The Prism-Pair: Simple Dispersion Compensation and Spectral

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Review Article: Open Access is usedfortuned,precisecontrolofdispersionandspectralphase. through aprism-pair,anddemonstrateshowthismajorcomponent of thetotalwavelength-dependentphaseaccumulatedbylightpassing [23-27]. This mini-review presents a simple and intuitive formulation low lightlevelspectroscopy[20-22],andquantumopticsexperiments is oftenthemain’toolofchoice’forintra-cavityapplications[17- 19], mask inthedispersivearm.Duetolowlossofprism-pair, it order), along with simple amplitude control using a slit or transmission two ordersofdispersion(GDDandsometimesanadditionalhigher pairs can provide tuned compensation with ultra-low loss of up to the Brewsterprism-pair[16]-subjectofthismini-review.Prism- commonly used,suchasthegrating-pair[14],chirpedmirrors15]or general pulse-shaperareaburden,andsimplerconfigurations applications, the high internal loss and technical complexity of a pulse whenpassingthroughopticalmediaandsetups.Forthese compensate forthedispersiveeffectexperiencedbyanultrashort the groupdelaydispersion(GDD)andhigherorderto However, mostapplicationsrequireonlymuchsimplercontrolof with aspatiallightmodulator[11,12]ordeformablemirrors13]). phase and amplitude for each frequency component of the light (e.g. shaper [10]canbeused,whichprovidesindependentcontrolofboth amplitude and phase is required, a general Fourier-domain pulse- oriented experiments.Whencompletecontrolofthespectral shaping techniquesandconfigurations are common inspectroscopy- spectral propertiesofthepulses.Forthisend,manytypespulse- manipulation of molecular dynamics the [7-9]bycontrollingthe enable toobservemolecularmotionintime[4-6],but allow also they field of molecular spectroscopy [3]. Not only that ultrashort pulses credited. author andsourceare the original in anymedium,provided and reproduction License, whichpermitsunrestricted use,distribution Copyright: Received: January27,2016: Ramat Gan52900,Israel, E-mail: *Corresponding author: Avi Pe'er, Physics departmentand BINA center forBarIlanUniversity, nanotechnology, Introduction 2 1 Yaakov Shaked ing ofUltrashortPulses Shap- and Spectral Compensation Dispersion Simple The Prism-Pair: Ophthalmology division,LumenisLTD,Yokneam2069204,Israel Bar-IlanUniversity,Ramat-Gan52900,Israel Department ofphysicsandBINACenterforNano-technology, Ultrashort Pulses. IntJExpSpectroscopicTech 1:007 Citation: The generation of ultrashort pulses [1,2 ] revolutionized the VIBGYOR © 2016 Shaked Y, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution Commons the termsof Creative under article distributed Y, etal.Thisisanopen-access © 2016Shaked

Shaked Y, Yefet S, Pe’er A (2016) The Prism-Pair: Simple Dispersion Compensation and Spectral Shaping of and SpectralShaping Compensation Shaked Y,Yefet S,Pe’erA(2016)ThePrism-Pair: SimpleDispersion demonstrating the flexibility of the prism pair to provide tuned, low-loss control of the dispersion and spectral phase experiencedbyultrashortpulses. and dispersion the of control low-loss tuned, provide to pair prism the of flexibility the demonstrating spectral total prisms, Brewster-cut The of pair a through passing component. after calculated is field experimental light broadband a by experienced a major phase of description unified a with spectroscopy of field the in students andbeginners using ultrashortpulses.Thisreviewaimstoprovide experiments spectroscopy-oriented in - Amostcommoncomponent for theBrewsterprism-pair is reviewed formulation and intuitive A simple Abstract 1 , ShaiYefet Accepted:April04,2016: [email protected] 2 andAviPe'er Published: 1 * refraction anglesobey matches withtheBrewsterangleθ figure 1,resultingin: of refractionthroughtheprismaresymmetric[28],asillustratedin When araypassesthroughprismatminimumdeviation,theangles A SinglePrism

propagates throughtheprismparalleltoitsbase. At minimum deviation,theentranceandexitanglesareequal andtheray Figure 1: Symmetry relations betweenthe refraction anglesina prism. April 07,2016 Spectrosi T Experimntal Inter where The analysisstartsbyreviewingthegeometryofasingleprism. θθ θθ 14 23 = = = natiol Jur α istheapexangleofprism.Ifentrance α 2 [1] Shaked etal.IntJExpSpectroscopicTech2016,1:2 nal of B (foracertainwavelengthλ echniqus ISSN: 2631-505X 0 ), the

