2. E 1 p , may not be 0. If ⊥ •• 8 ⊥ T = . A polarizer with perfect extinction has T is a familiar result. Because cos () ⊥ v T 2 vv • / 22 1/2 . When it is “crossed”, L cos L () L T T cos sin with sufficient accuracy to find the v () T || = = v = = 2 = T =+ 21 2|| T Tpe rrEe ETppEeT EppE 0, TT T = ratio is e where the phase shift of the transmitted field has been ignored. where the phase shift of the transmitted field has A real polarizer has a pass transmission, In the above, the phase shifts along the two directions must be retained. In the above, rejected beam. If v Similar expressions could be arrived at for the direction, then The above equation shows that, when the polarizer is aligned so that The v equation predicts that measurement at v thus Then the transmitted field the transmitted Then LINEAR that creates a state from an A linear polarizer is a device by removing the component orthogonal to the arbitrary input. It does this rejected plastic sheet polarizers which absorb the Unlike selected state. cube polarizers and thin-film plate polarizers beam (which turns into heat), Still others may refract creating two usable beams. reflect the rejected beam, thereby separating them. at different angles, the two polarized beams prism polarizers. and Rochon Examples are Wollaston of the polarizer is determined by unit vector Suppose the pass direction of the rejected beam, angle between the field E mum as a function of orientation angle, setting a polarizer at a maximum setting mum as a function of orientation angle, One has to either map of transmission is generally not very accurate. the cos 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.43 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.44

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.44 Figure 1.51 the boundary. The extraordinaryraywillbedeviated;theordinary a raynormallyincidentonbirefringentsurfacewillbedividedintwoat ization state, orthecrystallinesurfaceisperpendiculartoanopticalaxis, different The dispersioncurvefortheextraordinaryrayisafamilyofcurveswith unique curvewhentheindexofrefractionisplottedagainstwavelength. axis withinthecrystal.The dispersioncurveforordinaryraysisasingle, The twoindexesofrefractionareequalonlyinthedirectionanoptical of refractionfortheordinaryrayisconstantandindependentdirection. for extraordinaryraysisalsoacontinuousfunctionofdirection.The index velocity oftheserayschangeswithdirection.Thus, theindexofrefraction These raysneednotbeconfinedtotheplaneofincidence. Furthermore, the The and velocity. material aredistinguishedfromeachotherbymorethanjustpolarization the beamsoverlapuponemergence. The twonewbeamswithinthe polarized beams(seeFigure 1.51).The beamwillbeunpolarizedwhere collinear withthebeam,radiationwillemergeastwoseparate, orthogonally If thecrystalisaplane-parallelplate, andtheopticalaxisdirectionsarenot speed, dependingonwhetherthebirefringentcrystalisuniaxialorbiaxial. crystal inwhichthebeamwillremaincollinearandcontinueatsame speeds. There willbeonlyoneortwoopticalaxisdirectionswithinthe beams usuallypropagateindifferentdirectionsandwillhave monochromatic lightintotwobeamshavingoppositepolarization.The A birefringentcrystal,suchascalcite, willdivideanenteringbeamof Polarization Definitions unpolarized linearly polarized output beamA Fundamental Optics curves fordifferentdirections. Unlessitisinaparticularpolar- Double refraction inabirefringent crystal outputb ordinary rays arereferredtoasextraordinary(E)andordinary(O). unpolarized input beam ray eam ray extraordinary b irefring output beamB linearly polarized material ent beam intotwoorthogonallypolarizedbeams. changes inpropagationdirectionareoptimizedtoseparateanincoming other cases, suchasWollaston andThompson beamsplittingprisms, one ofthepolarizationplanes, forexample, inGlan-typepolarizers. In difference inrefractiveindexisusedprimarilytoseparateraysandeliminate to createbirefringentcrystalpolarizationdevices. Insomecases, the The differencebetweentheordinaryandextraordinaryraymaybeused upon emergencetoformanellipticallypolarizedbeam. If thecrystaliscutasaplane-parallelplate, thesebeamswillrecombine any value. Itisalsopossiblethatallenergywillgointooneofthenewbeams. The energyratiobetweenthetwoorthogonallypolarizedbeamscanbe the originalorientationofvectortocrystal. beam energy, whichwillbedividedbetweenthenewbeams, dependson angles totheotherandwilltravelindifferentdirections. The original beam willbedividedintotwoseparatebeams. Eachwillbepolarizedatright crystal alongadirectionnotparalleltotheopticalaxisofcrystal, If abeamoflinearlypolarizedmonochromaticlightentersbirefringent indices ofrefractionthematerial. smaller) extraordinaryrayindexn not. The ordinaryrayindexn degrees oflinearpolarizationmaybeachieved.This polarization method making thenumberofplateswithinstacklarge(morethan25),high reflected ateachsurface, andallthoseparalleltoitwillberefracted. By angle, somevibrationsperpendiculartotheplaneofincidence willbe If anumberofplatesarestacked parallelandorientedatthe polarizing a single-surfacereflectionissmall. transmitted (theother15percentisreflected).The degreeofpolarization from of incidenceistransmitted.Only85percenttheperpendicularlight is 100 percentofthelightwhoseelectricvectoroscillatesparalleltoplane For a linearly polarized. transmissive dielectricsuchasglass, theemergingrefractedrayispartially When abeamofordinarylightisincidentatthepolarizingangleon POLARIZATION BYREFLECTION of 100mmareavailable. field ofviewislarge(uptograzingincidence),anddiametersinexcess material The transmittedbeamislinearlypolarized.Polarizers madeofsuch , sothatabsorptionishighinoneplaneandweaktheother. dyed. The dyemoleculesselectivelyattachthemselvestoalignedpolymer is stretched,aligningmoleculesandcausingthemtobebirefringent,then ufactured withorganicmaterialsimbeddedintoaplasticsheet.The sheet during transmissionthroughamaterial.Sheet-typepolarizersareman- isselectiveabsorptionofonepolarizationplaneovertheother DICHROISM are veryusefulforlow-powerandvisualapplications. The usable single surface(withn , andthemostextreme(whethergreateror e , aretogetherknownastheprincipal Fundamental Optics = 1.50) atBrewster’s angle, www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.45 /4 at 633 nm due www.cvimellesgriot.com l to /4 at 633 nm or better Fundamental Optics Fundamental Fundamental Optics significant problem only when the polarizer is being significant problem only Laser Grade deformation of l with a wavefront due to striae only. Optical Grade deformation of 1l Calcite with a wavefront to striae only. Scatter calcite crystal account for the main source Small inclusions within the In general, bubbles. may appear as small cracks or They of scatter. scatter presents a that can be tolerated amount of scatter centers The used with a laser. beam size and power. is partially determined by CVI MELLES GRIOT CALCITE GRADES CVI MELLES GRIOT grouped the most applicable calcite qualities, CVI Melles Griot has selected into two grades: with a trace of yellow is acceptable. This yellow This with a trace of yellow is acceptable. essential to use colorless calcite. For near-infrared For essential to use colorless calcite. absorbed when the polarization vector is aligned with the absorbed when the polarization those that exhibit no optical defects, are difficult to find and are those that exhibit no optical defects, typically fall into laser applications or optical research. typically fall into laser applications or optical operties of the film. Peak absorption can be selected for any wave- operties of the film. Peak coloration results in a 15-percent to 20-percent decrease in transmission below 420 nm. Distortion (Striae) Wavefront are fluctuations in the refractive index of calcite, or streaked Striae, can cause distortion They caused by dislocations in the crystal lattice. is particularly passing through the crystal. This of a light wavefront troublesome for interferometric applications. Spectral Properties can as well as lattice defects, amounts of chemical impurities, Trace visible light cause calcite to be colored, which changes absorption. For it isapplications, material applications, CALCITE is found in a rhombohedral crystalline form of calcium carbonate, Calcite, is a naturally Since calcite such as limestone and marble. forms various highest The occurring material, imperfections are not unusual. materials, Applications for calcite com- more expensive than those with some defects. ponents CVI Melles Griot offers calcite components in two quality grades to meet those needs. various are three main areas of importance in defining calcite quality. There THIN METAL FILM POLARIZERS FILM THIN METAL on small, elongated metal particles will be Optical radiation incident preferentially is utilized in CVI Melles Griot polarizing beamsplitter cubes which are beamsplitter cubes CVI Melles Griot polarizing is utilized in thin films on the interior dielectric many layers of quarter-wave coated with two separates an incident laser beam into beamsplitter This prism angle. polarized beams. perpendicular and orthogonally CVI Melles Griot infrared polarizers utilize this effect long axis of the particle. polarizers are considerably These the near-infrared. polarizers for to make polarizers. more effective than dichroic thin films are formed by using the patented Slocum process to Polarizing spheroids onto a polished deposit multiple layers of microscopic silver prolate these spheroids determine the exact dimensions of The glass substrate. optical pr length from 400 to 3000 nm by controlling the deposition process. process. length from 400 to 3000 nm by controlling the deposition method. Other CVI Melles ratios up to 10,000:1 can be achieved with this high as 100,000:1. Griot high-contrast polarizers exhibit contrasts as 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.45 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.46

