16-1 Aberrations OPTI 509

Lecture 16 Monochromatic aberrations; causes of aberrations, coordinate system; wave aberrations, tangential and sagittal rays, transverse and longitudinal ray aberrations 16-2 What is an aberration? Act of straying from the normal way, or deviating from the usual or natural type, esp. from a moral standard or natural state PFailure of a mirror, refracting surface, or lens to produce exact point- to-point correspondence between an object and its image PNeed to study aberrations because first order optical theory no longer suffices PTwo main types of aberrations studied here are • Chromatic • Monochromatic PChromatic aberrations arise because the index of refraction is a function of wavelength PMonochromatic aberrations occur even when if light is monochromatic (or quasi-monochromatic) • Spherical aberration, coma, and astigmatism blur the image • Petzval field curvature and distortion deform the image 16-3 Cause of aberrations The primary cause of aberrations discussed in this cours is violation of paraxial assumption PManufacture of the optical components can create aberrations PThe shape of the curved surfaces selected for the system regardless of perfect manufacture • Recall that Fermat’s principle implies that if we want to go point to point, we are better off with an ellipse or paraboloid • Spherical shape does not allow us to go from one point to another point while having the same optical path PPlacement of those components into the system (optical design) 16-4 Cause of aberrations The aberrations covered here are not caused by defects in the optics PAn aberration is the difference between the reference wavefront and the actual wavefront PAn unknown defect can give the illusion of an aberration, but the optics are actually performing as they should for its construction PFor example, the Hubble space telescope was launched with spherical aberration • Everyone knew it would be launched with some aberrations • Spherical aberration should have been minimal • Optical design did an excellent job at limiting aberrations • Actuators on the mirror could be used to fix small changes in the mirror due to launch, lack of gravity, and space environment • An error in the characterization of the primary mirror prevented the design from operating the way it was anticipated • Thus, the image quality from Hubble was poorer than specified 16-5 Paraxial approximation - A reminder Rays arriving at small angles relative to the optical axis are paraxial rays PAssuming small angles simplifies the relationship between object and image location PUsing the law of cosines and Fermat’s principle we can show that nR()sin()sin x+ R φ nRx R − φ 12object − image 0 = 22d object d image

• Where n1 and n2 are the indexes of refraction on the object side and image side respectively • R is the radius of curvature of the spherical surface • N is the angle between the optical axis and the point of intersection between the object ray and the spherical surface • x is the distance between the object or image and the vertex of the surface • d is the distance along the ray between the object or image and the surface 16-6 Paraxial assumption Diagram below illustrates the variables in the previous viewgraph PIf we assume that N is small then cosN.1 and sinN.N PAlso assume that d.x then n n nn− 1221+= xobjectx image R

All values positive in this case

d object dimage N R

xobject ximage n n1 2 16-7 Paraxial approximation The formulation just shown is independent of the intersection of the ray from the object with the surface PNo dependency on N

PAlso implies that there is a perfect image of the object at ximage PThe approximation relied on the fact that the sine and cosine can be expanded as a series 246 cos φ =−+−+1 φφφ !!246! K =−357 + − + sin φφ !!!357φφφ K PKeeping only the first terms in each gives us the paraxial assumption • One way to view aberrations is that the paraxial assumption is no longer valid and we have large off-axis angles • Now need to include the higher order terms 16-8 Perfect world versus real world In a perfect world, spherical wavefronts from the object are spherical in image space and converge to a point PThis is not the case in the real world • Cannot create perfect spherical wavefronts in image space • May not satisfy paraxial conditions PResult is that the waves no longer converge to a point 16-9 Five third order aberrations Including the first two terms in the sine expansion allows aberrations to be described/understood/characterized PThis course examines the five third-order aberrations • Spherical • Coma • Astigmatism • Field curvature • Distortion PEach is caused by not satisfying the paraxial approximation • Image point can be far from the optical axis • Object point can be far from the optical axis • Lenses/optics are actually curved PAnother way to view this is there have been two critical assumptions: 1) simplification of Snell’s law; 2) assumed flat surfaces • Should use full form of Snell’s Law and not approximating sin2 by 2 • Surfaces are not flat but are curved (they have sag) 16-10 Third order aberrations summary Can summarize the causes of the five primary aberrations to illustrate concept of aberrations PSpherical aberration • Rays striking refracting surface further from the optical axis are focused closer to the vertex of the surface • Simplistically, the focal length depends on the aperture PComa • In reality our principle planes are curved surfaces • Coma evident when off-axis rays are imaged to an off-axis location PAstigmatism • Object rays a large distance from optical axis strike asymmetrically • Asymmetry causes different focal lengths for the different rays PField curvature • Flat plane object will have a curved in-focus image • Off-axis points from the object are not in focus in the image plane PDistortion is due to magnification being a function of off-axis location 16-11 Coordinate systems - Meridional plane Meridional plane is the y-z plane containing the object point and the paraxial image Pz is location along the optical axis yp Px and y are coordinates perpendicular to the optical axis with the y axis being “up-down” xp Px and y can also be equivalently described in cylindrical coordinates Pupil by D and 2 y y Px=0 for the p y’ meridional plane • Object point is located at (0,h) z z’ • Intersection with Object Pupil Image the pupil at (0, yp) 16-12 Tangential rays and meridional plane Tangential rays are defined as those rays confined to the meridional plane PWill not have symmetry along the y-axis about the x-axis PIn paraxial case, all rays shown would meet at the same point on the y’ axis (image) yp x y p z Object h x

Pupil 16-13 Sagittal Plane Sagittal plane is created by the special case of skew rays for which the polar angle is 90 degrees PSkew ray starts at the object (x=0, y=h) and travels to an arbitrary point in the entrance pupil yp

PThat Is, xp does not have to be zero PAlways start at (0, h) y x By rotating coordinate p z system to force this Object situation h x

PReiterate - the object will always be located Pupil at (0, h) because the coordinate system can be rotated to force this situation 16-14 Sagittal plane Most likely have not previously encountered the sagittal plane because systems were rotationally symmetric PThe sagittal rays will be symmetric about the y-axis PDifferent results could be obtained, however, for different sagittal planes • That is, we can have one case with an object oriented in a certain way relative to the entrance pupil • A different orientation leads to different image properties PPrevious systems have assumed a rotationally symmetric situation • Then, all meridional planes are equivalent • Only need the radial distance between the object point to the optical axis in these cases 16-15 Tangential and Transverse Rays Left diagram below is the tangential (or meridional) case and right diagram is the transverse (sagittal) case PReiterate that there is not symmetry along the meridional plane PExamination of previous diagram and that below should make it clear that there is symmetry along the sagittal plane PThe on-axis object (y=h=0) and the object point is at (0,0)) leads to the tangential and transverse rays being the same

yp xp {not y}

z z

View from the side View from the top 16-16 Coordinate system - General Figure shows a general case for a ray leaving an object point and traveling through the system to an image point

PNew term here is rc which is the radius of the pupil y’ PUse the term longitudinal to y refer to the distance p along the z-axis x’ y z’ (xp, yp) (0, h) x z h rc (x’, y’) xp Image Pupil Object 16-17 Pupil coordinates Derivation of third order aberrations uses normalized pupil coordinates such that xp, yp, and D vary from 0 to 1 PDefine D to be the radial coordinate in cylindrical coordinates 2 2 ½ • Then D=(xp + yp ) or xp=Dsin2 and yp=Dcos2 • Will use xp* and yp* for non-normalized values and then xp*=rcxp and yp*=rcyp yp PDiagram illustrates these terms • Normalization factor (xp*, yp*) is the pupil radius r • Will rely on this notation D* 2 c throughout the discussion of aberrations xp 16-18 Field coordinates H is position in the image plane referred to as the normalized field height, the field vector, or simply the field PNormalized to vary from -1 to 1 with the value of 1 at the edge of the field stop PH>0 implies a positive image height

• For ,y>0, ray is moved further from the optical axis when H>0 • If H<0, the positive value for ,y means moving towards optical axis PTypical values for H that are used in studies of optical systems are H=0, 0.7, and 1 • Value of zero is along the optical axis • Value of 1 is for the edge of the field • The 0.7 value is selected because it covers approximately ½ of the circular area of the stop PAssume that the optical system is axially symmetric so that pupils and stops are circular 16-19 Pupil and field illustration

Diagram illustrating H, D, and 2 y’

x’ D z’

yp 2 H

xp H=1 D=1

Pupil Image 16-20 Measuring error Characterize the errors caused by aberrations in terms of 1) ray errors and 2) wavefront errors PRay error is the difference between the actual ray position and that from the unaberrated case PWavefront error is the difference between the actual wavefront to a perfect wavefront of a reference spherical wave • Wavefront error is also called the optical path difference (OPD) • Reference sphere is relative to the image point as shown below Reference Image Point

z z

Reference Image Point Exit Pupil Exit Pupil 16-21 Wavefront error Reference wavefront can be used to determine the optical path difference (OPD) PNotation is either OPD or W • Units are in distance (usually mm or :m) • Sometimes in radians • Coordinates set up to ensure that the reference and aberrated wavefronts pass through the optical axis at the same point PThe wavefront error is measured as a function of the position in the pupil for a given object point PThen =− Wx(,)pp y W App (,) x y W Rpp (,) x y

• WA is the aberrated wavefront and WR is reference wavefront • Positive values indicate the aberrated wavefront has a smaller radius of curvature (bent more) PChanging the reference point alters W PReference point often selected as the intersection of the paraxial chief ray with the paraxial image plane 16-22 Wavefront error Wavefront notation and terms for the on-axis case

Reference Wavefront Aberrated Wavefront

Reference Image Point

W > 0 Exit Pupil 16-23 Wavefront error Similar approach is used for the off-axis case Reference Image Point

W > 0

Exit Pupil 16-24 Ray error Ray error is effectively an error in the position of the image relative to the paraxial location

PUse ,x and ,y for the image plane

PUse ,z for the change focus on the optical axis relative to paraxial

Paraxial Focus

Ray aberration, ,y

Ray aberration, ,z due to different W > 0 pupil locations focusing at Exit Pupil different points on the optical axis 16-25 Why two errors? Specific approach used to describe an aberrated system depends upon the application PRay error gives visual clues to the effect of aberrations on the location of rays at the image plane • Introduce spot diagrams to give quick insight into distribution of light on the image plane • Ray fans give a similar insight • Ray error knowledge aids in image plane location and shape to reduce the effects of aberrations PWavefront error, on other hand, gives an idea of what type of optical modification is needed to improve wavefront leaving the exit pupil PCan show that the two errors are related to one another according to dW ε = K []xyorz,, dxyorz[],, • K is a constant of proportionality • This is a general expression valid for all wavefront aberration • Will later show this empirically for the case of defocus 16-26 Transverse ray aberration Transverse ray errors/aberrations refer to those that occur parallel to pupil plane and orthogonal to optical PSeparate the errors between theaxis x and y axes • x axis is the sagittal (transverse) plane • y axis is the meridional (tangential) plane PThe transverse ray errors are measured Actual ray relative to the reference ray of the position on unaberrated case y’ image plane • The reference ray is determined from a specified location on the pupil plane ,y • Errors are a function ,x of x and y p p Reference image point is x’ for pupil

location (xp,yp) 16-27 Longitudinal ray aberration Longitudinal ray errors/aberrations occur orthogonal to the pupil plane and parallel to the optical axis PAgain separate the errors between the x and y axes • Lefthand figure shows y-axis case • Right-hand the x-axis case PLongitudinal ray errors are measured relative to reference ray of the unaberrated case • Reference ray determined from a location on the pupil plane

• Errors are a function of xp and yp but are treated separately yp Reference xp Actual Reference Actual image ray image ray point for position point for position pupil pupil ,z location location ,z (xp,yp) (xp,yp) z’ z’ 16-28 Basic approach to evaluate aberrations Exact solution to the aberrated wave front is not trivial, thus the derivation here relies on power series expansion PThis is what caused problems to start with • Write sines and cosines as power series • Paraxial approximation ignores all but first term PKeep as many terms as needed to study the aberrated wave front at the required accuracy PVariables in the formulation are the object (x, y) and pupil coordinates (xp, yp) PRequire rotational/axial symmetry about the optical axis • That is, the aberration must not vary when we rotate about z • Keep only variable combinations that are invariant to rotation • The rotation variable is N (rotated counterclockwise) Px2+y2 is invariant on rotation (equation of a circle) 2 2 Pxp +yp is also invariant 16-29 Coordinate system and variables The last variable combination that is invariant to rotation is xxp+yyp PThe graph shown here is y for the object plane but an yrot identical one can be developed for the pupil axes as well