Shaked et al. Int J Exp Spectroscopic Tech 2016, 1:2 ISSN: 2631-505X | • Page 2 of 4 •

Figure 2: Geometry of a prism-pair, defined using two degrees of freedom: the separation between the prisms R and the prism insertion H. Figure 4: Complete optical path through a prism-pair. Following a non- continuous ray by using the concept of wavefronts, the optical path through the entire system can be reduced to the segment EC.

Figure 3: Different wavefronts (black dashed lines) of equal phase in a plane wave beam passing through a prism pair.

Figure 5: Definition of the exit angles in the 1st prism. θB = θθ14 = π θθB +=2 [2] wavelength λ. Since the wavefront R1 is common for all wavelengths, 2 shifting its position only results in a simple optical delay, hence, the and the relation between the apex angle α and Brewster angle can position of R1 can be chosen freely to pass through the point A. be expressed as The same argument applies also to R5, which can be chosen to pass through point C. π αθ==22 2 −=− θBB πθ2 [ 3] 2 Consequently, for calculating the geometric dispersion, it is sufficient if the optical path through the entire prism-pair is reduced The Optical Path through a Prism-Pair to the segment: Our aim now is to calculate the total optical path (and phase) P= AA′′ + CC [4] through a prism-pair, which will allow to derive the dispersion experienced by the passing optical pulse. The coordinate system Note that although this path is composed of free-space rays used for the prism-pair has two degrees of freedom (Figure 2): R without refractions, it already takes into account refraction of the is the separation between the prisms (segment AB), and H is the planar beam in the material. As illustrated in figure 4 (which shows penetration of the prism into the beam (segment BC). The red line the rays without the prisms for clarity), the optical path in Eq. 4 is represents a Brewster ray (at the design wavelength λ ) that enters and identical to the segment EC, where the point C is the vertex of the 0 nd exits both prisms at Brewster angle θ . The blue line is the deviated ray 2 prism. The conclusion is that the optical path of the deviated ray B through the entire prism pair is simply given by: at λ ≠ λ0, which deviates from the Brewster ray by an angle δθ due to the prism dispersion. P= EC = ED + DC [5]

Wavefront Calculation where ED= R cosδθ , and DC= H cosθθHH (= ∠DCB ). To calculate the phase accumulated by an arbitrary wavelength Since EC is parallel to AA' and R2 is parallel to R4, one can see component λ (blue line) through the prism-pair, one usually calculates thatθH is also the angle between R2 and the deviated ray (blue line) in the optical path of the deviated beam along a continuous ray, starting the 1st prism, as illustrated in figure 5: from point A, through multiple refractions in the 2nd prism until a απ nd θ=+− θ − δθ [6] final reference plane perpendicular to the beam, located after the 2 HB22 prism where all the colors are parallel [29,30]. This method results Substituting Eq. 3 into Eq. 6 obtains: in somewhat complicated expressions for the optical path, leaving little room for intuition. An elegant alternative (outlined in Figure 3) θHB=−−=− π2 θ δθ α δθ , [7] that avoids calculations of multiple refractions is to use the concept indicating that DC= H cos(α −= δθ) H cos α cos δθ +H sin2 θB sin δθ . of wavefronts for a non-continuous ray. This concept was originally presented in [31] and calculated to include only the prisms separation By substituting the above relations into Eq. 5, the total optical R. The following generalizes the calculation to include also the prism path through the prism-pair is: insertion H. P=( RH++cosα) cos δθH sin 2 θB sin δθ [8] Each of the rays within a plane-wave beam experiences the same and the optical phase experienced by frequency ω is optical path (and phase) through the prism-pair, as illustrated in ω ϕ = P [9] figure 3, starting from reference plane R1 to R5 (all the rays have the c same wavelength λ0). Note that the optical phase is constant across Note that Eq. 8 was a generalization of the method presented in each of the reference planes (dashed black lines), since each of them [31] in which only the special case of H = 0 is presented. coincides with the wavefront perpendicular to the beam. The above The expressions in equations [8] and [9] for the optical path P argument is true not only for the Brewster ray λ0, but for any other

nsinβ20=+− ( nn δ) sin ( β0 δβ )

