Diffraction gratings affect the absolute phase of light in a way that is not obvious from the usual derivation of optical paths using the grating equation. For example, consider light which encounters first one and then the second of two parallel gratings. If one grating is moved parallel to its surface, the phase of the light diffracted from the grating pair is shifted by 2π each time the grating is moved by one grating constant, even though the geometric path length is not altered by the motion. This additional phase shift must be included when incorporating diffraction gratings in interferometers

Chen, Y.

Deshpande, A. J.

Mueller, G.

Quetschke, V.

Reitze, D. H.

Tanner, D. B.

Whiting, B. F.

Wise, Stacy

English

Scholarworks@UTRGV Univ. of Texas RioGrande Valley

University of Texas Rio Grande Valley ScholarWorks @ UTRGV Physics and Astronomy Faculty Publications and Presentations College of Sciences 2005 Phase Effects in the Diffraction of Light: Beyond the Grating Equation Stacy Wise Volker Quetschke The University of Texas Rio Grande Valley A. J. Deshpande G. Mueller D. H. Reitze See next page for additional authors Follow this and additional works at: https://scholarworks.utrgv.edu/pa_fac  Part of the Astrophysics and Astronomy Commons, and the Physics Commons Recommended Citation Wise, Stacy; Quetschke, Volker; Deshpande, A. J.; Mueller, G.; Reitze, D. H.; Tanner, D. B.; Whiting, B. F.; and Chen, Y., "Phase Effects in the Diffraction of Light: Beyond the Grating Equation" (2005). Physics and Astronomy Faculty Publications and Presentations. 352. https://scholarworks.utrgv.edu/pa_fac/352 This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Physics and Astronomy Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact justin.white@utrgv.edu, william.flores01@utrgv.edu. Authors Stacy Wise, Volker Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, and Y. Chen This article is available at ScholarWorks @ UTRGV: https://scholarworks.utrgv.edu/pa_fac/352 Phase Effects in the Diffraction of Light: Beyond the Grating EquationStacy Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. WhitingDepartment of Physics, University of Florida, Gainesville, Florida 32611, USAY. Chen*Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125, USAA. Tu¨nnermann, E. Kley, and T. ClausnitzerInstitute of Applied Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D-07743 Jena, Germany(Received 30 December 2004; published 27 June 2005)Diffraction gratings affect the absolute phase of light in a way that is not obvious from the usualderivation of optical paths using the grating equation. For example, consider light which encounters firstone and then the second of two parallel gratings. If one grating is moved parallel to its surface, the phaseof the light diffracted from the grating pair is shifted by 2 each time the grating is moved by one gratingconstant, even though the geometric path length is not altered by the motion. This additional phase shiftmust be included when incorporating diffraction gratings in interferometers.DOI: 10.1103/PhysRevLett.95.013901 PACS numbers: 42.79.Dj, 04.80.Nn, 42.60.DaOur understanding of diffraction gratings relies on thegrating equation [1,2], which says that if light of wave-length  is incident on a grating with grating constant d atan angle  (relative to the grating normal), then the dif-fracted light leaves the grating at a wavelength-dependentangle  satisfying dsin sin m  0, with m aninteger indicating the diffraction order [3]. One obtains thisresult by requiring that the phase of the diffracted lightfrom adjacent grating rulings or slits differs by 2m(which has the same effect as zero difference) so that thediffracted waves from all slits interfere constructively inthe direction of the diffracted beam.Given this understanding, one can imagine constructinga gadget like the one shown in Fig. 1, where a pair ofparallel and identical face-to-face gratings is used to dif-fract light twice, so that the outgoing waves are parallel to,but laterally displaced from, the incoming waves.According to the grating equation, the longer wavelengthlight is diffracted through a larger angle than the shorterwavelength light, so that the red light strikes the secondgrating to the left of the green while the blue light is to theright of the green. One can see by inspection that the totalfree-space path taken by a red ray is longer than that of agreen ray, which is in turn longer than the blue ray’s path.