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics where The slowandfastaxisphaseshiftsaregivenby: to thefastcomponent. axis andappendstheslowphaseshifttoit.Itdoesasimilaroperation right, thewaveplate takes thecomponentofinputfieldalongitsslow shows explicityhowthewaveplate actsonthefield.Readingfromleftto negative uniaxialcrystalsh the opticaxisiscalledslowaxis, whichisthecaseforcrystalquartz.For fast axes, andt The equationforthetransmittedfieldE where 1.46 the opticaxisanordinarywave. Ifthecrystalispositiveuniaxial, the opticaxisanextraordinarywave andthecomponentperpendicularto perpendicular totheopticaxis. This makes thefieldcomponentparallelto Suppose awaveplate madefromauniaxialmaterialhaslightpropagating STANDARD WAVEPLATES: LINEARBIREFRINGENCE is theoperationmodeofpolarizationrotators. phase shiftdiffersforleftandrightcircularlypolarizedcomponents. This waveplates. Withcircularbirefringence, theindexofrefractionandhence polarized linearpolarizationstates. This istheoperationmodeofstandard of refraction(andhencethephaseshift)differsfortwoorthogonally There aretwotypesofbirefringence. Withlinearbirefringence, theindex the conversionofonepolarizationstateintoanother. orthogonally polarizedfieldcomponentsofanincidentwave, causing Waveplates usebirefringencetoimpartunequalphaseshiftsthe Waveplates Let E wavelength. plate atagivenwavelength isneverexactlyahalfwaveplate athalfthat the birefringenceisveryimportantinwaveplate design;aquarterwave- an anglev In theabove, D lost inmeasuringintensity, andassigntheentirephasedelaytoslowaxis: To furtheranalyzetheeffectofawaveplate, wethrowaway aphasefactor fDhl l phl hl f ff EEe E h qphllfhqq phllfhqq EEe Ee 1 21 ss ff 11 21 = =−= be initiallypolarizedalong h s = = = = s 2 and and Fundamental Optics sff ss s sff ss p () with thex () () () f •• h •• are unitvectorsalongtheslowandfastaxes. This equation f f () are, respectively, theindicesofrefractionalongslowand is thethicknessofwaveplate. h ct tc ( ct tc 2/ l // // tt i ) isthebirefringenceh f i / f s l ⎣ ⎡ = = + axis. This orientationisshowninFigure 1.52. + 2 2 s () () () e f s < − () () h 1 o , theopticaxisiscalledfastaxis. f x () , andletthewaveplate slowaxismake i f f ⎦ ⎤ 2 t , intermsoftheincidentfieldE s ( l ) -h f ( l ). The dispersionof h e > h (1.102) (1.107) (1.106) (1.105) (1.104) (1.103) o , then 1 is: For ahalf-wave : waveplate withrespect toanx Figure 1.52.Orientationoftheslowandfastaxesa For afull-wave waveplate: and theplatethickness. function ofthewavelength, the birefringenceisafunctionofwavelength Note thatv ers thetransmissionsaregivenby: When thewaveplate isplacedbetweenparallelandperpendicularpolariz- For aquarterwaveplate, as apolarizationrotator. were rotatedthroughanangleof2 This transmissionresultisthesameasifaninitiallinearlypolarizedwave table below. between theslowandfastaxes. There arefourpossibilities listedinthe circularly system suchthatsxf= the slowandfastaxisunitvectorssfformarighthandedcoordinate To analyzethis, wehavetogobackthefieldequation.Assumethat Phase Shift f f = = f f f TEy TEx || 3 p ⊥ = = = p /2 ∝=− ∝= /2 (2m 2mp (2m || = |/ || = polarized light,linearlylightmustbealignedmidway 2 2mp is onlyafunctionofthewaveplate orientation,and 2 = = , 2mp T 222 22 22 1)p 1)p || = i sin 2 sin 1 sin 1, andT , T /2 (i.e., anoddmultipleof || f = 2sin cos = v ⊥ 2 z, thedirectionofpropagation.To obtain v f = 2 v 0, regardlessofwaveplate orientation. Y , andT f nu il ln InputField Along Input Field Along v . Thus, ahalf-wave waveplate findsuse 2 / (s 2 ⊥ RCP -polarized inputfield LCP = = E sin f 1 )/ v √ 2 p 2 _ 2 Fundamental Optics /2). v . www.cvimellesgriot.com s X (s RCP LCP 4 f f is onlya )/ (1.111) (1.110) (1.109) (1.108) √ _ 2 Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.47 960 1000 www.cvimellesgriot.com Fundamental Optics Fundamental 10° incidence angle. 10° incidence angle. 880 920 8 Fundamental Optics QWPO-800-XX-2 840 800 WAVELENGTH (nm) WAVELENGTH 760 720 Zero-order crystal quartz half-wave Zero-order . The two plates may be either air-spaced or optically contacted. plates may be either air-spaced two . The 680 0