PCompute xxp+yyp with the (x, y) standard x-y axes (xrot, yrot)

xrot PAlso compute the xxp+yyp for the case of a rotated

axes in terms of xrot and yrot PInvariance on rotation means N that the resulting equations should x be identical 16-30 Rotationally invariant

xxp+yyp for the rotated coordinate axis in terms of the x-y axis and N is shown below y PCan see from the graph that yrot =+φ φ xxrot cos y sin =−φφ yyrot cos x sin (x, y) PLikewise for the pupil plane we have (xrot, yrot) =+φ φ xrot xxrot_ p pcos y p sin =− yyrot_ p pcosφφ x p sin N PThen x =++22φ φ φφ φφ + xrot x rot p_ xx pcos yy p sin xy p cos sin x p y cos sin

=+−22 − yrot y rot p_ yy pcos xxφ p φ φφ sin xy φφ p cos sin x p y cos sin

PAnd the sum of xxp+yyp is +=+++22φφ xrot x rot__ p y rot y rot p xx()cos()sin p yy p xx p yy p =+ xxpp yy 16-31 Variables and expansion The variables that are invariant to rotation are x2+y2, 2 2 xp +yp , and xxp+yyp PHave a similar set of equations related to the image point 2 2 2 PAlso recognize that xp +yp = D which is the radial position on the pupil plane PIn the case of x’=0 then y’2=h2 • This limits the problem to the y-z plane • Realize that h2 is also related to H2

PRecognize that y’yp is the same as HDcos2

PConvert the wavefront expansion to be in terms of H, xp, and yp PActually use H2, D2, HDcos2 to maintain rotational independence

22ρρθ ρ=++ ρθ2 2 + WH(,, H cos) aH1 a2 aH3 cos 22 ++22 2 + bH1() b2 () bH3 ( cos) ++ + 22 2 2 bH4 ()() bH5 ( cos)() bH6 ( cos)() H higher order terms ρρθ ρρθρρθ 16-32 Terminology There are a number of ways to refer to level/order of aberrations PAdopt the nomenclature of the power of the ray error PRecall that the expansion of the wavefront used H2 and D2 • First term in the wavefront expansion is simply a constant • Second set of terms are related to the invariant variables to the first power (H2 and D2) • Third set of terms is related to the second power (H4 and D4) • Fourth set to the third power (H6 and D6) PRay error is proportional to derivative of wave error with respect to the pupil variable • The constant term goes to zero • First order aberrations refer to terms related to the first set of terms in the wavefront expansion series (which give powers of D1) • Third order refer to the second set of terms (powers of D3) • Fifth order refer to the third set (powers of D5) 16-33 Aberration coefficients The a and b coefficients in the expansion series are the aberration coefficients PIndicate the relative importance of one term to another • Larger value for a coefficient implies a larger aberration • Not exactly true in all cases, but conceptually is okay PRelabel the coefficients in terms of the exponents of the H, D, and cos2 variables • Use the letter w with three subscripts for the power of the variable

• Then have w H D cos2

PThe constant term, for example, becomes w000 3 Pw131 is related to the HD cos2 PNote that the w coefficients are different from Seidel coefficients which are often given instead • Will see how to convert between the two later • Use the w coefficients for now 16-34 In terms of aberration coefficients Rewrite a and b coefficients in terms of the notation for the aberration coefficient PAlso include the next level of higher order terms 22ρρθ ρ=++ ρθ2 2 + WH(,, H cos){ w200 H w020 w111 H cos} 4 ++4 2 + {(cos)wH400 w040 w222 H ++ + 22 3 3 wH220 ++ wH131 cos wH311 cos } +++ 6 5 42 2 42 33 {wH600 wH511 cos+++ wH422 cos wH420 wH331 cos 33 3ρρθ24 2 24 1 wH333 ρρθρθcos wH242 cos wH240 wH151ρθ5 ++ ρ6 cosw 060 } even higher order terms PRecall that ray errors areρθ relatedρθ to ρθderivative ρ θ ρ ρ of ρ the θ wavefront error • Terms related to the square order in wavefront are first order in ray error • Fourth order are related to third order ray aberrations • Sixth order wave aberrations are related to fifth order ray aberrations 16-35 Aberration pyramid A schematic method for looking at the different orders is the aberration pyramid PZeroth and First order w000

w020 w200 w111 PThird order also referred to as primary aberrations and Seidel aberrations w040 w220 w400 w131 w311 w222 PFifth order (which are not discussed in this course

w060 w240 w420 w600 w151 w331 w511 w242 w422 w333 16-36 Labels for the aberrations

w000 Piston

Defocus Field dependent phase

w020 w200 Tilt Field curvature w111 Spherical Field dependent phase w w w Coma 040 220 400 Distortion Astigmatism w131 w311 w222 5th order field curvature Cubic coma - elliptical Sagittal Oblique SA Field dependent phase 5th order SA w060 w240 w420 w600 5th order linear coma w151 w331 w511 5th order distortion Tangential Oblique SA w242 w422 5th order astigmatism w333 Line coma - elliptical 16-37 Another view of the “pyramid” for ray error Field H0 H1 H2 H3 H4 H5 0 Distortion Distortion Dp Tilt primary 5th order 1 Astigmat. Astigmat. Dp Defocus Fld. Curv Fld. Curv primary 5th order 2 Coma Elliptical Dp Coma primary 5th order 3 Spherical Oblique Dp Spherical primary 5th order Linear D 4 Coma p 5th order Spherical 5 Dp 5th order 3 311 3 311 2 2 2 2 cos cos } 3 131 131 222 +++ ++ 131 ++ θ θ ++ 222 and ray errors in terms of the with respect to the pupil positions 111 ++ sin (sin ) ] cos ( cos ) ]

++ ρθρ ρρθ θ ρ θ ρ sin

ρρθρθ cos cos 2 4 2 3 22 2 3

ρρθ 020 020ρθ ρθ 111 ρθ c 040 040 220 020 20 220 040 2 220 wwH w wH wH wwH wH wH wH wH wH wH {(cos) 40 220 42 222 =+ + Seidel aberrations =− + + =− + + aberration coefficients can be written

θ θ

ρρθ ρ ρθ ρ ρ 22 HRrwwH HRrw (,, cos){ cos} Ray error is the derivative of W Includes a factor of -R/r ( , ,cos ) ( / )[ cos ,cs (/)[ sin (,,cos) At this point the wave front yc xc P P

WH Hε w w H ε 16-38 17-1 Aberrations OPTI 509

Lecture 17 Ray fans, spot diagrams; RMS spot size 17-2 Ray and wave fans and others Introduce several methods to evaluate the level of aberrations (and types) in a system PRay fans • Plots showing the ray error as a function of pupil position • Typically also show a variety of field positions PWave fans • Plots showing the wave error as a function of pupil position • Typically also show a variety of field positions PSpot diagrams • Shows the ray error that is not limited to one or the other pupil axes • Surrogate for what the “image” might look like PAll of these, and often others, can be given by optical design packages • Allows quick visual evaluation of the system • Can provide quantitative information about the aberrations that are present 17-3 Ray fans Ray fan is a plot of the transverse or longitudinal ray errors as a function of pupil position (xp, yp) PPlots here illustrate the tangential ray fan (,y) • Ray fan diagram for the transverse error for the meridional plane

• Lefthand plot is versus the meridional pupil position (yp) • Right-hand plot is versus the sagittal pupil position (xp)

PTypically see the lefthand plot for the case of ,y

,y ,y

yp xp 17-4 Tangential ray fans Previous plots show the difference in the location of the actual ray to that of an unaberrated ray PRemind ourselves this is the difference in location of the actual ray from the pupil to to the image plane to that of an unaberrated ray PFigure illustrates the tangential ray fan y’ x’ y p z’

,y Case for xp=0

xp

yp

Image Pupil 17-5 Sagittal ray fans Sagittal ray fan is a plot of the transverse ray errors along the sagittal plane as a function of pupil position (xp, yp) PPlots below show the sagittal ray fan

• Lefthand plot is versus the meridional pupil position (yp) • Middle plot is versus the sagittal pupil position (xp) PTypically see the Right-hand plot since symmetry allows us to show only one-half of the plot ,x ,x ,x

yp xp xp 17-6 Sagittal ray fans Repeating the plot we examined before illustrates the sagittal ray fan ,x PThe case we showed before was for xp=0 PThus all of our points will be on the x-axis y’

xp x’ yp z’ Varying yp

xp

Image

Pupil 17-7 Sagittal ray fan

The illustration below shows the case of yp=0 but varying the x pupil position PThese are only examples to illustrate the ray fan concept PKeep in mind that we can have an array of these corresponding to different yp positions y’

x’ yp z’ , xp x

xp Image

Varying yp Pupil 17-8 Longitudinal ray fan The longitudinal ray fan allows for rapid inspection of the ray fan error along the optical axis PShows the focus shift in the zonal direction PPlots below show the longitudinal ray fan

• Lefthand plot is versus the meridional pupil position (yp) • Middle plot is versus the sagittal pupil position (xp) PLongitudinal ray fan usually used to examine radially symmetric aberration and thus the Right-hand plot can be used x yp p

,z ,z 17-9 Longitudinal ray fan

Case below looks again at the case of xp=0 and illustrates where each ray crosses the optical axis PHere the reference point is an unaberrated Case for x =0 focal point p y’ yp PAs we move off of the optical axis, the aberration increases x’ yp ,z z’

xp

Image

Pupil 17-10 Ray fans There are a large number of possible ray fans depending on object point, pupil locations, and reference position PTypically use a subset of the possible diagrams

• Tangential ray fan for xp=0 • Sagittal ray fan for yp=0 PCan be done for various locations on the optical axis but common practice is to use paraxial image plane PThe longitudinal ray error is ,y ,x also referenced against the paraxial case Problem is that there P yp can be ambiguity to a xp given ray fan • What was object point? • What is the orientation of the sagittal plane? 17-11 Ray fans - ambiguity Ambiguity arises from the fact that we are converting complicated 3-D problems to a 2-D representation PNormalization can give identical ray fans for two differently-sized systems PDifferent systems in normalized space can give identical ray fans by using different objects

PRay fans are tools to understand the aberrated y’ system ,x x’ PNeed to make sure that yp parameters used to derive z’ x the ray fan p x are understood p PGraph at right gives identical ray fan as a previous Image graph but the rays are considerably different Pupil 17-12 Wave fans Wave fans are similar to ray fans except they show the difference between reference and aberrated wave fronts Reference Wavefront PRecall wavefront error is Aberrated Wavefront difference between the reference spherical wave Reference and the aberrated wave Image Point PPlot the wavefront error as a function of pupil position PExample here has the largest wavefront error for pupil positions that are farthest W W > 0 W from the optical axis Exit Pupil POften the case since we are no longer satisfying x y the paraxial approximation p p 17-13 Wave fans Determine wavefront location as a function of reference wave radius and the location in the pupil PUsing the figure below, and the equation of a circle {x2+y2=r2} to get ρ 222+− = z ()(rRzRc ) ρ 22+− 2 2 += 2 rRc 2 RzzR • x=R-z • y= r r D c D c R PThen if z is small z’ 2 (z is even smaller) R ρ 22= r rRzc 2 c R ρr 22 z = c R2 PThis gives the z-axis position of the spherical wavefront for a given pupil location 17-14 Wave fans What is wanted is the aberrated wavefront which is the difference from the value of z just found PThen the distance zaberrated from yp is z 22 ρr WA z = c aberrated 2(Rz− ) R δ Drc PThe wavefront error is z’

W=Waberrated -Wreference

=zaberrated -zreference rc

PSubstituting gives *z ρ 22 22 22 r cccr rz⎛ ⎞ W = −=⎜ ⎟ 222()− ρρ δ2 ⎝ − ⎠ Rzδ R RRz PAnd this is the wavefront error due to curvature of the reference wavefront in terms of a focus shift at a given pupil location δ 17-15 Spot diagram The spot diagram is based on creating an array of points in the entrance pupil plane from a given object point PEmphasize again - rays from a single point from the object are traced to an array of locations of the entrance pupil PDetermine the transverse error on the image plane for each of the entrance pupil points yp

yp x 1 y p 0.5 Object 0 h x -0.5 -1 x -1 -0.5 0 0.5 1 p

Pupil 17-16 Spot diagrams Fill the pupil plane then see where the rays go on the image plane relative to the Gaussian image

yp x y p Object h x

Pupil 17-17 Spot diagram Using a uniform grid in the entrance pupil allows an equal amount of energy to be assigned to each ray PArea/dot = Constant PEnergy/dot = Constant PNumber of spots used depends on the desired accuracy

yp 1 1 ,y

0.5 0.5

0 0

-0.5 -0.5 , -1 x -1 xp -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 17-18 Spot diagram - Entrance pupil The appearance of the transverse error in the image plane depends upon the grid in the entrance pupil PHave three basic types • Square • Hexapolar • Dithered PEach is used depending upon the expected aberration, whether we want equal area, or equal angles Square Hexapolaryp Dithered 1 1 1