(In short-pulse laser applications, such arrangements ofparallel gratings are often used in pulse compressors [4].)Finally, then, if one puts mirrors at the places shown inthe figure, and adjusts the lengths of the common paths andthe intergrating spacing D correctly, it should be possibleto arrange for each color that the ratio of free-space path tothe wavelength is the same integer value. If this were so,and if the gratings had no other effect on the phase of thelight waves, then the device shown would be a cavityresonant for all wavelengths, a ‘‘white-light’’ cavity.Detailed calculations [5] show that the bandwidth of thiscavity would in fact be finite (because of the nonlineardispersion of the gratings) but would be many orders ofmagnitude larger than the bandwidth of the typical Fabry-Perot cavity, such as the ones in the arms of the LIGOdetector [6,7].In this Letter we describe measurements of the phaseshift of light by such a parallel grating set which show thatthe above concept is almost completely wrong. Instead, thepair of gratings provides a wavelength-dependent phaseshift nearly canceling the phase from the additional free-space path length shown in Fig. 1 [8]. We proved this toourselves by measuring the resonance bandwidth of Fabry-Perot cavities containing high-efficiency gratings and con-figured to be ‘‘white-light’’ cavities. We found no enhance-ment of bandwidth. Here we show why the expectedM1M2yαβD0xFIG. 1 (color). A pair of identical, parallel, and face-to-facediffraction gratings and two mirrors (M1 and M2) form aresonant cavity. The light rays approaching and leaving thegrating pair are parallel for all wavelengths but take differentpaths after the first grating. (From left to right the rays are red,green, and blue.)PRL 95, 013901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending1 JULY 20050031-9007=05=95(1)=013901(4)$23.00 013901-1  2005 The American Physical Societyenhancement does not occur. Moreover, we also show thatthe phase depends not only on the intergrating spacing, butalso on the exact relationship between the grating featuresas seen by the light. This dependence leads to the non-intuitive result that the phase is modulated strongly if oneof the gratings is translated parallel to its face, even thoughthe optical paths in Fig. 1 are wholly unaffected.The error lies with an inappropriate mix of geometricaland physical optics. Consider the effect of the parallelgrating pair on infinite plane waves. In Fig. 1, parallelreflective gratings are located in the y  0 and y  Dplanes. We will calculate the electric field at the twogratings and in a plane normal to the outgoing light (wherethe right-hand mirror in Fig. 1 is located). The light fieldimpinging from the left on the first grating isE1;in  E0eikx siny cos: (1)The grating at y  0 bestows a spatial phase modulation onthe incoming plane wave. The phase factor is eikGxx00; theperiodic function Gx x00 represents the grating profile,with origin at x00. This phase factor may be expanded in aFourier series:eikGxx00 XmCmeimgxx00; (2)where g  2=d.Each term of the series is a diffraction order. In thefollowing we consider only the m  1 order, set C1 1, and chose x00  0 for the first grating. The light fieldleaving that grating isE1;out  E0eik singxky cos; (3)where we have used the grating equation to substitutek sinmg for k sin< 0. When the light reaches thesecond grating, on the y  D line, it again receives aspatial phase modulation eimgx0x. The quantity x0 is thex offset of the second grating’s periodic modulation withrespect to that of the first grating. Note that the secondgrating is reversed relative to the first, so that its localcoordinate runs in the x direction. We again use m 1, making light of all wavelengths leave the secondgrating parallel to the incident light. The outgoing electricfield at the second grating isE2;out  E0eikx sinD cosgx0: (4)When the light finally arrives at a point x; y on the right-hand mirror of Fig. 1, the electric field will beEem  E0eikfx sinyD cosD cosggx0: (5)We now consider the phase, !; x; y  !=cx sin yD cosD cos  gx0, at M2. Thismay be written as!  !cL!  gD tan x0; (6)whereL!  x sin yD cosD1cos tan sin(7)is the total, frequency-dependent, geometric path from thefirst grating (at the origin) to the end mirror, as sketched inFig. 1. We next compute the dispersion @=@!, using thegrating equation to eliminate @=@!, and find@@! L!c: (8)Equation (8) makes it clear that the variation of phasewith frequency cannot be set to zero. Earlier calculationsby some of us [5] used only the geometric path lengthcontribution to the phase, !L!=c— the first term inEq. (6)— to predict (incorrectly) that @=@! could be-come zero, thus allowing for the possibility of a white-lightcavity. Missing from Ref. [5] was the second term in theright-hand side of Eq. (6), present due to the position-dependent phase shift that light receives upon reflectionfrom (or transmission through) the grating pair.The result of Eq. (8) is familiar to short-pulse laserphysicists as the group delay [9]. The exact form of theadditional phase shift is rarely a concern, as the pulsecompressor does not depend upon absolute phases. Toour knowledge, direct experimental verification of thephase shift’s form has never been published. The D andx0 dependence of the phase in Eq. (6) may be expressed ina particularly direct way as!  !cL0 Dcos cos  gx0 (9)where L0  x sin y cos is the perpendicular distancefrom the origin to the plane of M2. Analyzing the gratingcompressor with plane waves reveals the origin and sig-nificance of the position-dependent phase shift on reflec-tion from the gratings. We may now resume calculationswith the geometrical optical path, so long as we do notneglect the additional phase associated with the gratings.This theory makes a very specific prediction, which may beexperimentally confirmed, about how the one-way phaseshift depends on the distance D between gratings and thespatial offset x0 between grating profiles.To test that the phase shift does have the specific form ofEq. (9), we incorporated a pair of gratings into one arm of aMach-Zehnder interferometer, as shown in Fig. 2. We usedreflective gratings with 1500 grooves=mm and a designinput angle   42 (so that   68) We also used aninput angle of   50 (  57:3)in some of our trials.The gratings had a high efficiency, with 94–96% of theincident light diffracted in first order. We placed the secondgrating of our grating pair on a two-axis translation stage.This stage allowed us to vary the parameters x0 and D ofthe grating pair. We aligned the grating face with one of thePRL 95, 013901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending1 JULY 2005013901-2two orthogonal axes of the stage, which we will call x0 andy0, as well as was possible with a naked human eye. A smallmisalignment angle  inevitably remains between the grat-ing axes x; y and the stage axes x0; y0. The axes arerelated by bx0  bx cos by sin. Assuming the construc-tion of the translation stage is better than our ability toplace the grating, we also have by0  bx sin by cos. Asthe stage moves along the x0 direction, it will produce acombination of the effects on phase due to the phase’s x0and D dependence; however, because  is small, the influ-ence of x0, with a period equal to the grating period, willdominate. The converse is true when one moves the stagealong y0.To calibrate the displacement of the translation stage, weattached to it the end mirror of a simple Michelson inter-ferometer illuminated by a helium-neon laser. In fact, thereare two mirrors (and two interferometers) set perpendicularto the two motions of the stage. To ensure that perpendi-cularity, we move the stage in the orthogonal direction, sothat the mirror slews crabwise across the He-Ne beam, andadjust its angle relative to the stage until we reduce thenumber of output intensity fringes to a minimum. Again,this technique relies upon good inherent perpendicularityin the stage’s crossed axes. Whenever the stage moves, wemonitor the output intensity of both interferometers. Wequantify the stage motion by counting the He-Ne fringes.The input to the Mach-Zehnder interferometer is a1064 nm-wavelength grating-stabilized diode laser. Forgood contrast, the physical lengths of the two arms arenearly equal. We move the second grating along the x0 andy0 axes and observe the intensity fringes of the infraredinterferometer and compare with theory. While the lightinput angle  and the grating period d are known, remains as a fitting parameter. The output intensity of theMach-Zehnder is fit toIx0  A B cosx0  C; (10)wherex0  2cos cosx0 sin 2dx0 cos; (11)for the x0 motion, or toIy0  A B cosy0  C; (12)withy0  2cos cosy0 cos 2dy0 sin; (13)for the orthogonal direction. The quantities A, B, and C arerather unimportant fitting parameters; the period of theoutput fringes determined by  is key. We make a least-square fit of the theory to our data by adjusting A, B, C, and. Figure 3 shows examples of typical results for a trialwith   50.The top panel of Fig. 3 shows the interference seen formovement along the x0-direction, i.e., when the gratingmoves parallel to its face. This motion gives strong fringes;the measured fringe contrast is in the 92–94% range. Now,motion parallel to the grating face has no effect on thegeometric path lengths inside the interferometer. Thus, ourinitial expectation (based on the geometric path length)was that the light phase would be unaffected by thismotion. In contrast to this expectation, the phase of thelight goes through a full cycle as the grating is translated byan amount d.In the bottom panel of Fig. 3, we show the interferencesignal observed for motion along y0. We also plot, in0 1 2 3 4 5 6 7 8 9 1000.20.40.60.81Displacement along x´(  m)µIntensity0 1 2 3 4 5 6 7 8 9 1000.20.40.60.81Displacement along y´ (  m)µIntensityFIG. 3 (color online). (Top) Measured (crosses) and theoretical(solid line) data for motion parallel to the grating face. The lightis incident at 50. The fit to Eqs. (10) and (11) required amisalignment angle of   0:2  0:2. (Bottom) The crossesshow the data for motion perpendicular to the grating face. Ofthe two calculations, the results favor the one using Eqs. (12) and(13), with the additional phase shift on reflection (solid line) overone based on geometric path length alone (dotted line). The fitrequired a misalignment angle of   0:10  0:07.y´ actuatorMto He-Ne interferometerMstageθy´x´xGRGRMBSBSMMto photodiodelaser inputx´ act.FIG. 2 (color online). The second of a pair of parallel gratingswithin one arm of a Mach-Zehnder interferometer is placed onan x-y translation stage. Michelson interferometers with He-Nelasers monitor the motion of the stage.PRL 95, 013901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending1 