≈+nn00 sinβ δ sin β 0 − δβ n 0 cos β 0

=+−nn00 cosθBB δ cos θ δβ n sin θ B [15]

sinθout = sin( θB += δθ) sin θB cos δθ + cos θB sin δθ

≈+ sinθBB δθ cos θ Equating both sides yields

nn00cosθB+− δ cos θ B δβ n sin θ BBB =+ sin θ cos θ δθ [16] sinθ Figure 6: Definition of the refraction angles for small angle approximation of Dividing Eq. 16 by B provides the angular dispersion. δ n δθ 11+−n0δβ =+ nn00 and optical phase φ are exact and can be easily used for numerical δ−=2 δβ δθ calculation of the spectral phase φ(ω) and its frequency derivatives nn0 [17] GD= dϕω/ d , GDD= d22ϕω/ d and higher order dispersion terms. The Substituting Eq. 13 into Eq. 17 yields refraction angle δθ for each wavelength/frequency can be calculated according to Snell’s law by using the prism’s apex angle and the δθ=2 δ n = 2( nn − 0 ) [18] Sellmeier formula for either n(λ) or n(ω). Finally, Eq. 18 can be substituted into Eq. 8 and Eq. 9 to retrieve Approximation of the Prism Angular Dispersion the total wavelength dependent phase through the prism-pair. Since the wavelength/frequency dependence is now only inδλn( ) , a simple Although Eq. 8 and Eq. 9 already allow complete calculation of expression for the GDD= d22ϕω/ d can be derived: the dispersion properties, much intuition can be gained by further 2 d 22ϕω24dn d n A dn developing the expressions with the assumption that the angle δθ(λ) =−δω +− 22(B2 An) 2  [19] is small. Replacing cosδθ≈− 1 δθ 2 / 2 and sinδθ≈ δθ yields: dω c dωω d cd ω

2 P≈( RH +cosα) ⋅−( 1 δθ / 2) +(H sin 2θB ) ⋅ δθ [10] where ARH= + cosα and BH= sinθB . The wavelength dependent refractive index of the prisms n can be obtained from the Sellmeier The angle δθ can now be expressed using Snell’s law assuming formula. small angles, such that: cosδθ ≈ 1andsinδθ≈ δθ . It is assumed that the beam enters the 1st prism at the minimum deviation angle for a The exact spectral phase and any of its derivatives (ddmmϕω/ ) can be easily calculated using the chain rule [32]: certain wavelength λ0; that the entrance angle matches the Brewster angle θ for λ ; and that the prism refractive index for λ is n(λ ) = n . 2 B 0 0 0 0 dnλ dn nn= +δλ n = − The refractive index for the deviated beam (λ ≠ λ0) is: 0 ( ) . dω2 πλ cd dn23λλ dn dn 2 The angle of refraction inside the prism for the Brewster beam = + 2 222 [20] dωπ22 cd λ d λ is β0 (equal at both faces of the prism). As illustrated in figure 6, the β= β + δβ angle of refraction for the deviated beam inside the prism is 10 For typical optical bandwidths it is safe to assume that B 2 Anδ . It is easy to show that the deviated beam will hit the exit face of the and that RH , since R∼−10 50 cm and H∼10 mm are common values. β= β − δβ prism at an angle of 20 . The exit angle for the deviated beam Thus, it is intuitive to think of A as the distance between the prisms θ= θ + δθ tanθ = n will be out B . In addition, since at Brewster angle B 0 and ( AR≈ ), and of B as the propagation distance inside the prisms. θπ=/2 − β sinθβ= cos cosθβ= sin B 0 , the following relations hold: B 0 and B 0 . Further approximation can be made since for most practical optical 22 Following Snell’s law from the entrance face to the exit, an materials 2/⋅⋅(dndωω)  ( dnd / ω) . Hence, the total GDD of the prism pair is the sum of two parts: the accumulated GDD from the approximation of the deviation angle δθ can be obtained. At the entrance: material dispersion due to propagation through the prisms and an additional negative GDD resulting from the geometric dispersion

sinθβB = n sin1 [11] from the path between the prisms: 2 Expanding the right hand side yields d22ϕω24 B d n A ω dn ≈− [21] dωω22 cd c d ω nsin β10=++( nn δ)sin ( β0 δβ )