0.5 0.5 0.5

0 0 0

-0.5 -0.5 -0.5

-1 -1 -1 xp -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 p y p x ½ spot size ½ spot y , y , Spot size N i =− ′ yyi centroid, size, RMS size 11 == N i ∑∑ 11 NN Spot diagram - Figures of merit == =− xxi ′ yyy xxx εεεε εεε εεε Each has its use depending upon type of aberration and what is the to be understood about the trying system The spot diagram centroid is simply the mean and indicates the average ray error The spot diagram size is determined by the maximum and minimum difference between all rays and the centroid Further characterize our spot diagrams by examining the P P P 17-19 17-20 Spot diagram - Figures of merit Root-mean squared, or RMS, spot size gives a measure of the spread of the spot size PAvoids overly weighting outliers PFirst approach is to compute the sample standard deviation N N εε εε2 2 ∑∑()xi x− ()yi y − s = i ==1 s i 1 = x N 11− y N − PAlternatively (and the accepted method), is to integrate over the pupil area and normalize by B to ensure a unit area ππ 112 1 2 1 RMS=−()2 d d RMS () =−2 d d yyyxxxπ ∫∫ ∫∫ 0 0 εερρθ 0 0 PDefining εε' =− ε then

2ππ1 2 1 ==112 2 RMSyy∫∫(') d d RMS xx (') ∫∫ d d 0 0 π 0 0 ε ε ρρθ π ερρθ

π ε ρρθ i N i 2 2

θθ

and εερρθ () D 1 0

π of 0 ∫∫ 2

1 π 11 == N =− i ∑∑ rrr =+ =− =− +− 11 NN 222 ryx 22 22 == RMS d d rrryyxx rri εεεε εεεεεεε RMS RMS RMS (')()( )( ) Spot diagram - linear versus radial The previous figures used linear coordinates previous The Converting to radial coordinates give the centroid as The RMS spot size in radial coordinates is The relationship between the linear and radial RMS is Can do our spot diagram in terms of x and y or P P P P 17-21 17-22 Spot diagram - examples Plots below show the simplistic case of a uniformly distributed ray error of varying size and centroid PThe uniform distribution indicates a linear ray fan PThe changing centroid shifts the circular spot from the center Centroid (0.1,0.1) spot size 0.7 Centroid (0.1,0.1)1 spot size 0.3 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0 ,y ,y ,y ,y

-0.5 -0.5 -0.5 -0.5 , ,x -1 x -1 -1 -1 y -1 -0.5 0 0.5 1 y -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 p -1 -0.5 0 0.5 1 p Centroid (0.1,0.1) spot size 0.3 Centroid (0.1,0.1) spot size 0.7 1 1 1 1

0.5 0.5 0.5 0.5

0 0 ,y ,y 0 , 0 ,y y -0.5 -0.5 -0.5 -0.5 ,x -1 -1 , y -1 x -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 p -1 yp -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 17-23 “Real” example The following attempts to illustrate the ray fans, wave fans, and spot diagrams for a “real” lens PLens is a biconvex lens with 20 mm focal length PDiagram is for the case of a 5 mm aperture stop PObject is a point source on the optical axis at a distance of -40 mm 17-24 Ray fans Meridian and sagittal ray fans as typically shown without axes labels PMeridian case shows ,y as a function of yp PSagittal case shows ,x as a function of xp PNote that the ray errors get larger for pupil locations that are further from the optical axis PMakes sense since these points are further from paraxial PShape of the ray fan gives information about the type and magnitude of the aberrations 17-25 Wave fans Plot here is the accompanying wave fan for the ray fans shown on previous viewgraph PMost likely that the wave fan is a smoothly varying function PWave fans can likewise be used to understand the aberrations that are present as well as the magnitudes 17-26 Spot diagrams Spot diagrams are shown for several longitudinal locations relative to the paraxial focus PLocation indicated by .0 mm is the paraxial focus PIndication is that the ray errors are smaller for focus positions closer to the lens 17-27 Decreasing Aperture stop size to 2 mm

PNote how the upper ray fan shows less ray error for the 2 mm stop size PAgain makes sense since this closer to paraxial

Original 5-mm stop size 17-28 2-mm aperture stop case Show the sagittal wave fans for the 2 mm stop size (on the left) and the 5 mm stop size (on the right) P2-mm case shows smaller wave error than the 5-mm case for the same pupil position PNo aberrations would be shown by a flat line in the wave error at zero 17-29 Longitudinal ray fans Include the longitudinal ray fans as well for the 2-mm case (left) and 5-mm case (right) PSystem is rotationally symmetric • Due in part to the object being on axis • Lens is circular

PX-axis is ,z

PY-axis is either yp (meridian) or xp (sagittal) 17-30 Move the source closer to the lens Distance is now -30 mm and the original 5-mm stop size

-30 mm object distance

-40 mm object distance 17-31 -30 mm object distance

PAs expected spot diagram is larger PWave fan error larger 17-32 Moving the object off axis Placing the object 10 degrees from the optical axis with the 2-mm stop size increases aberrations 17-33 Off-axis case 18-1 Aberrations OPTI 509

Lecture 18 Spherical aberration, variation with bending; high-order spherical aberration 18-2 First order Third order aberrations are examined one by one in some detail

PIgnore terms without pupil dependence (any term with wj0k) PPostpone discussion of defocus and tilt • These two are used extensively for correction of aberrations • Thus, use them when discussing image quality PAlso show that tilt and defocus are not aberrations in the same sense as the third order aberrations • Can still cause tilt and defocus even in the paraxial case • Tilt and defocus appear in a similar fashion to the other five so treat them the same for simplicity Defocus Tilt 22ρρθ ρ ρθ=++2 2 + WH(,,cos){ H w200 H w020 w111 H cos} 4 ++4 2 + {(cos)wH400 w040 w222 H 22 ++3 3 wH220 wH131 cos wH311 cos }

ρρθ ρρθρθ 18-3 Piston

Still the w000 term which is referred to as piston

PThere is no ray error for piston PThere is no change in curvature of the wavefront PExists partially as a result of our method for deriving wavefront error z • Wavefront error is a WA power series expansion in rotationally invariant variables Drc • First term in the series is that z’ related to all of r the variables to the c R zeroth power

Reference 3 311 3 311 2 2 2 2 cos cos } 3 131 131 222 +++ ++ 131 ++ θ θ ++ 222 111 ++ sin (sin ) ] cos ( cos ) ]

++ ρθρ ρρθ θ ρ θ ρ sin

ρρθρθ cos cos 2 4 2 3 22 2 3

ρρθ 020 020ρθ ρθ 111 ρθ 040 040 220 020 20 220 040 2 220 wwH w wH wH wwH wH wH wH wH wH wH {(cos) 40 220 42 222 =+ + =− + + =− + + Wave and ray aberrations

θ θ

were derived for the rotationally invariant case ρρθ ρ ρθ ρ ρ 22 HRrwwH HRrw (,, cos){ cos} Recall from previous lecture that wavefront aberrations Can also use this to derive the ray aberrations following set of equations the Obtain ( , ,cos ) ( / )[ cos ,cs (/)[ sin (,,cos) yc xc P P

WH Hε w w H ε 18-4 18-5 Wave and ray aberrations Previous equations already omitted all terms that did not vary in D PDefocus and tilt are covered later PStart going through the equations on previous viewgraph term by term • Start with spherical • Show this for the wavefront aberration equation 22ρρθ ρ=+ ρθ2 + WH(,,cos){ H w020 w111 H cos} 4 ++2 {(cos)wwH040 222 22 ++3 3 wH220 wH131 cos wH311 cos } ρρθ

ρρθρθ 3 311 3 311 2 2 2 2 cos cos } 131 3 131 222 ++ +++

θ ++ 131

θ ++ 222 sin (sin ) ] 111 ++ cos ( cos ) ] ρ ρθ ρ θ sin

ρ ρθ ρ θ 2 ++ cos cos 3

ρρθρθ 2 location 3 020 4 ρθ 22 2 ρρθ 020ρθ ρθ 111 040 220 20 040 220 2 040 w wH wH 020 220 wwH wH wH wH wwH 40 220 wH wH wH 42 222 {(cos) =+ + Spherical aberration =− + + =− + +

θ θ

ρρθ ρ ρθ ρ ρ 22 HRrw HRrwwH ,cs (/)[ sin (,,cos) ( , ,cos ) ( / )[ cos (,,cos){ cos} xc

yc ε Spherical aberration has a dependence only on the pupil ε WH H w w H 18-6 18-7 Spherical aberration The first of the “real” aberrations discussed is spherical aberration PSpherical aberration is caused by the fact that rays at the edge of the optical system come to a focus at a different location than those near the center PAgain, not caused by a defect in the optical system 18-8 Transverse and longitudinal In the case of spherical, there is a transverse aberration and a longitudinal aberration PThe transverse aberration (TA) can be related to the longitudinal aberration (LA) using the angle between the ray and the optical axis W PThen yp TA = -LA tan(") Longitudinal Aberration Aberrated LA

" Paraxial

TA

Reference 18-9 Spherical Aberration caustic The caustic is the envelope of the rays from the pupil to the optical axis PCaustic is rotated about the optical axis POnly the “Upper” caustic is shown here for clarity yp

Caustic Marginal ray

Paraxial focus Marginal focus 18-10 Spherical aberration minimum circle Spherical aberration blurs the image or, equivalently, causes a point object to be imaged as a fuzzy blob PMinimum circle is located where the marginal ray crosses the yp Marginal rays external caustic beyond marginal focus Marginal focus PLocation of the minimum circle depends on the optical system • For a biconvex lens the minimum circle is always inside paraxial Paraxial focus • For a biconcave the minimum circle is outside paraxial External caustic TA

Minimum circle 18-11 Spherical aberration - minimum circle Use the case of a biconvex lens with 20 mm focal length, the object at -40 mm, and a pupil radius of 5 mm

Paraxial 18-12 Spherical aberration - spot diagram Spot diagrams here are for different locations of the image plane along the longitudinal axis PNote that the smallest sized spot diagram is inside focus PShow later that • Marginal focus is most uniform spot • Paraxial is the brightest central core • Midfocus is between the two 18-13 Spherical aberration - wave fans

4 The wave error goes is w040 D which leads to the characteristic bowl-shaped wave fan PTransverse wave fans for meridional and sagittal case shown here • F/5 system with 50 mm focal length and 5 mm radius pupil • Wave error is 1/10 wave in spherical aberration (W=Wspherical=w040 =0.10 waves)

PAll three values 0.1 0.15 for H (0.0, 0.7, 1.0) 0.1 are shown but 0.05 there is no 0.05 dependence 0 on H 0 0.5 1 -1 -0.5 0.5 1 x_p -0.05 y_p -0.05 -0.1 H=0.0 H=0.7 H=1.0 -0.1 -0.15 18-14 Spherical aberration - ray fans Show the ray fans for the same case as on previous viewgraph PRay error is proportional to the pupil position to the third power 3 • For xp=0 then ,y=-4(R/rc)w040yp 3 • For yp=0 then ,x=-4(R/rc)w040xp PNow the focal length and pupil 2 1 size play a role 1.5 PWavelength is 1 0.5 0.500 :m 0.5 x_p y_p PAgain, no H 0 dependence -0.5 0.5 1-1 -0.5 0.5 1

-1 H=0 -0.5 -1.5 H=0.7 -2 H=1.0 -1 18-15

Spherical aberration dependence on w040

Lefthand plots have w040=0.01 waves and right hand are for w =-0.01 waves 0.02 040 0.02 0.015 0.015 0.01 0.01 0.005 0.005 0 0 -0.005 -0.005 -0.01 -0.01 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 -0.015 -0.015 -0.02 -0.02 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 -0.1 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 -0.15 -0.15 -0.2 -0.2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 18-16 Longitudinal spherical aberration Typical case for a positive lens is under corrected spherical aberration PWe’ll see shortly how to correct for spherical PLongitudinal aberration has the same sign as the transverse aberration 1 PShow here the longitudinal ray error for the case shown on the previous viewgraph for the positive wave error 0.5 PThe focus shifts inside of paraxial focus for this case