60 80 40 20

100

parallel

(%) T 10 nm. Changes with temperature and drive conditions cause wave- 8 Figure 1.54 Figure waveplate for 800 nm WAVEPLATES POLYMER since they are true offer excellent angular field of view waveplates Polymer 1.55 compares the change in retardance as Figure zero-order waveplates. A polymer waveplates. function of incidence angle for polymer and quartz changes by less than 1% over a waveplate concern. change is often of key Retardance accuracy with wavelength tolerance an off-the-shelf diode laser has a center wavelength example, For of polymer waveplates These length shifts which may alter performance. performance even with minor shifts in the maintain excellent waveplate temperature sensitivity of laminated polymer The source wavelength. is about 0.15 nm/°C, allowing operation over moderate tem- waveplates perature ranges without significantly degrading retardance accuracy. types and their dependence on A comparison of different waveplate is shown in figure 1.56. wavelength the fast axis of the other. The net phase shift of this zero-order net phase The of the other. the fast axis of one against is p waveplate between waveplate 800-nm zero-order half-wave transmission of an The Its 1.54 using a 0-10% scale. in Figure parallel polarizers is shown 100:1 over a bandwidth of about 95 nm centered extinction is better than at 800 nm. multiple order and zero order crystal quartz CVI Melles Griot produces and 2100 nm. Virtually between 193 nm any wavelength at waveplates in stock, and custom wavelength kept are all popular laser wavelengths time. with short delivery parts are available solution for low- zero-order waveplate are an inexpensive Mica waveplates power applications and in detection schemes. l 8 l 4 / 0, we have a l

= 33 560 580 2 / l

17 4 / l

35 9 l mm thick, which is too thin for 520 540 4 / l

18. Multiple-order waveplates are 18. Multiple-order waveplates 37 2 / = l

500 19 4 / /2 half waveplate at 488.2 nm and a 10 /2 half waveplate l

l WAVELENGTH (nm) WAVELENGTH 39 480 10 l 4 / l

2 / 41 l

460 21 Transmission of a 0.5-mm-thick crystal Transmission 440 0 phase shift at 800 nm. If made from a single plate of crystal

60 80 40 20

100

parallel (%) T / 4 waveplate at 500 nm with m / 4 waveplate l inexpensive, high-damage-threshold retarders. Further analysis shows that high-damage-threshold retarders. inexpensive, this same 0.5mm plate is a 19 transmission of this plate between par- at 466.5 nm. The waveplate full-wave The 1.53 as a function of wavelength. allel polarizers is shown in Figure Note how quickly points is also shown. key retardance of the plate at various wave- multiple-order Because of this, the retardance changes with wavelength. plates are generally useful only at their design wavelength. ZERO-ORDER WAVEPLATES are not useful with tunable multiple-order waveplates As discussed above, ). A zero-order femtosecond or broad bandwidth sources (e.g., can greatly improve the useful bandwidth in a compact, high- waveplate damage-threshold device. waveplate half-wave consider the design of a broadband As an example, centered at 800 nm. Maximum tuning range is obtained if the plate has a single p zero order waveplate. crystal quartz near 500 nm is approximately 0.00925. birefringence of The simple calculation A Consider a 0.5-mm-thick crystal quartz waveplate. for 500 nm; in fact, it is a shows that this is useful as a quarter waveplate 37 1.53 Figure quartz waveplate between parallel polarizers would be about 45 quartz, the waveplate two crystal quartz is to take solution easy fabrication and handling. The plates differing in thickness by 45 mm and align them with the slow axis MULTIPLE-ORDER WAVEPLATES MULTIPLE-ORDER examples given in standard waveplate quarter-wave the full-, half-, and For m given by the integer m. For is the order of the waveplate waveplates, > 0, m For waveplate. is termed a multiple-order the waveplate Sometimes, waveplates described by the second line above are called 3/4 line above are called described by the second waveplates Sometimes, Melles Griot permits the CVI order waveplates, multiple For waveplates. to satisfy the requirements classes of waveplates use of either of the above waveplate. of a quarter-wave 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.47 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 7/6/20092:02PMPage1.48