=(nn00++δ)(sin β cos δβ cos β 0 sin δβ ) The geometric dispersion in the second part attributes always a negative GDD for any material (independent of the sign of dn/dω), ≈+(nnδ)(sin β + δβ cos β ) [12] 0 00 whereas the first part depends on the material dispersion, which may =++βδ βδββ nn00 sin sin 0 n 0 cos 0 be either positive (as is usually the case in the visible or NIR range), or

=++nn00 cosθδBB cos θδβθ n sin B negative (as is usually the case for the IR in most materials). Hence, a simple prism-pair offers tunable GDD, both negative and positive, for

Substituting into Eq. 11 and dividing by sinθB (and noting that visible and NIR wavelengths, but for the IR range it will commonly tanθB = n0 ) provides provide only negative GDD. Tuned positive dispersion in the IR range can still be obtained by inserting a 1x1_telescope between the prisms δ n that can flip the sign of the geometric dispersion by imaging the 11++n0δβ = n0 first prism beyond the 2nd prism, effectively generating a “negative 2 distance” R between the prisms [23]. nn0δβ= − δ [13] Following similar logic at the exit face of the prism, obtains Summary The performance of the Brewster prism-pair was reviewed - a nsinβθ2 = sinout [14] common major component of ultrafast spectroscopy apparati. The Expanding both sides of Eq. 14 yields total spectral phase was calculated as accumulated by broadband light

Citation: Shaked Y, Yefet S, Pe’er A (2016) The Prism-Pair: Simple Dispersion Compensation and Spectral Shaping of Ultrashort Pulses. Int J Exp Spectroscopic Tech 1:007 Shaked et al. Int J Exp Spectroscopic Tech 2016, 1:2 ISSN: 2631-505X | • Page 4 of 4 • in passage through the prism-pair. Following a simple and intuitive 15. Vered RZ, Rosenbluh M, Pe’er A (2012) Two-photon correlation of broadband- approach using the concept of wavefronts, it was showed how the amplified spontaneous four-wave mixing. Phys Rev A86: 043837. total optical path through a prism pair can be reduced to the path of 16. Gershgoren E, Bartels RA, Fourkas JT, Tobey R, Murnane MM, et al. (2003) Simplified setup for high-resolution spectroscopy that uses ultrashort pulses. a non-continuous beam, hence avoiding unnecessary computations Opt Lett 28: 361-363. of multiple refractions inside the prisms. Finally, the total phase and 17. Arissian L and Diels JC (2007) Carrier to envelope and dispersion control in any of its derivatives can be easily calculated. Specifically, by careful a cavity with prism pairs. Phys Rev A75: 013814. choice of the two degrees of freedom of the prism pair R and H, it is 18. Fork RL, Cruz CB, Becker P, Shank CV (1987) Compression of optical pulses possible to compensate two orders of dispersion simultaneously (e.g. to six femtoseconds by using cubic phase compensation. Opt lett 12: 483-485. d2φ/dω2 and d3φ/dω3), which is crucial when using ultra-broadband 19. Spence D, Sibbett W (1991) Femtosecond pulse generation by a dispersion- light to maintain high temporal resolution [23]. compensated, coupled-cavity, mode-locked ti: sapphire laser. JOSA B 8: 2053-2060. References 20. Ogilvie JP, Debarre D, Solinas X, Martin JL, Beaurepaire E, et al. (2006) Use 1. Hargrove LE, R. Fork L, Pollack MA (1964) Locking of hene laser modes of coherent control for selective two-photon fluorescence microscopy in live induced by synchronous intracavity modulation. Appl Phys Lett 5: 4-5. organisms. Optics express 14: 759-766. 2. Spence DE, Kean PN, Sibbett W (1991) 60-fsec pulse generation from a self- 21. Entenberg D, Wyckoff J, Gligorijevic B, Roussos ET, Verkhusha VV, et al. mode-locked ti: sapphire laser. Opt Lett 16: 42-44. (2011) Setup and use of a two laser multiphoton microscope for multichannel intravital fluorescence imaging. Nat protoc 6: 1500-1520. 3. Zewail AH (2000) Femtochemistry: Atomic-scale dynamics of the chemical bond. J Phys Chem A 104: 56605694. 22. Kobayashi T, Yoshizawa M, Stamm U, Taiji M, Hasegawa M (1990) Relaxation dynamics of photo-excitations in polydiacetylenes and polythiophene. J Opt 4. Tannor DJ, Kosloff R, Rice SA (1986) Coherent pulse sequenceinduced Soc Am B 7: 1558-1578. control of selectivity of reactions: Exact quantum mechanicalcalculations. J chem phys 85: 5805-5820. 23. Shaked Y, Yefet S, Geller T, Peer A (2015) Octave-spanning spectral phase control for single-cycle bi-photons. New Journal of Physics 17: 073024. 5. Shapiro M, Brumer P (1996) Coherent control of collisional events: 24. Shaked Y, Pomerantz R, Vered RZ, Pe’er A (2014). Observing the non- Bimolecular reactive scattering. Phys rev lett 77: 2574. classical nature of ultra-broadband bi-photons at ultrafast speed. New 6. Hansch TW (2006) Nobel lecture: passion for precision. Reviews of Modern Journal of Physics 16: 053012. Physics 78: 1297. 25. Sensarn S, Yin GY, Harris SE (2010) Generation and compression of chirped 7. Dhar L, Rogers JA, Nelson KA (1994) Time-resolved vibrational spectroscopy bi-photons. Phys Rev Lett 104: 253602. in the impulsive limit. Chem Rev 94: 157-193. 26. Peer A, Silberberg Y, Dayan B, Friesem AA (2006) Design of a high power continuous source of broadband down-converted light. Phys Rev A 74: 053805. 8. Assion A, Baumert T, Bergt M, Brixner T, Kiefer B, et al. (1998) Control of chemical reactions by feedback optimized phase-shaped femtosecond laser 27. Bessire B, Bernhard C, Feurer T, Stefanov A (2014) Versatile shaper-assisted pulses. Science 282: 919-922. discretization of energy-time entangled photons. New Journal of Physics 16: 033017. 9. Aharonovich I, Peer A (2016) Coherent amplification of ultrafast molecular dynamics in an optical oscillator. Phys rev lett 116: 073603. 28. Hecht E (1987) Optics (Addison Wesley).