PBecause it is feasible to correct for ,z (:m) spherical aberration in a 0 straightforward fashion, any -0.004 -0.002 0 0.002 0.004 spherical aberration is considered to be under corrected when nothing is being done 18-17 Balancing spherical aberration Can balance spherical aberration using lens bending and higher order spherical PCreate multiple lens shapes all with the same power PThe specific shape that minimizes spherical aberration depends upon the object position POrientation of a lens can also play a role • Cases at right show the same lens • Upper case suffers less spherical aberration • Conceptually, the reason is that both surfaces in the upper case provide power 18-18 More lens bending The power needed and refractive index of the glass determines the optimal shape of the lens PFocal length “on-axis” is the same regardless of the shape PShape determines the focal length for an off-axis object ray PSketch here assume 10 a 10-mm focal length lens 9.8 • Curve is the 9.6 focal length for a fictitious object ray 9.4 striking at the edge 9.2 of the pupil • Can also alter the 9.0 shape so that different pupil positions give little to no aberration 18-19 Higher order balancing Lens bending has limits and another option is to use higher order spherical aberration to balance the total PIncluding the fifth order spherical aberration gives =++ρρρ2 4 6 Ww020 w040 w060 ερθρθρθ =−R +3 5 + y 246(ww020 cos 040 cos w060 cos ) rc =− + + ερθρθρθR 3 5 x 246(sinww020 040 sin w060 sin) rc PSetting the defocus to zero leaves the third and fifth order PFurther assume our goal is to balance the aberration at the edge of the pupil meaning ,y(yp=1)=0

PThen w040 = -(3/2)w060 PIncluding defocus can also improve the image and this is discussed later with defocus PThis only considers the spherical aberration and attempting to solve the spherical aberration could lead to other aberrations or issues 18-20 Higher order balancing Consider 20 mm focal length, 5 mm radius pupil, 1/10 wave 3d order spherical aberration (left), -1.5/10 wave 5th order, and sum 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 Epsilon_y (micrometers) Epsilon_y Epsilon_y (micrometers) (micrometers) Epsilon_y Epsilon_y -1.5 (micrometers) Epsilon_y -1.5 -1.5 -2 -2 -2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p y_p

H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0

0.1 0.1 0.1

0.05 0.05 0.05

0 0 0

-0.05 -0.05 -0.05 W_y (in number of waves) W_y (in number of waves) W_y (in number of waves) -0.1 -0.1 -0.1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p y_p H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 19-1 Aberrations OPTI 509

Lecture 19 Distortion 3 311 3 311 2 2 2 311 2 cos cos } 3 131 131 222 ++ +++ ++ 131 θ ++ 222 111 sin (sin ) ] ++

cosρ (ρθ cos ρ ) θ ] sin ++ 2

ρρθρθ ρθ ρ3 θ cos cos 4 2 020 22 2 ρθ ρρθ 3 020ρθ ρθ 111 040 220 20 040 020 w wH wH 220 040 2 220 wwH 40 220 wwH wH wH wH wH wH wH {(cos) 42 222 Distortion - w =+ + =− + + dependence on the field variable =− + + θ

ρρθ ρ ρθ ρ 22 HRrw ,cs (/)[ sin (,,cos) HRrwwH Distortion is a non-blurring aberration that has cubic (,, cos){ cos} xc ,cs (/)[ cos (,,cos)

yc ε

WH Hερθ w w H ρθ 19-2 19-3 Distortion Magnification change occurs as image point is located further from the optical axis (that is, as H increases) PThere is a field dependent magnification PExamine the cause of this later

PWill not derive an analytic formula for w311 • Take the philosophy that value for w311 is either given or can be derived from a ray fan or wave fan diagram • Concentrate on the outcome of the aberration at this point, its effect, and how corrections can be made 3 3 PWdistortion=w311H Dcos2=w311H yp 3 P,y=-(R/rc)w311H

P,x=0 PRay error has no dependence on pupil but a cubed dependence on field PNo error for the sagittal case 19-4 Distortion - impact on image If there is a circular object (H=constant) there is also no noticeable effect PThere is still an aberration but the image has the “wrong” magnification PSchematic below assumes that the system magnification is unity • Smaller H leads to smaller aberration • Effect is not known unless the unaberrated image size is known y y’ p yp y’

xp x’ xp x’ 19-5 Distortion - image effects Consider the case of the object as a set of concentric circles PStretching of the object in image space would be apparent PAll of the circles still hold their shape y’

yp

xp x’ 19-6 Distortion - impact on image Any straight line through the origin of the image plane will be a straight line in the image plane

PThe image height in the paraxial case is h’=y’maxH PImage height in the non-paraxial case (the aberrated case) is

• h’=y’maxH + ,y 2 • In terms of w311 it is h’=H(y’max -(R/rc)w311H )

PIn the above, y’max can be related to the magnification 2 • Error in magnification can be related to -(R/rc)w311H • Error in magnification goes as the square of the field location y y’ Object at pupil plane p

Image plane

xp x’ 19-7 Distortion - impact on image Now consider an object for which H is not constant but does not go through the optical axis PFor example, a linear object located some distance from the optical axis PEnds of the object have largest values for H • Largest distortion happens at these points • The ends of the object move furthest from (or towards) the optical axis yp yN

xp xN 19-8 Pincushion Distortion The classic demonstration of distortion is the pincushion distortion

PGet pincushion distortion when w311<0 • Then ,y>0 when H>0 • And ,y<0 when H<0 • That is, the aberrated ray moves away from optical axis in both situations y’ • Image expands y but does so more at larger H

x x’ 19-9 Barrel distortion Barrel distortion is similar to pincushion except the sign of w311 is positive PShrinking the image relative to the unaberrated PThe corners of the image still move the furthest except now they move towards the optical axis

PBarrel distortion when w311>0 • Then ,y<0 when H>0 • And ,y>0 when H<0 y y’ • Aberrated ray moves toward the optical axis in both situations x x’ • Image shrinks but does so more at larger H 19-10 Distortion - cause A common cause of distortion is location of the stop relative to optical elements PStop can be used to reduce the size of the entrance pupil to bring the system closer to paraxial and thus reduce spherical aberration PProblem is that adding a stop can change the location of the pupil and thus the behavior of the chief ray PThe case below shows a coincident aperture stop and lens • Chief ray goes through the center of both without deviation • No distortion in this case 19-11 Distortion - cause Moving the stop relative to the optical element in this simple example causes the chief ray to be refracted PChief ray travels through the center of the stop PPosition of the stop determines the sign of the distortion • Image position determined by chief ray • Effect does not depend on stop size (recall the lack of dependence in the ray error on pupil position)

Pincushion

Barrel

PThus, distortion is not truly caused by deviation from the paraxial case - it is not an aberration in the strictest sense 19-12 Distortion - example Photograph from an inexpensive digital camera of a grid of straight lines 19-13 Distortion - examples Images here suffer from distortion 19-14 Distortion wave fans Recall that the wave error for distortion is 3 W=w311H Dcos2 and this gives a linear wave fan 3 3 PConsider the meridional case, Wy=w311H Dcos2=w311H yp PWave fan plots the wave error as a function of the pupil position PConsider an f/5 system 0.02 H=0 H=0.7 H=1.0 • f/#=(focal length/pupil 0.015 diameter)=R/2rc • For example, a biconvex 0.01 lens that is 5 mm in 0.005 radius and has a focal 0 length of 50 mm. -0.005 PPlot at right assumes -0.01 w311 = -1/100 of a wave PLater assume that the -0.015 wavelength is 500 nm (and -0.02 w311= -5 nm = -0.005 :m) -1 -0.5 0 0.5 1 y_p 19-15 Distortion wave fan The dependence on H in distortion is only apparent when multiple H values are plotted on the wave fan PThe H3 dependency is not plainly evident but can be seen by the fact that there is a non-linear change at -1 and +1 PWave error is largest for light 0.02 H=0 H=0.7 H=1.0 coming from the edge of the pupil 0.015 PWave error is largest for imaging 0.01 at the edge of the field 0.005 PThis plot will not change if 0 • Aberration coefficient does -0.005 not change -0.01 • f/# is constant -0.015 -0.02 PAgain, it is assumes that either -1 -0.5 0 0.5 1 • The aberration coefficients are given y_p • Or, the ray and/or wave fans an be used to infer the coefficients 19-16 Distortion - wave fan Show the sagittal wave fan for the same f/5 system

3 PW=w311H Dcos2 0.02 H=0.0 H=0.7 PThen Wx=0 0.015 • Graph shows the wave fan H=1.0 plots for H=0, H=0.7, H=1.0 0.01 • All three lines are on top of 0.005 each other at a value of zero 0 PThen there is no wavefront error for rays for which yp is zero -0.005 • This is regardless of the xp location -0.01 • Regardless of the aberration coefficient -0.015 PThe f/5 does not play a role in the wave fan plots will play a role in -0.02 the ray fans 00.51 x_p 19-17 Distortion - ray fans Now consider the meridional ray fan for the case just shown PRay error is proportional to the derivative of the wave error with respect to the pupil position 3 • , = -(R/rc)w311H cos2 3 0.04 • ,y = -(R/rc)w311H • And in terms of f/# 0.02 3 ,y= -2(f/#)w311H 0 PThis leads to a set of horizontal lines for ,y -0.02 as shown -0.04 PThere is no dependence on pupil location -0.06 H=0 H=0.7 H=1.0 -0.08 -1 -0.5 0 0.5 1 y_p 19-18 Distortion -fan diagrams

0.02 0.02 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 0.015 0.015 0.01 0.01 PPincushion 0.005 0.005 0 0 (w311<0) is shown on -0.005 -0.005 the right -0.01 -0.01 -0.015 -0.015 PBarrel is on -0.02 -0.02 the left -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p

0.04 0.04

0.02 0.02

0 0

-0.02 -0.02

-0.04 -0.04

-0.06 -0.06 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 -0.08 -0.08 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 19-19 Spherical and Distortion Show the case here where the aberration coefficients for both distortion and spherical are 0.005 :m 0.2 0.1 PMeridional case 0.15

• Distortion tips the wave 0.1 0.05 fans and gives variation 0.05 with H 0 0 • Also now have a shift -0.05 -0.1 -0.05 of the ray error -0.15 PSagittal case looks -0.2 -0.1 -1 -0.5 0 0.5 1 00.51 identical to spherical alone y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 1 2 • Distortion has no xp dependence 1.5 0.5 1 • This gives a clear clue as 0.5 to the aberrations present 0 0 -0.5 -0.5 -1 -1.5 -1 -2 -1 -0.5 0 0.5 1 00.51 y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 19-20 Spherical and distortion Balanced third order spherical with higher order spherical, how about balancing distortion and spherical? PNo good way to use these two aberrations to balance each other

PIn fact, no way at all to improve the xp effect of spherical PCan reduce the effect of spherical aberration at one pupil edge using distortion but total aberration worsens at other edge of the pupil

0.2 0.1 0.15

0.1 0.05 0.05

0 0 -0.05

-0.1 -0.05 -0.15

-0.2 -0.1 -1 -0.5 0 0.5 1 00.51 y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 20-1 Aberrations OPTI 509

Lecture 20 Field Curvature and Astigmatism 20-2 Field curvature Field curvature is caused primarily when the focal plane is not curved PA planar object is imaged on a planar surface in the paraxial case PNon-paraxial case means an off-axis object is further from the pupil and this brings the image to a focus at a point closer to the pupil PThe surface giving the best image is the Petzval surface 3 311 3 311 2 2 2 2 cos cos } 3 131 131 222 ++ +++

θ ++ 131

θ ++ 222 111 sin (sin ) ] ++ cos ( cos ) ] ρ ρθ ρ θ ρθ ρsin θ 2 ++ ρ cos cos 3

ρρθρθ 2 3 4 020 22 2 ρθ 020 111 dependence ρρθ ρθ ρθ 2 040 220 20 040 220 2 040 H 020 w wH wH 220 wwH wH wH wH wwH 40 220 wH wH wH {(cos) 42 222 Field curvature =+ + =− + + =− + +

θ θ

ρρθ ρ ρθ ρ ρ 22 HRrw HRrwwH ,cs (/)[ sin (,,cos) ( , ,cos ) ( / )[ cos (,, cos){ cos} xc yc Field curvature is the only third order aberration with an ε ε WH H w w H 20-3 20-4 Wave fan diagram for field curvature 20-mm focal length, 5-mm pupil radius, 0.5-:m wavelength PShow field curvature for 1/10 wave aberration coefficient PH2 dependence in both the meridional and sagittal cases shows up as a separation of the three curves PShape of the curve is dominated by the D2 dependence 0.2 0.2 PSimilar to spherical 0.15 0.15 except with lower 0.1 0.1 power on the pupil 0.05 0.05 dependence 0 0 • Means field curvature -0.05 -0.05 is “worse” than -0.1 -0.1 spherical aberration W_x (in number of waves) of number (in W_x W_y number(in waves) of -0.15 -0.15 • Assuming the -0.2 -0.2 aberration coefficients -1 -0.5 0 0.5 1 0 0.5 1 for both are equal y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 1 p 2 0.5 x_p H=1.0 220 wHx H=0.7 c 0 R r 1 0 -1