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.48 wavelength range. retardances. Retardationtoleranceisbetterthanl/100overtheentire Three wavelength rangesareavailable inbothquarterandhalfwave achromatic performance. are comprisedofcrystalquartzandmagnesiumfluoridetoachieve ACWP-series achromaticwaveplates fromCVIMellesGriot(seeFigure 1.57) over amuchwiderrangethaninwaveplates madefromonematerial. materials. T waveplate, cancellationcanoccurbetweenthedispersionsoftwo If twodifferentmaterialsareusedtocreateazero-orderorlow-order is requiredforbandwidthsupto300nm. design whichcorrectsfordispersiondifferencesoverthewavelength range increases toabout100nmatacenterwavelength of800nm.Adifferent retardation At 500nm,acrystalquartzzero-orderhalf-wave waveplate hasa ACHROMATIC WAVEPLATES quarter waveretarders. Figure 1.56 and polymerwaveplates Figure 1.55

RETARDANCE () RETARDANCE (waves) 0.225 0.250 0.275 0.300 0.200 0.30 0.40 0.50 0.70 0.60 Fundamental Optics .009 .011 1.20 1.10 1.00 0.90 0.80 hus, thenetbirefringentphaseshiftcanbeheldconstant tolerance ofl 05 Wavelength performanceofcommon Retardance vsincidenceangleforquartz 10 INCIDENCE ANGLE(degrees) RELATIVE WAVELENGTH (l/ /50 overabandwidthofabout50nm.This slow axis around fast axis tilt around 20 30 polymer quartz multiple order achromatic zero order 40 c ) 0 applications throughtheuseofthinplates. In addition,wemanagelowgroupvelocitydispersionforultrashortpulse high degreeofachromatizationisachievable bythedualmaterialdesign. linear polarizerparalleltotheinitialpolarizationstateshouldbezero. A perfect retardation toleranceisbetterthanl 33% and67%.(Inallbuttheshortestwavelength design,quarter-wave waves, andtransmissionthroughalinearpolarizermustbebetween For quarter-wave waveplates, perfectretardanceis amultipleof0.25 Figure 1.57 function ofthethicknessdifferenceenablingwidebandwidthperformance. temperature stability. The retardationofthecompound waveplate isalsoa thickness differencebetweenthewaveplates, resultinginexcellent In thisconfiguration,thetemperaturedependenceisafunctionof axes orthogonaltooneanother, effectivelycreatingazero-orderwaveplate. Another approachistocombinetwoquartzwaveplates withtheiroptical operate bestoveranarrowbandwidthandtemperaturerange. high orderwaveplate. Therefore, thesedual-wavelength waveplates and retardationconditions. This oftenresultsintheselectionofarelatively careful One way toachievethemultiple retardationspecificationsisthrough or triplinglasersourcessuchasNd:YAG (1064/532/355/266). leaving theotherunchanged.This frequentlyoccursinnonlineardoubling ing beamsplitterbyrotatingthepolarizationofonewavelength by90°,and common applicationisseparationofdifferentwavelengths withapolariz- Dual-wavelength waveplates areusedinanumberofapplications. One DUAL-WAVELENGTH WAVEPLATES

transmission (%) selection ofmultiple-orderwaveplates whichmeetbothwavelength retardance is0.5waves, whileperfecttransmissionthrougha 0.00 0.25 0.50 0.75 1.00 400 ACWP-400-700-10-2 5 0 5 0 5 700 650 600 550 500 450 wavelength (nm) /100.) For half-wave waveplates, Fundamental Optics www.cvimellesgriot.com

0.46 0.48 0.50 0.52 0.54 retardation in wave in retardation Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.49 (1.113) (1.114) (1.115) (1.116) 90 100 www.cvimellesgriot.com 80 Fundamental Optics Fundamental is the phase shift is the phase Fundamental Optics 60 70 REFLECTIVITY (%) 50 is given by R -1 40 R Reflectivity finesse vs. coating reflectance of Reflectivity finesse vs. coating reflectance in cm in nm in Hz cos F FSR R d d d R 2 30 d 1 c h h h hv = l − 0 2 2 2 60 80 40 20 p 1 p

l 100 = =

= 2 is the reflectance of each surface; d is the reflectance =

E SS FINE TIVE C = REFLE R is the etalon spacing or thickness is the etalon spacing or thickness R is the refractive index (e.g., 1 for air-spaced etalons) 1 for air-spaced is the refractive index (e.g., is the angle of incidence

d FSR h d v F FWHM where, where, (FSR) of the etalon is given by free spectral range The F reflectivity finesse, The Figure 1.59 Figure each surface Here, Here, Figure 1.59 shows the reflectivity finesse as a function of the coating reflec- Figure tivity. bandwidth (FWHM) is given by The the above applies to theoretical etalons which are assumed to be However, show defects that limit theoret- even the best etalon will perfect. In reality, in a real etalon, the actual finesse Therefore, ically expected performance. will usually be lower than the reflectivity finesse. (1.112) R F FSR nd c 2 ) 2 d ( FREQUENCY FSR = 2 FWHM = sin / 1 2 ) R R 4 are a special type of solid etalon in which the cav- − 1 ( + consist of pairs of very flat plano-plano plates separated 1 Transmission characteristics of a Fabry-Perot Transmission are made from a single plate with parallel sides. Partially are made from a single plate with parallel sides. 0 inc 50 trans I 100 I