10. Weiner AM, Heritage JP, Kirschner EM (1988) High-resolution femtosecond 29. Yang S, Lee K, Xu Z, Zhang X, Xu X (2001) An accurate method to calculate pulse shaping. J Opt Soc Am B 5: 1563-1572. the negative dispersion generated by prism pairs. Optics and Lasers in Engineering 36: 381-387. 11. Dayan B, Peer A, Friesem AA, Silberberg Y (2005) Nonlinear interactions with an ultrahigh flux of broadband entangled photons. Phys rev lett 94: 043602. 30. Arissian L, Diels JC (2007) Carrier to envelope and dispersion control in a cavity with prism pairs. Phys Rev A75: 013814. 12. Weiner AM Ultrafast (2011) optical pulse shaping: A tutorial review. Opt 31. Fork RL, Martinez E, Gordon JP (1984) Negative dispersion using pairs of Comm 284: 3669-3692. prisms. Opt Lett 9: 150-152. 13. Weinacht T, Bartels R, Backus S, Bucksbaum P, Pearson B, et al. (2001) 32. Cojocaru E (2003) Analytic expressions for the fourth- and the fifth-order Coherent learning control of vibrational motion in room temperature molecular dispersions of crossed prisms pairs. Appl Opt 42: 6910-6914. gases. Chem Phys Lett 344: 333-338. 14. Oron D, Dudovich N, Yelin D, Silberberg Y (2002) Narrow-band coherent anti- stokes Raman signals from broad-band pulses. Phys Rev Lett 88: 063004.

Citation: Shaked Y, Yefet S, Pe’er A (2016) The Prism-Pair: Simple Dispersion Compensation and Spectral Shaping of Ultrashort Pulses. Int J Exp Spectroscopic Tech 1:007