0.8 0.6 0.4 0.2

-0.2 -0.4 -0.6 -0.8 Epsilon_x (micrometers) Epsilon_x H=0.0 1 px 0.5 2 the ray fans H=1.0 220 0 y_p wHy c R r H=0.7 22 -0.5 =− =− y H=0 εε -1 1 0 -1

0.8 0.6 0.4 0.2

-0.2 -0.4 -0.6 -0.8 Ray fan diagram - Field curvature (micrometers) Epsilon_y Linear in pupil position and quadratic in field location Same case as shown on previous viewgraph except for P 20-5 20-6 Wave fan diagrams Compare field curvature, third order spherical, and distortion all with 1/10 wave aberration 0.2 0.2 0.2 H=0 H=0.7 H=1.0 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 -0.05 -0.05 -0.05 -0.1 -0.1 -0.1

W_y W_y waves) (in number of -0.15 W_y waves) (in number of -0.15 -0.15 -0.2 -0.2

W_y (in number of waves) number (in W_y -0.2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p y_p H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 -0.05 -0.05 -0.05 -0.1 -0.1 -0.1

W_x W_x number of (in waves) -0.15 -0.15

W_x W_x waves) (in number of -0.15

-0.2 -0.2 W_x (in number of waves) -0.2 0 0.5 1 0 0.5 1 0 0.5 1 x_p x_p x_p H=0.0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 20-7 Ray fan diagrams Same as previous viewgraph except now with the ray fans Field Curvature Spherical Distortion 1 1 1 0.8 H=0 H=0.7 H=1.0 0.8 H=0 H=0.7 H=1.0 0.8 H=0 H=0.7 H=1.0 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p y_p 1 1 1 H=0.0 H=0.0 H=0.0 0.8 0.8 0.8 H=0.7 H=1.0 H=0.7 H=1.0 H=0.7 H=1.0 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -1 -1 -1 00.51 00.51 00.51 x_p x_p x_p 20-8 Wave fan diagrams

Sum of everything to this point - distortion (w311), spherical (w040), and field curvature (w220) PAll aberrations set at 1/10 wave aberration and summed PSummation includes distortion, 3d order spherical, and field curvature POf course, this is somewhat artificial but still illustrates the concept 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 -0.1

W_x (inW_x number of waves) -0.15 W_y (in numberof waves) -0.15 -0.2 -0.2 -1 -0.5 0 0.5 1 0 0.5 1 y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 20-9 Ray fan diagrams Show the ray fans associated with wave fans for the sum of distortion, spherical, and field curvature PThe sagittal case shows that there must be something besides spherical because of the H dependence PThe meridional shows an H dependence but also a “tilt” to the curves indicating distortion 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1 -0.5 0 0.5 1 00.51 y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 20-10 What aberrations are present here? Start with the fact that there is only a combination of spherical, distortion, and/or curvature PThe wave fan for H=0 clearly shows spherical PThe variation in H for the sagittal means that curvature must be present PBecause the spherical aberration wave fan gets “better” implies that there is a negative curvature 0.2 0.2 H=0 H=0.7 H=1.0 H=0.7 H=1.0 0.15 0.15 H=0.0 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 -0.1 -0.15 -0.15 -0.2 -0.2 -1 -0.5 0 0.5 1 00.51 y_p x_p 20-11 What aberrations are present Show the ray fans for the same case of a possible combination of spherical, curvature, and distortion

PDistortion cannot be present because the ,y ray fan does not separate at yp=0 (distortion has no ,x and is a constant in ,y(yp)) PRay fan for H=0 shows spherical aberration PThe “twisting” of the ray fan indicates the presence of curvature and this is further shown by the symmetric effect on the wave fan 1 1 H=0.0 0.8 0.8 H=0.7 H=1.0 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 H=0 H=0.7 H=1.0 -0.8 -1 -1 -1 -0.5 0 0.5 1 00.51 y_p x_p 20-12 What aberrations are present? Look at another example combining the three aberrations we have seen so far 0.2 0.2 PNon-zero Wx that has H=0 H=0.7 H=1.0 H=0.0 0.15 0.15 no dependency on H H=0.7 H=1.0 0.1 implies there is spherical 0.1 0.05 0.05 and no field curvature 0 0 PWave fan & ray fan for H=0 -0.05 -0.05 also clearly shows spherical -0.1 -0.1 -0.15 -0.15 PSeparation of spherical -0.2 -0.2 aberration-shaped ray fan -1 -0.5 0 0.5 1 00.51 y_p x_p points to distortion as does 1 1 H=0.0 0.8 0.8 tilt of the wave fan H=0.7 H=1.0 0.6 0.6 PThe upward bowl shape 0.4 0.4 indicates w >0 0.2 0.2 040 0 0 PSeparation of the ray fans -0.2 -0.2 -0.4 -0.4 indicates that w311>0 -0.6 -0.6 -0.8 H=0 H=0.7 H=1.0 -0.8 -1 -1 -1 -0.5 0 0.5 1 00.51 y_p x_p 20-13 Aberration balancing - first exposure Eventually want to find methods for improving the image quality of our system POne method will be aberration balancing PFor example, field curvature has a D2 dependence • Note that defocus and astigmatism aberrations have D2 dependence as seen in the wavefront and ray error equations • Could offset field curvature with defocus of equal value and opposite sign (but only for H=1) PLikewise, spherical and defocus have the same H and cos2 dependence (that is, no dependence) • Combine defocus and spherical without worrying about impacting the system at different values of H and cos2 • This was the philosophy of balancing third-order spherical with higher order spherical already seen PWhat happens only distortion, spherical, and curvature are used to balance aberrations? 20-14 Aberration balancing - first exposure Assume that we have a system with 1/10 of a wave of

spherical aberration 0.2 H=0 H=0.7 H=1.0 PSubtracting 4/10 of a wave of field 0.15 curvature gives the wave & ray fans shown 0.1 PThere is no way with distortion or field 0.05 curvature to fix the spherical aberration 0 at the center of the field -0.05 -0.1 PIn this case, the spot size is smaller for -0.15 the H=0.7 position than without curvature -0.2 -1 -0.5 0 0.5 1 PNothing has been made worse y_p 1 • H=0 is the same as before 0.8 • The spot size for H=1.0 is slightly 0.6 larger than for spherical only 0.4 0.2 PIncluding distortion will not help unless 0 -0.2 the goal is to improve aberrations over -0.4 half of the field at the expense of the -0.6 -0.8 other half H=0 H=0.7 H=1.0 -1 -1 -0.5 0 0.5 1 y_p 20-15 Petzval surface The surface, or image plane, giving the best image is the Petzval surface PAssume that all aberrations are absent except field curvature PPetzval surface it the shape of the image plane to ensure that field curvature is absent PThe shape and location of the Petzval surface depends upon the properties of the lens • Power distribution • Index of refraction PPetzval surface curves inward for positive lens PPetzval surface curves outward for a negative lens d ff 1212 between the two lenses where between =+− index of refraction and the goal fff is the focal length of jth lens 111 0 j = ⎞ ⎟ ⎠ 1 jj 0 nf 1 = m j 11 11 2 2 ∑ nf nf 2 ⎛ ⎜ ⎝ 2 h 2 += =− 2 h = Petzval condition =+ z 11 2 2 1 2

ε z ε nf nf then f f /d where d is the distance 2 1 is the index of jth lens and f j can reduce field curvature using a multi-lens system Where h is the image location from the optical axis the is h Where n In terms of the focal lengths m lenses, deviation from In paraxial is • • Simplifying to a two-lens case gives Assuming the lenses have same Leaves f=f is to have the ray error to be zero gives to is the power of a two-lens system is The Petzval surface depends only on the lens used and P P P P 20-16 20-17 Cause of Astigmatism The rays in both the meridional and sagittal planes are symmetric for an on-axis object y PMarginal rays from the object to p z’ the lens travel the same distance when going to both yp=-1 and +1 PLikewise, for the sagittal y case, marginal rays from xp the object to the lens travel the same distance

when going to both xp=-1 x and +1 p

PThen the focal length will y be the same for all rays p (ignoring spherical z’ aberration) z’ 20-18 Cause of astigmatism Astigmatism is caused when rays from an off-axis object enter the optical system asymmetrically PMarginal rays from the object to the lens travel the same distance y when going to both xp=-1 and +1 p z’ PHowever, the marginal rays from the object to the lens for the meridional case travel different distances when xp going to both yp=-1 and +1 xp PThe focal length is y

different for rays in the z’ meridional plane (even if spherical aberration y is zero) p

z’ 20-19 Isn’t this field curvature? Astigmatism and field curvature are similar, but there are key differences PBoth are caused by changes in focal length due to the distance from the object to the pupil changing as the object moves off axis PField curvature can be present even when astigmatism is not

PAstigmatism will not have a sagittal ray error versus xp y p z’

y xp

yp

z’ 3 311 3 311 2 2 2 p 2 cos cos } 131 3 131 222 ++ +++

θ ++ 131

θ ++ term 222 111 sin (sin ) ] ++ cos ( cos ) ] 222 ρ ρθ ρ θ ρθ ρsin θ ρ 2 cos cos ++ 3 2 w ρρθρθ 3 020 4 2

22 ρθ 020 111 ρρθ ρθ ρθ 040 220 20 040 220 2 040 020 w wH wH dependence (makes sense since we are 220 wwH wH wH wH Astigmatism p wwH 40 220 wH wH wH 42 222 {(cos) =+ + =− + + =− + +

θ θ

ρρθ ρ ρθ ρ ρ 22 HRrw HRrwwH ,cs (/)[ sin (,,cos) ,cs (/)[ cos (,,cos) (,,cos){ cos} xc yc ε ε WH H w w H Note that there is no x astigmatism as a defocus in y of think Can always symmetric in the sagittal plane Going back to the wavefront expansion, now examine P P 20-20 20-21 Astigmatism wave fan 20 mm focal length, 5 mm pupil radius, 0.5 :m wavelength PShow the case of astigmatism with 1/10 wave aberration coefficient

PNo Wx PRight-hand plot is the meridional transverse wave fan for field curvature with 1/10 wave aberration coefficient PNote that astigmatism and field curvature are identical 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 -0.05 -0.05 -0.05 -0.1 -0.1 -0.1 W_y W_y (in number waves) of W_x W_x (in number waves) of -0.15 -0.15 waves) of number (inW_y -0.15 -0.2 -0.2 -0.2 -1 -0.5 0 0.5 1 0 0.5 1 -1 -0.5 0 0.5 1 y_p x_p y_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 20-22 Astigmatism wavefront error Wavefront error stereogram for astigmatism

PCan see the lack of wavefront error along the sagittal PSquared dependency for meridional is evident

3d order astigmatism

5th order astigmatism 20-23 Astigmatism ray fans Same case as shown on previous viewgraph except for the ray fans

PNo ,x component PSame result for astigmatism and field curvature PThe only difference between astigmatism and field curvature is that astigmatism has no sagittal term 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 Epsilon_y (micrometers) Epsilon_y Epsilon_x (micrometers) Epsilon_x -0.8 -0.8 Epsilon_y (micrometers) -0.8 -1 -1 -1 -1 -0.5 0 0.5 1 0 0.5 1 -1 -0.5 0 0.5 1 y_p x_p y_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 20-24 Ray fan for astigmatism & field curvature Can combine field curvature and astigmatism in such a way as to minimize the overall aberrations P1/10 wave of aberration for astigmatism & -1/10 wave for curvature P20 mm focal length, 5 mm pupil, 500 nm wavelength 1 1 1 0.8 0.8 0.8 0.6 Astigmatism 0.6 Curvature 0.6 Sum 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 H=0 H=0.7 H=1.0 -1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p y_p 1 1 H=0.0 1 0.8 0.8 H=0.7 H=1.0 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 H=0.0 -0.6 -0.8 -0.8 H=0.0 H=0.7 H=1.0 -0.8 H=0.7 H=1.0 -1 -1 -1 00.51 00.51 00.51 x_p x_p x_p 20-25 Astigmatism with curvature It is not possible to completely remove the aberrations due to astigmatism or curvature (or later with defocus) 1 PIn the previous example, the tangential ray 0.8 error was reduced to zero 0.6 0.4 • Called the tangential focus 0.2 • Still an aberration in the sagittal direction 0 -0.2 PCan reduce the sagittal ray error to zero -0.4 -0.6 • Called the sagittal focus -0.8 H=0 H=0.7 H=1.0 • Still aberration in the tangential direction -1 -1 -0.5 0 0.5 1 • Achieve this by being at paraxial focus y_p 1 H=0.0 then astigmatism is zero 0.8 H=0.7 H=1.0 0.6 PFinally, can also have the tangential and 0.4 sagittal ray errors equal but of opposite signs 0.2 • Called medial focus 0 -0.2 • The ray fans for this case are shown at -0.4 right -0.6 -0.8 -1 00.51 x_p 20-26 Tangential focus