== ION (%) ION SS MI S TRAN T reflecting coatings are then deposited on both sides. The cavity is formed by The reflecting coatings are then deposited on both sides. the plate thickness between the coatings. ity is formed by a deposited layer of coating material. The thickness of this The ity is formed by a deposited layer of coating material. required and can range deposited layer depends on the free spectral range cavity is sandwiched The from a few nanometers up to 15 micrometers. between the etalon reflector coatings and the whole assembly is supported on a fused-silica base plate. avoid Etalon plates need excellent surface flatness and plate parallelism. To peak transmission losses due to scatter or absorption, the optical coatings also have to meet the highest standards. incident on the etalon, the transmission of the etalon is a plane wave For given by: Solid Etalons Deposited Solid Etalons Figure 1.58 Figure etalon Air-Spaced Etalons Etalons are most commonly used as line-narrowing elements in narrow- line-narrowing elements most commonly used as Etalons are elements in and coarse-tuning or as bandwidth-limiting band laser cavities profile Further applications are laser line lasers. broadband and picosecond monitoring, diagnosis. type. in this section are all of the planar Fabry-Perot etalons described The Figure characteristics for this type etalon are shown in transmission Typical 1.58. Etalons by optically contacted spacers. The inner surfaces of the plates are coated inner surfaces The by optically contacted spacers. the outer surfaces are coated with antire- with partially reflecting coatings, flection coatings. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.49 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.50

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics The beamdivergencealsoinfluencestheactualfinesseofan etalon. 1.50 All threetypesofdefectscontributetothetotaldefectfinesse total defectfinesse Figure 1.60 (graphical representationsareexageratedforclarification). The defectsthatcontributetothisreductionareasshowninFigure 1.60 is thediffraction-limitedfinessecoefficient. where coefficient, FFF F F F F F 1 1 111 d R D S v F 2 = = = = R Fundamental Optics ++ =+ is thereflectivityfinesse, 1 p d M C c. b. a. 2 633nm 2 F − d A t a v S l l R R 2 n 2 is theincidentbeamdivergencefinessecoefficient,andF 2 Three typesofdefectscontributing tothe v l Parallelism Defec S S pheric pheric dg 2 al Irreg al Defec dp 2 F s is theplatesphericaldeviationfinesse ularities ( ts ( ts ( F F s ) d p ) F dg ) F d : (1.121) (1.120) (1.119) (1.118) (1.117) d Taking intoaccountallthesecontributions, theeffectivefinesse( ness andclearaperture. The examplesbelowshowhowtheeffectivefinessevaries withplateflat- especially whenahighfinesseisrequired. the absoluteclearaperture, butalsoontheusedapertureofetalon, The effectivefinesseauserseeswhenusingtheetalondependsnotonlyon is: etalon (withF mission values. tant touseveryhigh-qualityplatesensureahighfinesseandgoodtrans- These examplesillustratethat,forlarge-apertureapplications, itisimpor- Example 1: Example Example 3: Example 2: Example F F Beam divergence= Plate rms= Spherical /parallelismdefects= Spacer (airgap)= Used aperture= CA R Spherical /parallelismdefects Same parametersasexample1except: F Used aperture= Same parametersasexample1except: F Beam divergence:0.1mRad Plate rms= 11111 e R R R ======+++ 95% (8 25 mm 61, 61, 61, Air-spaced etalon, Air-spaced etalon, Air-spaced etalon, FFFF F F F R R 2 e e e 0.80 nm 0.40 nm being thereflectivityfinesseand = = = 1%) at633nm 40 (8 40 (8 10 20 mm 5 mm D 1 mm 22 0.1 mRad 8) 4) v = < l S /100 l 2 /20 Fundamental Optics F v the divergencefinesse) www.cvimellesgriot.com F (1.122) e ) ofan Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.51 www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics (1.123) ⎤ ⎥ ⎦ d T ∂ ∂ As the angle of incidence is increased, the As the angle of incidence d + h ∂ ∂ n 11 ⎡ ⎢ ⎣ ) Primarily used for solid etalons, temperature-tuning etalons, Primarily used for solid FSR ( : Air-spaced etalons can be tuned by increasing the pres- : Air-spaced =− ) refraction, and thus the effective spacing. TT FSR ∂ ( ∂ changes both the actual spacing of the reflective surfaces via expansion changes both the actual of the material, which changes the optical and the index of refraction can be given by tuning result spacing. The center wavelength of the etalon can be tuned down the . of the etalon can center wavelength Pressure tuning Temperature tuning: TUNING AN ETALON TUNING AN a limited range to alter their peak transmission Etalons can be tuned over are: techniques These wavelengths. Angle tuning or tilting the etalon: sure in the cavity between the optics, thereby increasing the effective sure in the cavity between the optics, index of plate above examples illustrate how critical the optical surface flatness, The performance of an etalon. parallelism and surface quality are to the overall that allows software At CVI Melles Griot we have developed sophisticated of an etalon. To us to simulate all effects that influence the performance are required. order an etalon, FSR, finesse and used aperture 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.51 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.52