JULY 2005013901-3addition to the predicted output from the theory above, theoutput intensity that would be observed if only the geo-metric path length were changed by the grating motion.Use of geometric path alone predicts a period for theinterference pattern that is different from the measuredone, whereas the theory that incorporates the position-dependent phase shift predicts the period that we mea-sured. The outcome makes clear that the setup cannot beunderstood with only the diffraction angle and the geomet-ric path length. The additional position-dependent phaseshift is real.Figure 4 shows an indication of the agreement betweenexperiment and the theory presented here. In it, we plot themisalignment angle  of the grating for a number of trials.Eleven of the twenty-three measurements are derived fromgrating motion parallel to its face, and 12 from perpen-dicular motion. In every case, we obtained high-contrastfringes with agreement with theory comparable to what isshown in Fig. 3. The error bars on each datum reflectuncertainty in ; d; ; , and the motion of the stage.Clusters of specific values for  indicate the systematicerror in the alignment of the grating on the stage, but theoverall errors are very small. The quality of the fit to themeasured interference pattern is evident in Fig. 3 and in thesmall values for the misalignment angles in Fig. 4, averag-ing   0:03  0:12, a reasonable value for alignmentby human eye.The phase of light reflected by or transmitted through adiffraction grating cannot be deduced from the gratingequation alone. That equation omits the curious result,derived above, that the absolute phase is proportional tothe distance along the grating face at which the lightstrikes. Indeed, the flat gratings behave as mirrors tiltedat angles tilt  sin1m=2d relative to the x axis shownin Fig. 1. We confirmed this theory by testing the depen-dence of light phase on the position of the grating. For agrating-compressor setup, we found good agreement be-tween this theory and the change of light phase as themirror moved both parallel and perpendicular to its face.Our result shows that white-light cavities cannot be builtfrom grating pairs. In fact, one might have conjectured thatcausality should prevent white-light cavities from beingbuilt in a much wider class of nondissipative systems—notjust grating pairs. Indeed, we have found that a pair ofprisms has a similar effect to the gratings on the phase oflight passing through them. Finally, we note that the phaseeffect discussed here is not unique to the grating pair andwould arise in an experiment utilizing a single grating. Inour arrangement, the first grating is fixed and serves topreserve the beam width and to keep the angle of the lightleaving the second grating constant as wavelength is ad-justed. Otherwise, it is equivalent to a mirror. Except for aloss of contrast, we expect that the data of Fig. 3 would beidentical if the first grating were replaced by an appropri-ately oriented mirror.We thank Albrecht Ru¨diger and Bill Kells for a carefulreading of the manuscript. Our research is supported by theNational Science Foundation through Grant No. PHY-0244902.*Present address: Max-Planck-Institut fu¨rGravitationsphysik, Am Mu¨hlenberg 1, 14476 Golm,Germany[1] M. Born and E. Wolf, Principles of Optics (PergamonPress, Oxford, 1970), 4th ed.[2] Erwin G. Loewen and Evgeny Popov, Diffraction Graingsand Applications (Marcel Dekker, New York, 1997).[3] The choices made for the signs of all the terms in thegrating equation vary widely. We take the angles aspositive when measured in the usual counterclockwisesense from the grating normal and the order to be positivewhen the diffracted wave gains parallel wave vector fromthe experience. In Fig. 1, all angles are positive and theorder is m  1. This convention is the most common.(See Ref. [2].)[4] E. B. Treacy, IEEE J. Quantum Electron. 5, 454 (1969).[5] S. Wise, G. Mueller, D. Reitze, D. B. Tanner, and B. F.Whiting, Classical Quantum Gravity 21, S1031 (2004).[6] B. Abbott et al., Nucl. Instrum. Methods Phys. Res., Sect.A 517, 154 (2004).[7] P. Fritschel, Advanced LIGO Systems Design ReportNo. LIGO-T-010075-00-d, LSC Conference 2001.[8] S. G. Rautian, Opt. Spectrosc. (USSR) 93, 934 (2002).[9] S. D. Brorson and H. A. Haus, J. Opt. Soc. Am. B 5, 247(1988).0 5 10 15 20 25−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5trialangle (degrees)FIG. 4 (color online). Misalignment angle  as determinedfrom fits of measured interference data to theory. The meaningof the symbols is as follows. Crosses: motion along x0 with  50; Circles: motion along y0 with   50; Asterisks: motionalong x0 with   42; x’s: motion along y0 with   42. Thedata in Fig. 3 are from trials 2 and 8.PRL 95, 013901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending1 JULY 2005013901-4

Phase Effects in the Diffraction of Light: Beyond the Grating Equation

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Phase Effects in the Diffraction of Light: Beyond the Grating Equation

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