,y =0 at tangential focus and ,x increases with field position PA point object creates a line that is perpendicular to the radial line from the optical axis

PLength of the line is proportional to the aberration coefficient w222 PLength of the line is proportional to H2 PWill see later that defocus can be used to counteract the curvature and astigmatism y y y p z’ p z’ p z’

xp xp xp 20-27 Sagittal focus Sagittal focus is effectively the same as tangential except for the orthogonal axis PSee later this is the case where defocus is zero PLength of the line has the same dependency on aberration coefficient as does the tangential case PPoint object creates a line image that is parallel to any radial line PEffect goes as H2

yp y y z’ p z’ p z’

xp xp xp 20-28 Medial focus Aberration in any given direction (sagittal or tangential) is smaller than for either tangential or sagittal focus PThe tradeoff is that it is larger for the orthogonal direction • Neither ray error is zero but equal in orthogonal directions • Circular spot is created by a point object PSize of the spot is proportional to H2 and the aberration coefficient PDiameter of spot is 1/2 the length of the tangential and sagittal focus spot diagram lengths Circular spot y y p z’ p z’

xp xp 20-29 Wavefront stereograms

Sagittal

Medial

Tangential 20-30 What happens in between? Spot shape varies between the “extremes” of a tangentially-oriented line and sagittally-oriented line PDepends on image plane location and with field location H PThe typical case has the y’-axis image plane vertically-oriented • Sagittal focus line is parallel to this y’-axis (no aberration in x’) • Tangential focus line is perpendicular to y’-axis (no aberration in y’) PMoving from tangential to medial to sagittal focus gives an image that goes from a horizontal line to a horizontally-oriented ellipse, then a circle, vertically-oriented ellipse and finally a vertical line PThe medial focus is also referred to as the circle of least confusion PSchematic below illustrates this for a given H value Tangential Medial Sagittal 20-31 More about the focus positions Lines at tangential & sagittal focus follow the same orientation relative to any radial line PAt tangential focus, the images from a set of point objects are lines that are perpendicular to a radial line (bottom left) PAt sagittal focus, the images from a point object are lines that are parallel to a radial line (bottom middle) PAt medial the images are circular dots with increasing size at increasing H PIn reality, there is also diffraction (and other aberrations) 20-32 Spoked-wheel example Concept of tangential and sagittal focus leads to the interesting result of the spoked wheel illustrated here PAt tangential focus the surrounding circle is in focus but radial lines are blurred PAt sagittal the radial lines are in focus but the circle is blurred PAll of the blurring increases with increasing H PMedial focus results in blur of all points on the image equal in all directions but increasing with H Object Tangential Sagittal 21-1 Aberrations OPTI 509

Lecture 21 Coma, stop-shift effects 3 311 3 311 2 2 2 2 cos cos } 131 3 131 222 ++ +++

θ ++ 131

θ ++ 222 111 sin (sin ) ] ++ cosρ (ρθ cos ρ ) θ ] sin 2 ++ ρ ρθ ρ θ cos cos ρρθρθ 3 2 020 4 3

22 ρθ 2

ρρθ 020ρθ ρθ 111 040 220 20 040 040 220 2 020 w wH wH 220 wwH wH wH wH wwH 40 220 wH wH wH Coma {(cos) 42 222 =+ + term in the wavefront expansion 131 =− + + =− + +

θ θ

ρρθ ρ ρθ ρ ρ 22 HRrw HRrwwH ,cs (/)[ sin (,,cos) ( , ,cos ) ( / )[ cos (,, cos){ cos} xc

yc ε ε WH H w w H which is clearly not as simple the other terms Now look at the w Non-symmetric Causes a blur in the image P P 21-2 21-3 Coma - cause Coma is caused by magnification that is a function of the pupil position POriginal assumption was that the principle planes are planar surfaces in paraxial space PThese surfaces are curved in reality PMagnification can become larger or smaller depending upon the optical layout • Negative coma has marginal rays with smaller magnification than paraxial rays • Positive coma has the marginal rays with larger magnification PRecall that distortion is due to magnification being a function of field position 21-4 Coma - schematically Diagrams shown here illustrate coma y p z’ Negative coma where magnification decreases with pupil xp y p z’

xp

Postive coma where magnification increases with pupil p 2 p 131 wHx c R r p 2 131 wHx R r Sagittal case 20 22 21 =90 degrees (cos) ( ) (sin ) 2 2 2 =0 p and ray errors in the sagittal 131 131 wH wH cc c R R r r =− + =− − =− =− = y x ερθ ερθ In this case, Equivalently, y The situation simplifies a bit when looking at the wave • Sagittal case has the sagittal case, ray error depends on x in that Note Also note that there is a squared dependence on the pupil position and the error is same for either +/-value • P P P 21-5 p 2 p 131 wHy c R r p 2 131 wHy R r case =0 2 20 22 213 Tangential case (cos) ( ) (sin ) 2 2 =0 p 131 131 wH wH cc c R R r r =− + =− + =− =− = y x ερθ ερθ Equivalently, x In this case Now look at the ray and wave errors for tangential • • Tangential case means that the tangential case, ray error depends on y in that Note Also note that there is a squared dependence on the pupil position and the error is same for either +/-value P P P 21-6 2 131 22 2 2 131 shape of the aberrated image Coma - spot diagram Rrw H Rrw H (/) ( cos) (/) (sin) =− + =− yc xc ερθερθ The spot diagrams are instructive in determining the The image from an off-axis object creates a circularly-shaped The circular image is displaced from the paraxial location P P 21-7 21-8 Coma - Spot diagram Each set of points from the object for constant D on the pupil plane leads to a circular region on image plane PA set of circles is created with each circle related to a different pupil position PCircles get larger for larger pupil positions PCircles become further from the paraxial image point for larger pupil positions 21-9 Coma - spot diagrams Reiterate, that as the object rays intersect at larger pupil locations the circle size increases and moves further from paraxial 21-10 Coma - Spot diagrams Look at the spot diagram related to the rays coming from a point object to a given radius on the pupil plane PSpot diagram shape can be discerned from the general equations for the transverse ray errors • These rays image to a circle on the image plane

• Plot below shows the case where -(R/rc)w131H=1 PShow the input pupil plane positions • Assumed D=1 • Upper plots show 0<2<90 and 270<2<360 • Full circle in the spot diagrams is present 1 3

2.5 0.5 2

0 1.5

1 -0.5 0.5

-1 0 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 pupil location - x_p epsilon_x 21-11 Coma - spot diagrams The same full circular spot diagram is created by rays from either the upper or lower portion of the pupil 3 PUpper plots are same 1 as just shown 2.5 0.5 2 PLower plots show 90<2<270 0 1.5 1 PDue to the squared -0.5 dependence in pupil 0.5 position -1 0 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 pupil location - x_p epsilon_x 1 3

2.5 0.5 2

0 1.5

1 -0.5 0.5

-1 0 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 pupil location - x_p epsilon_x 21-12 Coma - Spot Diagrams Variety of pupil positions leads to a set of circles in the transverse ray error

PAgain plots below show the case where -(R/rc)w131H=1 PPupil values take on values of 0.2, 0.4, 0.6, 0.8, 0.9, and 1.0 PEach color in the pupil corresponds to the like color in the spot diagram 1 3

2.5 0.5 2

0 1.5

epsilon_y 1 -0.5 pupil location - y_p 0.5

-1 0 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 pupil location - x_p epsilon_x 21-13 Coma - spot diagrams Error increases with pupil position and circular array of spots moves away from chief ray with increasing D PError maintains a circular shape for a constant pupil value PThis case is for a given H PLarger H gives larger errors PAll of these rays come from a single point on the object

1 3

2.5 0.5 2

0 1.5

epsilon_y 1 -0.5 pupil locationy_p - 0.5

-1 0 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 pupil location - x_p epsilon_x 21-14 Spot diagrams Summarize the key points of the shape of the aberrated image from a point on the object PA circle in the pupil plane maps to a “double” circle in the image plane (the effect of D2) PThe diameter and location of the 3 double circle varies with the f/#, field location, and pupil location 2.5 • Diameter of the circle is 2 2 2(R/rc)w131HD 2 • Displacement is -2(R/rc)w131HD 1.5

PDisplacement is always to one epsilon_y 1 side of the chief ray focus 0.5

0 -1.5 -1 -0.5 0 0.5 1 1.5 epsilon_x 21-15 Coma - spot diagrams The coma pattern is contained within a 60-degree wedge (full-sized) PRays from a point on the object fill the pupil space PImage plane is also filled within the 60-degree wedge out to the 3 edge of the circle from the marginal rays 2.5 2

1.5

epsilon_y 1

0.5

0 -1.5 -1 -0.5 0 0.5 1 1.5 epsilon_x 21-16 Coma - Wave fan Use the 20 mm focal length, 5 mm pupil radius case at 0.5 :m and a 1/10 wave aberration coefficient for coma 2 PRecall W=w131HD yp • Other way to view it is in terms of D3 3 • Tangential case is 2=0 and W=w131Hyp because xp=0 • Sagittal case is for 2=90 and W=0 PLooks like 0.2 0.2 no other 0.15 0.15 case we 0.1 0.1 had before 0.05 0.05 0 0 -0.05 -0.05 -0.1 -0.1 -0.15 -0.15 -0.2 -0.2 -1 -0.5 0 0.5 1 00.51 y_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 21-17 Coma - Ray fan The ray fans give a distinctive shape relative to other aberrations that have been covered PLinear dependency in H PQuadratic dependency on pupil position PMagnitude of the ray error for the sagittal case is zero

PPlot below right is for ,y versus xp 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 Epsilon_x (micrometers) -0.8 -0.8 Epsilon_y (micrometers) -0.8 -1 -1 -1 -1 -0.5 0 0.5 1 0 0.5 1 0 0.5 1 y_p x_p x_p H=0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 H=0.0 H=0.7 H=1.0 21-18 Coma - Ray fans It is instructive to examine how the ray fans correspond to the spot diagrams

PConsider the meridional/tangential case first in which xp=0

PThus, only values for yp give a ray error PCan see in the spot diagram at right the quadratic dependence 3 1

2.5 0.5 2

0 1.5

-0.5 1

0.5 -1 -1 -0.5 0 0.5 1 0 pupil location - x_p -1.5 -1 -0.5 0 0.5 1 1.5 epsilon_x 21-19 Coma - ray fans

Examine the sagittal case for which yp=0 and only a variation in xp P2=0 and thus sin2=0 PThere is no ray error in the sagittal direction for this case 3 1 2.5

0.5 2

0 1.5

pupil location - y_p -0.5 1

-1 0.5 -1 -0.5 0 0.5 1 pupil location - x_p 0 -1.5 -1 -0.5 0 0.5 1 1.5 epsilon_x 21-20 Coma - ray fans The size of the spot due to the sagittal ray fans can be computed in a similar fashion as done previously PAbout 55% of the light is contained between the paraxial image spot and sagittal “edge” PThis can be seen in the diagrams below

3

2.5 Tangential 2

1.5

1 0.5 Sagittal 0 -1.5 -1 -0.5 0 0.5 1 1.5 epsilon_x 21-21 Coma - stop shift The amount of coma is often dictated by the position of the stop PRecall a similar effect for distortion PThe idea for coma is similar in that the movement of the stop alters the position of the chief ray PMore importantly, the position of the stop alters the symmetry of the rays PDiagram below illustrates the stop coinciding with the lens PNote the symmetry of the rays being blocked and being let through 21-22 Coma - stop shift Moving the stop in the previous location alters what rays can now pass through the system PNow the lowest ray can pass through the stop but the upper is still blocked PThis creates an asymmetry in the light on the image plane because the lower ray is not “balanced” by the similar upper ray PIn addition, this lower ray is further from paraxial and more susceptible to aberration 21-23 Coma - stop shift example Example here uses a single 20 mm focal length lens

PNo aberration balancing is being done • Thus, there is definitely spherical aberration present • Curvature and astigmatism are also going to be present P3-mm stop at the lens PObject is 10 degrees off axis PStop is at the lens in this case 21-24 Coma - stop shift example