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics in timebytheconstantgroupdelay the reflectedpulseisscaledbyamplitudereflectance Thus, providedthephaseshiftislinearinfrequencyover thepulsebandwidth, replica oftheoriginalpulse. Consider ageneralinitialpulseshape of amirrorortransmissiveopticdoesthisimply? pulse notbebroadenedordistorted.Whatconstraintontheperformance Fourier componentsofthepulsemustalsobepreservedinorderthat dealing with 1.52 however, tosimplypreservethepowerspectrum polarization withanacceptabletransmissionoftheother. Itisnotenough, fairly constantreflection;apolarizermaymaintainitsrejectionofone have areflectivitygreaterthan99.8%;50%beamsplittermay This meansthat,overtheentirepulsebandwidth,acavitymirrormay laser mirrorareacceptableoverthebandwidthofafemtosecondpulse. Assume thatthepower, reflectivity, andpolarizationcharacteristicsofa are constantlyfieldingnewrequests. optics inthissectionhavebeentestedbyresearchersthefieldandwe created withdesirablecharacteristicsforfemtosecondresearchers. All of thephasecharacteristicsinmind.Newproprietarydesignshavebeen of theseeffects. Certaincoatingdesignshavebeenmodifiedwithcontrol pulse bandwidth.CVIMellesGriothasmadeanintensivetheoreticalstudy control thephasecharacteristicofopticalsystemoverrequisitewide The distinguishingaspectoffemtosecondlaseropticsdesignistheneedto Ultrafast Theory constant shift phase effectshavebeenassumedtobedescribablebyasingle that reflectivity For this“ideal”mirror, r reflectance: mirror is“ideal”,andusetheFourier transformofitscomplexamplitude Suppose thispulsereflectsoffofamirror. For thisexample, weassumethe ponents, itmaybeexpressedas: E E rre re

f(q) thatislinearlyproportionaltofrequencywithproportionality r 0 ( ( q ( r d trE Ed tE ) ) ) Fundamental Optics t = =− == = d . The reflectedpulseisthen: Ett t rE ∫ ∫ femtosecond pulses. The phaserelationshipamongthe + () i () Fq qq ( ( is assumedconstantoverthepulsebandwidth.All qq ) − ) d it −− q itt q () + d it q is arealconstantequaltotheamplitude d E t 0 d ( . Itis, otherwise, anundistorted t ). AsafunctionofitsFourier com- S(q) = r, anddelayed |E(q)| 2 (1.126) (1.125) (1.124) when general, the constantgroupdelayoverfrequencyneededforperfectfidelity. In Examined overalargeenoughbandwidth,noopticalsystemwillexhibit velocity dispersionF″(q in aTaylor seriesforfrequenciesnear center frequencyq where The dispersion. dominating contributiontophasedistortionisnon-zerogroupvelocity Gaussian pulsepassesthroughamedium,orisincidentonmirrorwhose frequency, considerwhathappenswhenanunchirped,transform-limited To illustratepulsedistortionduetothedependenceofgroupdelayon assist thoseinterestedinthemodelingofrealopticalelements. the actualphaseshiftfunction be insufficient.Afullnumericalcalculationmayhavetoperformedusing Note, however, thatforextremelyshortpulsestheexpansionabovemay soluble model,forthepropagationofatransform-limitedGaussianpulse. sured infsec pulse enteramediumorreflectoffofmirrorwithnon-zero These derivatives are, respectively, thegroupdelay 10 to100femtoseconds. theoretical on pulsepropagation.The graphsshowninFigure 1.61 representthe pulses, isneverthelessanexcellentmodeltostudytheeffectsof dispersion This result,valid onlyforinitiallyunchirped,transform-limitedGaussian will remainGaussian;theresultforbroadenedFWHMis: ened byitsencounterwithgroupvelocitydispersion.The powerenvelope the material.)The Gaussianpulsewillbebothchirpedandtemporallybroad- material, where tFqt tt E qF q q q F Fq Fq 10 ( ( =+ tt t b ) 0 ″ ) the phaseshiftnearsomecenterfrequency =− is theinitialpulseduration(FWHMofintensity).Let and ( = q broadening fromdispersionforinitialpulsewidthsranging ex 2 ⎣ ⎡ ) isthegroupvelocitydispersion(GVD)percentimeterof qqq Fq 142 Fqqq radians. (For acontinuousmedium-like glass, z p/ ′′ ′′′ () is thephysicalpathlength,incentimeters, traveledthrough () ⎣ ⎡ field envelopeofthepulseisassumedtobeform: () () 000 00 () n/ ln 00 0 2 ln2 + . This expansionisheuristicallyuseful,inanexactly () () ′ − ′′ − () 0 ), andthecubictermF′″(q ( 22 ) t () 2 0 3 / F(q). CVIMellesGriotwillbehappyto /3 ⎦ ⎤ 0 2 2! − 3! . 2 + ⎦ ⎤ 1/2 q + 0 : Fundamental Optics www.cvimellesgriot.com q 0 0 F may beexpanded ), evaluated ata F ′ ( q ″ ( 0 q ), thegroup F ) = ″ ( b q ″ (1.129) (1.128) (1.127) ), mea- ( q ) # z Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.53 ) 1.6 1.8 e 2 SF11 SF10 SF2 LaFN28 BK7 CQ (n C7980 Fused Silica CaF www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics 1.0 1.2 1.4 WAVELENGTH (mm) WAVELENGTH GVD for common glasses 0.2 0.4 0.6 0.8 0 500

-500

1500 1000 4000 3500 3000 2500 2000 -1000

m) c (fs²/ GVD Figure 1.62 Figure GROUP-VELOCITY & CUBIC DISPERSION GROUP-VELOCITY FOR VARIOUS for show the GVD and cubic dispersion respectively 1.62 and 1.63 Figures UV Some of the glasses can be used in the some common used glasses. pulse useful in estimating material dispersion and should be region. They check these calculations independently before Please distortion effects. using them in a final design. GVD (fs²) GVD (fs²) GVD (fs²) 60 fs 50 fs 40 fs 30 fs 20 fs 100 fs 90 fs 80 fs 70 fs 10 fs Output pulse width vs. GVD 0 400 800 1200 1600 2000 0 200 400 600 800 1000 0 40 80 120 160 200 5

85 65 95 75 85 75 55 35 65 45 40 30 20 35 25 15 10

115 105

S PUL OUTPUT E WIDTH (fs) WIDTH E S PUL OUTPUT E WIDTH (fs) WIDTH E S PUL OUTPUT E WIDTH (fs) WIDTH E Figure 1.61 Figure 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:31 PM Page 1.53 2:31 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:31PMPage1.54

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.54 greater than100mJ/cm CVI MellesGriotusesthreebasicdesigns;TLM1mirrorsforenergyfluence DISPERSIVE PROPERTIES OF MIRRORS Figure 1.63 pulse distortionproblemsandshouldbeusedwheneverpossible. incidence, mirrors at45°incidencewiththep nearly zerooverabroadwavelength range. Thus oneshouldavoidusing sensitive towavelength, whiletheGVDfors and (c)at45°incidence, theGVDof at thedesignwavelength, (a)GVDiszero, (b)thecubictermisminimized, and aredesignedforuseatnormalincidence45degrees. Note that, shown inFigure 1.64.Intheseexamples, themirrorsare centeredat800nm parameter, andcubicdispersionparameterforTLM2highreflectorsare pulses, andTLMBmirrorswhichareahybridofthetwo. The reflectivity, GVD

TOD (fs³/cm) Fundamental Optics 1000 2000 2500 1500 3000 500 s -polarization providesverybroadbandwidthandminimizes 0 . . . 0.8 0.6 0.4 0.2 Cubic dispersionforcommonglasses 2 , TLM2mirrorsforcwoscillatorsandlow-fluence WAVELENGTH (mm) -polarization. Ontheotherhand,at45° p CaF FusedSilica C7980 CQ (n BK7 LaFN28 SF2 SF10 SF11 . . 1.4 1.2 1.0 -polarization componentisvery 2 -polarization componentis e ) . 1.8 1.6 TLM2-800-0 andTLM2-800-45 Figure 1.64 extracavity tosatisfychirpcontrolrequirements. with off-the-shelfavailability, andcanbeemployedbothintracavity Negative GroupVelocity DispersionMirrors(TNM2)meetthese needs where thelasermustprovideacompact,stable, andreliable solution. circuit. This becomesmandatoryinindustrialandbiomedicalapplications pensation ofthebuilt-inpositivechirpencounteredinlaseroptical Ti: andotherfemtosecondlasersystemsneedprismless com-

CUBIC TERM = F”’(q), fsec3 GVD = F”(q), fsec2 REFLECTIVITY (%) 1000 -100 250 500 750 100 100 -50 50 80 85 90 95 0 0 0 5 0 5 0 950 900 850 800 750 700 0 5 0 5 0 950 900 850 800 750 700 0 5 0 5 0 950 900 850 800 750 700 Dispersion andreflectivity formirrors 45°P 45°P TLM2-800 WAVELENGTH (nm) WAVELENGTH (nm) WAVELENGTH (nm) Fundamental Optics 45°P www.cvimellesgriot.com 0° TLM2-800 0° TLM2-800 0° 45°S 45°S 45°S Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings - 1.55 - and p s component p 72°P 72°S www.cvimellesgriot.com TFPK-800 TFPK-800 TFPK-800 Fundamental Optics Fundamental Fundamental Optics WAVELENGTH (nm) WAVELENGTH Properties for one coated side of a TFPK Properties 700 750 800 850 900 950 0 40 80 60 20 0.2 0.1 0.0 100 -0.2 -0.4 -0.6 -0.8 -1.0 -0.1 -0.2

California, 1986), for a good discussion of linear pulse propagation. California, 1986), for a good

³ c fse , C UBI C P 72° MITTED S TRAN ION (%) ION SS MI S TRAN ² c fse GVD, P 72° MITTED S TRAN Figure 1.67 Figure polarizing beamsplitter optimized for 800 nm. Both sides coated for these properties. are regenerative amplifiers. The main emphasis is on linear phase characteristics. emphasis is on linear main The amplifiers. regenerative Mill Books, (University A. E. Siegman 9 of Lasers, See Chapter Valley, amplification, the pulse may have to pass In chirped pulse regenerative 10 to 20 can be There twice per round trip. through one or two polarizers is saturated and the pulse is ejected. At this round trips before the gain psec); however the phase shift at each stage the pulse is long (100–1000 maintained to minimize the recompressed pulse frequency must still be put trips of the pulse in the regenerative amplifier many round width. The the phase characteristics of the coatings. stringent requirements on power transmission curves for both 1.67 shows the Figure polarization and the transmitted phase characteristics of the polarization and the transmitted phase characteristics specify any wavelength for a TFPK optimized at 800 nm. (Users may 200 mW 80 fs pulses negative GVD mirror output coupler centered at 785 nm 3mm are ideal for intracavity use in femtosecond Typical optical setup incorporating low GVD Typical Typical optical set-up of negative GVD mirrors Typical negative mirror low GVD GVD mirror pump Figure 1.66 Figure OUTPUT COUPLERS AND BEAMSPLITTERS Output-coupler partial reflectors and beamsplitters behave similarly; behavior The consideration in their analysis. here is an additional however, of the transmitted phase of the coating and the effect of material dispersion into within the substrate on the transmitted beam have to be taken coating transmitted phase In general, the account in a detailed analysis. has similar properties and magnitudes of GVD and cubic to the reflected we recom- a beamsplitter, As usual, centering is important. As phase. As an output mend the 1.5 mm thick fused silica substrate PW-1006-UV. thick, 30 minute wedge we recommend the 3.0 mm coupler substrate, fused silica substrate IF-1012-UV. CVI Melles Griot has developed the TFPK Series Broadband Low Dispersion Beamsplitters to satisfy requirements for very-high-power, Polarizing These short-pulse lasers. Figure 1.65 Figure in an ultrafast application and Negative GVD mirrors In experiments using CVI Melles Griot TNM2 negative group velocity dispersion negative group velocity using CVI Melles Griot TNM2 In experiments were achieved in a centered at 785 nm 80-fsec pulses 200-mW, mirrors, configuration is shown in The oscillator. Ti:Sapphire prismless, simple, 1.65. Figure 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:32 PM Page 1.55 2:32 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:32PMPage1.56