PNow move the stop closer to the object PStop is the set of dots to the left of the lens 21-25 Coma - stop shift example

PMove the stop to the other side of the lens PStill the same size 21-26 Coma - spot diagrams

PExample shown here is a set of spot diagrams for varying H locations for a given aberration coefficient POne could view this as the image from an array of objects PExample here is for positive coma 21-27 Positive and negative coma The sign of the aberration coefficient determines whether the “comet’s tail” points away or towards the optical axis PPostive coma (aberration coefficient is positive) causes the ray error to be negative for positve H • Spots move towards the optical axis • “Tail” points to the center • “Head” is away from the center POpposite occurs for negative Positive coma coma

Negative coma 21-28 Aperture obstructions Mirror systems will typically have a portion of the aperture obscured to fold the system PThe obstruction then blocks a portion of the pupil plane PThis leads to the interesting result seen in the spot diagram shown below PThis case is a bit unrealistic in that the obscuration is from D=0 to 0.7 which means that nearly one half of the area of the No obstruction primary mirror is obscured

f1 f2 With obstruction 22-1 Aberrations OPTI 509

Lecture 22 Longitudinal chromatic aberrations of a thin lens, thin lens achromatic doublet, spherochromatism, secondary chromatic aberration; lateral chromatic aberration 22-2 Examples of Chromatic Aberration 22-3 Chromatic aberration Root cause of chromatic aberration is the dispersive nature of the lens materials used in the optical system PExample shown here is based on BK7 glass and a Schott formulation for dispersion aaaa λλ= +2 ++++3 4 5 6 naa() 12 2 4 6 8

PThis dispersion is useful 1.535 in sensor design as a 1.53λλλλ means of spectral selection (e.g., prisms) 1.525

PThis dispersion is now 1.52 a problem in image quality 1.515

1.51 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Wavelength (micrometers) is < 1 − − d FC n nn length notation, the Abbe number, =

υ =587.6 nm is the index of refraction for each wavelength the is 8 F =486.1 nm =656.3 nm 8 8 , n C dispersion of the refracting material Reminder - Abbe number , n d Where n d, F, and C - Yellow light d F - Blue light F Lettering scheme has heritage with labeling emission spectra scheme Lettering C - Red light • Use the typical notation for three wavelengths in the visible notation typical the Use • • Using this d, F, and C, wave • • • The Abbe number is used as a parameter to describe the P P 22-4 22-5 Abbe number - example Use the dispersion curve for BK7 that was shown in a previous viewgraph PGraph highlights the d, F, and C wavelengths

PIndexes are nd=1.517, nF=1.522, and nc=1.514 PThe above gives an Abbe number of <=64.2 • Large Abbe number 1.535 usually due to index F d C at F and c being similar 1.53 • Implies low dispersion 1.525 PMilcode number for BK7 is 517642 1.52 • Index is 1.517 • Abbe number 64.2 1.515 1.51 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Wavelength (micrometers) 22-6 More Abbe number Smaller Abbe numbers imply that the change in index with wavelength is larger PThe inverse of the Abbe number is known as the dispersive power PThen larger dispersive power implies greater dispersing power • Larger values are good for building a spectrometer (or rainbows) • Larger values are bad for limiting chromatic aberration POther parameters and/or 1.55 indexes that can be used instead of the Abbe number 1.54 • Abbe number may not be <=38.0 useful for spectral regions 1.53 outside of the visible <=64.1 • Used here as a means for 1.52 rapid evaluation of the <=95.9 dispersion 1.51 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Wavelength (micrometer) 22-7 Schott Glass Catalog Abbe Diagram

2.00

nd

1.45 85 < 20 22-8 Impact of chromatic aberration Major impact of chromatic aberration to image quality is that the focal length now varies with wavelength PGo back to the thin lens equation 1 11⎛ ⎞ =−()n 1 ⎜ − ()()nCC⎟ =−1 12 − f RR⎝ 1 2 ⎠ Pf=f(8) if n=n(8) and different colors focus at different locations PThe change in focal length due to a change in index is 1 1 ∆ fff=−=() − kj ⎛ ⎞ ⎛ ⎞ −−11⎜ ⎟ 11−−⎜ ⎟ n()kj1 ⎝ ()n⎠ 1 ⎝ ⎠ RR1 2 RR1 2 []−− − = ⎛1 nn()()jk⎞ 11−− ⎜ − ⎟ 11⎝ nn()()⎠kj11 1 2 − RR= [] ⎛ ⎞ −−⎜ −⎟ nnjk ⎝ ⎠ 11 nn()()kj11 RR1 2 22-9 Error in focal length Compute this error in focal length in terms of the F and c wavelengths PSubstituting for the index at F and c gives

nn[]FC− ∆f − = FC 11⎛ ⎞ −−⎜ −⎟ ()()nnFC11⎝ ⎠ RR1 2 PSubstituting with the definition of the Abbe number1 gives 1 1 n d []− nnFC[]− n d − ν ∆ = = f FC− 11 1 − 11 ⎛ n d ⎞ ⎛ ⎞ nn()()−−11⎜ −⎟ nn ()()11−−⎜ −⎟ FCRR⎝ 1 2 ⎠ FCRR1 2 ⎝ ⎠

PMultiply the top and bottom by (1/R1-1/R2) and 11⎛ ⎞ − 1 2⎜ ⎟ 11 nd 1 −RR⎝ ⎠ ∆f − = FC ν ⎛ nn()()⎞ −−⎛ ⎞ ⎜11− FC⎟ 1111⎜ − ⎟ ⎝ ⎠ ⎝ ⎠ RR1 2 RR1 2 22-10 Error in focal length Make use of the definition of the focal length in terms of the index and radii of a two surface lens PRecall 1 ⎛ 11⎞ =−()n 1 ⎜ −⎟ f ⎝ RR1 2⎠ 11 n() ⎛ ⎞ d 1 −−1 ⎜ 2 ⎟ PThen 11 RR⎝ 1ffFC ⎠ ∆f − = = FC νν⎛ ⎞ ⎛ f ⎞ ()−−11⎜ ()⎟ 11−−⎜ d ⎟ n F 1 ⎝ nC ⎠1 ⎝ ⎠ RR1 2 RR1 2 PMake the approximation that focal lengths at the three wavelengths are nearly the same

• That is fd = fC =fF are nearly the same • Allows for a simple interpretation of the Abbe number’s relationship with focal length f f −∆ 1 • Then ∆ ≈≈dFC f FC− ννthen f d 22-11 Error in focal length The conclusion is that the approximate %change in focal length is inversely proportional to the Abbe number PShould make sense since smaller 20.5 Abbe numbers are related to larger <=95.9 dispersive power 20 <=64.1 PGraph shows focal length as a 19.5 function of wavelength for three <=38.0 materials 19 PBased on the equation just derived 18.5 f dFCf −∆ 1 ∆ ≈≈0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 f FC− ννthen f d Wavelength (micrometers) The %changes in focal length are 2.6%, 1.6%, and 1.0% PThese changes can be used to analytically determine how the spot diagram will change with image location 22-12 Dispersion and focal length Upper graph below shows dispersion curve for BK7 (517642) 1.535 PLower graph shows how this F d C affects the focal length as a 1.53

function of wavelength 1.525

PAs mentioned before, the index 1.52 for blue light is higher than for red 1.515 PThus, the focal length for blue is 1.51 smaller (1.5% relative to the red) 0.4 0.45 0.5 0.55 0.6 0.65 0.7 PAlso note that the functionality Wavelength (micrometers) 20.2 is non-linear 20.1 20 19.9 19.8 19.7 19.6 19.5 F d C 19.4 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Wavelength (micrometers) 22-13 Dispersion and focal length Typical lens materials for the positive lens case will lead to blue light coming to focus closer to the lens PThis is illustrated below for red, green, and blue PCan use this diagram to understand how an image will appear

yp

z’ 22-14 Longitudinal Chromatic Aberration The chromatic aberration then creates image planes for which a specific wavelength is at focus P“Inside” of focus will have a blue image with red halo P“Outside” of focus will have red image with blue halo PSchematic diagram illustrates the haloes for blue, green, and red

focus yp

z’ 22-15 Spot diagrams 50-mm focal length silica glass biconvex lens with an object at -2000 mm and 4-mm stop size PPlots shown are for various image plane locations PLocations closest to lens are upper left and furthest is lower right

Blue focus Green focus

Red focus 22-16 Concave lens The effect is present in both convex and concave lenses and the effect is the same for both PThat is, blue light focuses at a point closer to the lens than that of longer wavelengths PFormulations previously developed work just as well for the convex case as long as sign conventions are consistent PThe aberration is also be visible away from the optical axis

z’ 22-17 Chromatic aberration wave fan Can show that the wavefront aberration should go as the 2 square of the pupil location and then W=8w020D PThe notation above is used to indicate that the aberration coefficient is a function of wavelength

PTypically use “d” as the reference wavelength, thus dw020/0 PWave fan is quadratic PNo wave error for d-wavelength (green line is at zero) PBlue is inside focus and this occurs For positive wave error POpposite case “outside” of d focus

-1 -0.5 0 0.5 1 y_p 22-18 Chromatic aberration and defocus Because chromatic aberration is similar to defocus, defocus will be used to balance chromatic aberration PDefocus is discussed in more detail later in overall image quality and aberration balancing 2 2 • Wtotal=Wdefocus+Wchromatic=w020D +8w020D PHave to realize that defocus is a function of wavelength PWill want defocus to be equal and opposite to chromatic aberration PSpot diagrams shown previously were for different values of defocus to obtain the focus for each color 22-19 Chromatic aberration ray fan Sagittal ray fan shown here is based on using an image plane at the focal point for d wavelength light (green) PThe ray fan is linear for a given wavelength, • Separate ray fan for each wavelength • Since d is reference wavelength it suffers ,x no aberration PShort wavelength light is inside of focus C • Ray error is negative for positive x p x • Negative ray error implies crossing p the optical axis closer to the lens d 1 PLonger wavelengths are outside of focus F 22-20 Chromatic aberration ray fan Meridional ray fan is also linear for the different wavelengths -all ray fans are for H=0 PThe amount of aberration is given by the change in focal length with wavelength

• Notation for chromatic aberration is 8w020 (also 8)w20 and 8w20) • Only depends on pupil position and wavelength PNote that the aberration is , smaller for C wavelength than y At H=0 F wavelength • Result of non-linear dispersion C • Equal focus error inside and outside leads to a ray fan of yp the same slope opposite in sign -1d 1

F 22-21 Fixing chromatic aberration It should appear that we could simply solve our chromatic aberration problem by moving the image plane PMoving the image plane inside of the C wavelength focus will begin to bring the d wavelength into focus PFurther movement will bring F wavelength into focus PCould stop down the optical system and that should also help PNo matter what is done, there will still be chromatic aberration at other wavelengths yp

z’ 22-22 Achromatization Simply moving the image plane is not satisfactory for correcting chromatic aberration PUse two lenses that can cancel the effects of chromatic aberration • Goal is to have F and C light come to focus at the same position • This is done by using lens elements with differing dispersions PProcess is referred to as Crown Glass achromatization Flint Glass PThe doublet is an achromat PThese are readily available as cemented pairs 22-23 Achromat design Postive and negative powers with different dispersion allows identical focal length at two wavelengths POne element is typically made from crown glass <>50 • Recall that large Abbe numbers implied low dispersive power • Thus this lens does not have a large change in focal length with wavelength • This lens is used to control image location • Chromatic aberration is still an issue and hence the reason to y attempt a correction p POther element from flint glass <<50 • Blue light has a much shorter z’ focal length than red light • This second lens is to correct for dispersion

z’ 22-24 Achromat design The first lens gives power and the second lens corrects for the dispersion of the first PThe combination gives a single focus for the two wavelengths PThus, the chromatic aberration is canceled - or is it?

yp

z’ z’

yp

z’ 22-25 Achromat design The achromat brings the F and C light to focus at the same point and at the desired location PThe problem is that the d wavelength (green line below) comes to a focus at a different location compared to F and C • In this situation the green comes to focus inside the F and C wavelength focus • Can only “fix” two wavelengths with two lenses we

yp

z’ 22-26 Achromatization example Start with bi-convex lens situation with a 20 mm focal length, the object at -40 mm and a stop size of 5 mm PThe graph shows the image location for this particular lens as a function of wavelength PIn this case, the only option would be to move the image plane to improve the image quality for a specific wavelength 22-27 Representation of Chromatic Aberration The curve on previous page shows the change in focal length as a function of wavelength for the given lens and object distance