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics increase ininsertionlossforthetransmitted increase thisataslightlossinbandwidth,incidenceangle, and The reflectivityfors-polarizationislimitedto75%.Variant designscan mechanical clearancesofthelaserbeamatsuchasteepincidenceangle. has tobesetproperlyandoptimized.Somethoughtgiven There aresomesubtletiesassociatedwiththeTFPK.The near72°angle performance isdominatedbythesubstrate. characteristics surface shouldbesquaredindeterminingthespecifications. The phase in Figure 1.67.Therefore, thes both sidesoftheoptichavecoatingwhosepropertiesaredescribed and havethesamelownonlinearitybroadbandwidth.Notethat s for the cubicphaseterm.Notshownarereflectedcharacteristics from 250nmto1550nm.)The phasecharacteristicsshownaretheGVDand 1.56 polarizers withp Figure 1.68 -polarization; theyaresimilartothep

REFLECTED BEAM CUBIC, fsec³ REFLECTED BEAM GVD, fsec² TRANSMISSION (%) Fundamental Optics -0.4 -0.2 0.2 0.6 0.8 2.6 3.0 0.0 0.4 2.2 2.4 2.8 3.2 3.4 40 45 55 60 50 0 5 0 5 0 950 900 850 800 750 700 show thatinallmodesofoperation,theTFPKpolarizer Transmission characteristicsforFABS series -polarized light WAVELENGTH (nm) - andp -polarization transmissioncurves, -polarization transmissionsper p FAB FAB FAB -polarized component. S-800-45P S-800-45P S-800-45P phase CVI MellesGriotcanproduceFABS inotherthan50:50withexcellent optics, thes by thesubstratematerialdispersion.Aswithvirtuallyalldielectriccoated linear pulsepropagationpropertiesofthesebeamsplittersaredominated splitters optimizedat800nm,areshowninFigures 1.68and1.69.The along withthecorrespondingreflectedphasecharacteristicsforbeam- Power transmissioncurvesforthes advantages overpartiallyreflectingmetalcoatings. T configurations. pump-probe experimentsandintheconstructionofantiresonantring band, 50%all-dielectricbeamsplitters. They areusefulinmanytypesof The FABS autocorrelatorbeamsplittersfromCVIMellesGriotarebroad- polarizers withs Figure 1.69

REFLECTED BEAM CUBIC, fsec³ REFLECTED BEAM GVD, fsec² TRANSMISSION (%) characteristics. -0.04 -0.02 -0.05 -0.03 -0.01 0.03 0.01 0.02 0.0 0.0 40 45 55 60 50 -polarized versionisbroaderthan 0 5 0 5 0 950 900 850 800 750 700 hey areessentiallylosslessandextremelydurable. Bothhave Transmission characteristicsforFABS series -polarized light WAVELENGTH (nm) - andp -polarized versionsoftheFABS, Fundamental Optics FAB FAB FAB www.cvimellesgriot.com p -polarized version. S-800-45 S-800-45 S-800-45 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:32 PM Page 1.57

Fundamental Optics

www.cvimellesgriot.com Fundamental Optics

ANTIREFLECTION COATINGS All CVI Melles Griot antireflection coating designs work well in femtosec- ond operation as the forward-going phasor is the dominant contribution to the phase shift; the AR coating is very thin and simply “fixes” the small Fresnel reflection of the substrate. asinBeam Optics Gaussian

PRISMS

A

red broadband blue red light blue

B

Figure 1.70 Brewster prism

Very-high-quality isosceles Brewster’s angle prisms for intra and extra- Optical Specifications cavity use are available from CVI Melles Griot. The design of these prisms satisfies the condition of minimum loss due to entrance and exit at Brew- ster’s angle. To calculate GVD at Brewsters angle, refer to Figure 1.70 and use the following equation:

2 3 2 2 λ d l d h dh = − (1.130) GVD 2 ψ ≈ 2 L 2 4l dω 2πc dλ dλ ω−ωl λl λl

where h = refractive index of the prisms (assuming the same material) l = tip to tip distance (AB) L = total avg. glass path w = spectral phase of the electric field Material Properties = ql ll 2pc (assumes Brewster prism at minimum deviation). For more on the Ultrafast phenomena, see J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Academic Press, 1996. Optical Coatings

Fundamental Optics 1.57 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:32PMPage1.58

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.58 Fundamental Optics How canwehelpyoumakeyourprojectasuccess? Your ordershipsintwoweeksorless • Coatingsfrom193nmto2300 • A widevarietyofin-stocksubstratesavailable(curves,flats,prisms,waveplates) • 2500 coatingoptions • 4000 uncoatedsubstrates • CVI MellesGriotofferstwo-weekdeliveryofopticsatcatalogprices. build-your-own, rapiddeliverysolutions. CVI MELLESGRIOT.YOURSOURCEFOR Lasers |LensesMirrorsAssembliesWindowsShuttersWaveplatesMounts Fundamental Optics www.cvimellesgriot.com