Focal lengths derived for specified wavelengths as shown on the graph 22-28 Achromatization example Replacing the lens of the previous example with an achromat gives the case here PThe achromat is selected to be similar to the bi-convex case with effective focal length (at d) of 20.7 mm PThe achromat is a cemented combination of a bi-convex and a plano-concave from the raytrace packages database • Can specify the desired overall focal length of the achromat • The achromat is purchased as a “single” package 22-29 Achromatization Show the focal length as a function of wavelength for the achromat PThere are a pair of wavelengths for which the focal lenght is identical for an achromat PThus, the image plane can be placed at this location and there will be no chromatic aberration at those two wavelengths 22-30 Achromat - design Examine the design of the achromat by assuming that two thin lenses are in contact

PThe power of a combination of two lenses is M=M1+M2 where M=1/f PWant the change in focal length to be zero between two wavelengths • Select the two wavelengths to be F (blue) and C (red) • Wavelengths between these two will suffer from chromatic aberration but it should be relatively small PThe design relies on two lenses with differing dispersion

PRecall that )fF-C.fd /< then )MF-C.Md /< PUsing this ) formulation for each individual lens gives

• )M=)M1+)M2 • Then in terms of F and C wavelengths ∆Φ=+ ∆Φ ∆Φ FC−−−12__ FC FC 22-31 Achromat design The goal is to have the two lenses such that the difference in power between F and C is zero

PSetting )MF-C =0 gives 1 __ddΦΦ2 =+=+∆Φ ∆Φ 0 12__FC−− FC νν 1 2 PNote that the dispersion of each lens is contained within the Abbe number

PSubstituting for )M=Md/< for each of the wavelengths gives ∆Φ=+= ∆Φ ∆Φ FC−−− FC12__ FC 0

PSolving for M1 gives 2_Φd Φ =−ν 11_d ν2 22-32 Achromat design Now have the power of the first element in terms of the Abbe numbers and the power of the second

PRecall that M=M1+M2 • In this case Md =M1_d+M2_d • Or M2_d=Md -M1_d and ΦΦ− Φ =−ν dd1_ ΦΦΦ−= 11_d then 11__ddd 21 1 ν2 Φ and 1_ d = 1 Φ −ν νν ν d 12 νν PTaking a similar approach for M2 Φ υ 2 _ d =− 2 Φ υυ− d 12 22-33 Achromat - design Thus, it has been shown that for a two lens case, the lenses must be negative and positive PSummarizing the power needed for the two lenses to ensure F and C are at the same focus is Φ υ Φ 1__d = 1 2 d 2 =− Φ υυ− and Φ − d 12 d 12 PRoute usually taken is to require a system focal length • Available glasses and their Abbe numbers areυ known • Recall the Schott-glass diagram υυ PDetermine powers needed for two lenses to get overall focal length • Lens with low dispersion (high <) has extra power • Low < element subtracts extra power & reduces overall dispersion

PCompact system with low curvature lenses requires L1-L2 be as large as possible 22-34 Spherochromatism One thing that not considered at this point is how chromatic effects impact other aberrations POne of the biggest problems is that now there is a chromatic effect in the spherical aberration , PIn the case of the doublet with no spherical y aberration the ray fan is that shown F C schematically at right yp PPut the image plane at the focus for the d -1d 1 wavelength • Then F and C are outside this focus F y • The positive ray error p for the positive yp shows this

z’ 22-35 Spherochromatism Recall the aberration due to 3d order spherical has a fourth power in wave and cubic in ray error PShow here the wave and ray fans now for three wavelengths PBlue light has a larger index thus has a larger spherical aberration PAll plots are shown for H=0 0.1 2 1.5 0.05 1 0.5 0 0 -0.5 -0.05 -1 -1.5 -0.1 -2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 22-36 Spherochromatism The spherical aberration adds error to the achromatized system 2 Achromat only PAdd spherical aberration to the chromatic 1.5 aberration that has been corrected 1 0.5 • Ray fan for achromat is upper right 0 • Ray fan for 3d order spherical is below -0.5 Adding the two gives the ray fan below right -1 P -1.5 -2 -1 -0.5 0 0.5 1 y_p

2 3d order spherical 2 1.5 1.5 1 1 Summation 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 22-37 Spherochromatism The design of the achromat for the d-wavelength means that this wavelength has 3d order spherical only PF wavelength has the same spot size as d in this particular case PC wavelength has the largest spot size 2 PThen in this case, the 3d order spherical aberration “improves” 1.5 both the F and C wavelengths 1 PThe achromat, however, does 0.5 not behave as the theoretical 0 design intended • That is, there is chromatic -0.5 aberration at all wavelengths -1 • This is a result of the 3d order -1.5 spherical -2 -1 -0.5 0 0.5 1 y_p 22-38 Secondary chromatic aberration An achromat gives a matched focal lengths at two wavelengths PThis was based on the design of the power of the lenses and the dispersion (Abbe numbers) PThen as illustrated, the F and C come to focus at the same point PHowever, d (and other wavelengths) still have a focus shift PThis focus shift is the secondary chromatic aberration yp f8-fd

z’

F d C 8 22-39 Secondary chromatic aberration - ray fan Can examine how the focus of the d wavelength shifts relative to others and determine a way to minimize this PMaking the difference less between d and the F-C focus (or better yet, make it zero) should give lower aberrations at all wavelengths PThe ray fan for the F and C wavelengths are identical PShow that the shift in focus for this case is )f/f=)P/)< • P is the dispersion ratio • Focus shift to F, C from d • Value for )f/f will be around 0.005 for typical lens materials yp F,C ,y F C z’ yp -1 d 1 d F 22-40 Secondary chromatic aberration Examine the difference in power between the d and C wavelengths

P)Nd-C=)M1_d-C+)M2_d-C PUse these two because the difference for d and F for an achromat will be the same PFollow the same approach used previously to examine the change in focal length due to wavelength ∆Φ =− −−− −=− − 1121212_dC− ()()()()()()nCCnCCnnCC d11 C d C PTrying to correct the secondary chromatic while not affecting the “primary” chromatic aberration • It is of interest to understand how the dispersion between all three wavelengths relate to each other • Introduce the partial dispersion ratio 22-41 Partial dispersion ratio Partial dispersion ratio, P, includes the index of refraction of all three wavelength

PP=(nd-nC)/(nF-nC) PThen the difference in power between d and C is written as nn11__FC− ∆Φ=− −= ∆Φ 111___dC− ()nn d C − ()CC12 P 11_ FC − nn11__FC

PUsing a similar formulation for )M2_d-C in terms of the partial dispersion for the second lens and computing the d-C difference in power gives ∆Φ=+ ∆Φ ∆Φ dC−−−PP11__ FC 2 2 FC 22-42 Secondary chromatic aberration Recall that for the simple positive singlet the difference in power is )MFC=M/L we can write the above as PThis allows the previous formula to be rewritten as ΦΦ ∆Φ =+PP11 22 dC− νν 1 2

PIncluding the result for the achromat design where M1/L1=-M2/L2 leads to PPΦΦΦ ∆Φ =−=11 21 1 () − dC− νννPP12 1 1 1 PFinally, substituting the achromat solution Φ υ ΦΦ 11= 1 = Φ υυ− ν υυ⇒ − 12 112 Gives PP− P ∆ ∆Φ= 12 Φ= Φ dC− νν− ν ∆ 12 22-43 Partial dispersion ratio Most glass falls along nearly a straight line for the plot of partial dispersion ration versus Abbe number

PA large Abbe number implies low dispersion (nF and nC are similar) PAs Abbe number increases, so does partial dispersion PIgnoring the odd cases (triangles in the graph), the slope of the line is 0.000476 0.315

0.31

0.305

0.3

0.295 R-square = 0.955 # pts = 61 y = 0.277 + 0.000476x 0.29 30 40 50 60 70 80 90 100 Abbe Number 22-44 Partial dispersion ratio The slope of the line is )P/)< and this gives the error in focus between d and C wavelengths for the achromat PThe singlet error in focus is 0.315 proportional to 1/< and for BK7 0.31 this value is 1/64=0.0156 0.305

PConsider the achromat using 0.3 • BK7 (517642) 0.295 • F5 (603380) R-square = 0.955 # pts = 61 y = 0.277 + 0.000476x 0.29 • The error in focus for 30 40 50 60 70 80 90 100 the achromat is Abbe Number (0.3079-.2943)/(64.17-38.03)=0.00052 • This value is close to the simplistic approach using the slope of the line PP− P ∆ ∆Φ= 12 Φ==0. 000476Φ dC− νν− ν ∆ 12 PCan select a specific achromat that minimizes secondary chromatic aberration 22-45 Transverse axial chromatic aberration Have only examined the longitudinal (or axial) chromatic aberration on axis to this point PThere is also a component of the longitudinal aberration in the transverse plane

PThe transverse axial chromatic aberration (TAch) is the lateral blur associated with the longitudinal aberration

PDerive the TAch assuming that the marginal rays for the three colors are effectively parallel at the image planes

rc " z’ f TAch )f yp 22-46 Transverse axial chromatic aberration Compute the transverse axial chromatic aberration in terms of the focal length change PThe geometry of the transverse axial chromatic aberration shows that r TA −=cchtan(α ) =− f ∆f

PThen rearranging and solving for TAch gives ∆f TA= r ch c f PBut, )f/f=1/< from our singlet solution r TA = c ch ν rc " z’ f TAch )f yp 22-47 Transverse axial chromatic aberration The solution just shown is for the singlet case but can go through a similar derivation for the achromat PNote that this singlet solution is independent of focal length

• Larger Abbe number (lower dispersion) gives smaller TAch • Larger radius lens gives larger spot size PBoth of these conclusions should make sense based on previous discussions PNow have a method for understanding the size of the spots from the spot diagrams just seen yp • 3d order spherical versus wavelength • Chromatic aberration z’ 22-48 Lateral chromatic aberration Lateral chromatic aberration is caused by the change in power as a function of wavelength for off-axis objects PThere is a magnification change in going from the reference wavelength (d) to other wavelengths (F and C) • The chief ray varies with wavelength C yp d • See later that this is F r very similar to tilt c • In fact, often referred z’ to as chromatic tilt PFigure at right illustrates this schematically • Assumes that the typical case of index of refraction increasing with decreasing wavlengths • Blue light (F) is refracted more and thus is “below” C 22-49 Lateral chromatic aberration -wavefronts Write the wavefront error in a similar fashion to the tilt

wavefront case -W(8)=8w111HDcos2 PWe will define dw111/0 0.2 H=0 PTypically, Fw111 >0 and Cw111<0 PEffect is larger for larger H F POpposite slopes of wave fans indicate the d relative location of the blue and red light Wavelength C relative to the green image point -0.3 -1 -0.5 0 0.5 1 PThere will also be radial symmetry y_p

0.2 H=0.7 0.2 H=1

F F

d d

Wavelength C -0.3 -0.3 Wavelength C -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 22-50 Lateral chromatic aberration - ray fans

Write the ray error as ,y(8)= - 8w111HR/rc 0.5 0.4 H=0 PRay error increases with field position 0.3 0.2 PF wavelength is always below 0.1 F the d and C 0 -0.1 Wavelength d -0.2 PThe ray error for the d wavelength C is set to be zero -0.3 -0.4 PVery similar to distortion and will be -0.5 -1 -0.5 0 0.5 1 similar to tilt y_p 0.5 0.5 0.4 H=0.7 0.4 0.3 0.3 H=1 0.2 0.2 F 0.1 0.1 F 0 0 Wavelength d -0.1 -0.1 Wavelength d -0.2 -0.2 C -0.3 -0.3 C -0.4 -0.4 -0.5 -0.5 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 y_p y_p 22-51 Lateral chromatic aberration Ray fans mean that the lateral color is linear in H and there is rotational symmetry PIgnore the axial aberration for this discussion PFigure at right shows the image plane • Assume we have a white object • The object is imaged at the indicated field locations PColors shown are meant to represent what is happening with H • Only show the blue and red at the edges of the image • No attempt to blur & mix the colors • For example, the center of an image will be white with the outer edge transitioning smoothly to orange then red PFigure is not quantitative 22-52 Combined lateral and axial What will the focused image look like if there is both axial and lateral chromatic aberration PThe off-axis axial aberrated rays for each of the colors is effectively parallel PLateral aberration is zero on axis • At d focus, the focus spot is green in the center with an “even” mix of reds and blues surrounding it • At F focus, spot is blue to green to red • At C focus, spot is red to green to blue • Showed this on previous viewgraphs PFigure attempts to show what can be expected for the lateral and axial case • d wavelength is always in “focus” • Then blue and red are blurred spots offset from green