Analysis of phase sensitivity for binary computer-generated …€¦ · Analysis of phase...

Analysis of phase sensitivity for binarycomputer-generated holograms

Yu-Chun Chang, Ping Zhou, and James H. Burge

A binary diffraction model is introduced to study the sensitivity of the wavefront phase of binarycomputer-generated holograms on groove depth and duty-cycle variations. Analytical solutions to dif-fraction efficiency, diffracted wavefront phase functions, and wavefront sensitivity functions are derived.The derivation of these relationships is obtained by using the Fourier method. Results from experimentaldata confirm the analysis. Several phase anomalies were discovered, and a simple graphical model of thecomplex fields is applied to explain these phenomena. © 2006 Optical Society of America

OCIS codes: 050.1380, 090.2880, 120.2880.

1. Introduction

Computer-generated holograms (CGHs) are diffractiveoptical elements synthesized with the aid of comput-ers. CGHs use diffraction to create wavefronts of lightwith desired amplitudes and phases. As a null correc-tor, CGH is widely used in aspheric surface testing.1,2

A common type of CGH is a binary hologram, whichconsists of patterns of curved lines drawn onto oretched into glass substrates. The patterns in a binaryCGH may be interpreted as bright and dark interfer-ence fringes. A binary CGH can store both the ampli-tude and phase information of the complex wavefrontsby controlling positions, widths, and groove depths ofthe recorded patterns. The design of a CGH is sepa-rated into two parts that allow the independent deter-mination of amplitude and phase of the diffractedlight. The structure of the lines, such as the groovedepth, or duty cycle, is set to give the appropriatediffraction efficiency. The overall pattern of lines ischosen to give the correct shape of the wavefront. Forprecise measurement, the error influence must be wellknown and characterized. The CGH pattern errors andsurface figure errors of the substrate have been spec-ified and separated by the absolute testing method.3,4

Nevertheless, the sensitivity of phase to the duty cycle

and groove depth, which is also very important foroptical testing, is less known.

We investigate the effects of duty-cycle and phasedepth variations of CGHs on the diffracted wavefrontphase. A binary diffraction model, which is assumed tobe in the scalar regime, is used for analysis. Diffractionof the wavefront phase is expressed mathematically asa function of the duty cycle and phase depth. Analyt-ical solutions of wavefront phase sensitivity to duty-cycle and phase depth deviations are derived. The datafrom the experiments confirm the analysis results ofthe model. Also, a graphical representation of diffrac-tion fields is introduced, which provides an intuitivemethod of diffraction wavefront analysis. Both ampli-tude and phase values of the diffracted wavefront canbe easily obtained from this graphical representation.

2. Diffraction Model

The simplest form of a hologram is a linear diffractiongrating, where the spatial frequency of the gratingpattern is constant over the entire hologram. A CGHwith variable fringe spacing may be viewed as a col-lection of linear gratings with variable spatial fre-quencies. By controlling the spatial frequencies ofthese linear gratings across the CGH, the incidentlight can be deflected into any desired form. The per-formance of a CGH may therefore be directly relatedto the diffraction characteristics of a linear grating.5Linear gratings are often used for studies on CGHproperties in order to avoid mathematical difficultiesin modeling complicated holograms. They are alsochosen as the model for our work.

The linear grating model used in this study is as-sumed to have binary amplitude and phase distribu-tions. A binary linear grating has a surface relief

Yu-Chun Chang is with KLA-Tencor Corporation, One Technol-ogy Drive, Milpitas, California 95035. Ping Zhou ([email protected]) and J. H. Burge are with the College of Optical Sci-ences, University of Arizona, Tucson, Arizona 85721.

Received 30 June 2005; accepted 30 November 2005; posted 16December 2005 (Doc. ID 63100).

0003-6935/06/184223-12$15.00/0© 2006 Optical Society of America

20 June 2006 � Vol. 45, No. 18 � APPLIED OPTICS 4223

profile that may be described as an infinite train ofrectangular pulses with a uniform width. The wave-length of the incident light is assumed to be muchsmaller than the grating period (S), so the scalardiffraction approximations can be applied.6 For a pla-nar wavefront at normal incidence, the output wave-front immediately past the grating, either reflected ortransmitted, can be expressed as a simple product ofthe incident wavefront function and the grating sur-face profile function. The complex amplitude of the

field, immediately after the grating, can be written as

u�x� � A0 � �A1 exp�i�� A0�rect�xb�1

S comb�xS�, (1)

where A0 and A1 correspond to the amplitudes of theoutput wavefronts from the peaks and valleys of thegrating, respectively. The values of A0 and A1 aredetermined by the amplitude functions of the reflec-tance or the transmittance coefficients at the gratinginterface, using Fresnel equations.7 The phase func-tion � represents the phase difference between lightfrom the peaks and that from the valleys of the grat-ing. The form of the output wavefront function,shown in Fig. 1, resembles the shape of the gratingprofile.

Based on Fraunhofer diffraction theory, the far-field diffraction wavefront may be related to the orig-inal wavefront by a simple Fourier transformrelationship. Hence the far-field wavefront function

of a normally incident planar wavefront upon thegrating described in Eq. (1) is

U�� u�x�� A0�� A1 exp�i��

� A0�b sinc�b�� comb�S��

�A0�� A1 cos��

� A0�D sinc�DS�� m��

�

�� mS ��

� iA1 sin��D sinc�DS�� m��

�

�� mS ��;

(2)

looking at one order at a time,

where � gives the component of the diffracted light inthe direction of � � m��S; the duty-cycle D of thelinear grating is defined as D � b�S; and m is thediffraction order. Equation (2) shows that the diffrac-tion wavefront function U�� has nonzero values onlywhen � has a value that is an integer multiple of 1�S.This behavior describes the existence of multiple dif-fractive orders. This equation also serves as a foun-dation for the rest of this study.

A. Diffraction Efficiency

The diffraction efficiency of a specific diffraction orderdescribes the efficiency of the hologram in deflectinglight into a particular direction. Diffraction efficiency� is defined as the ratio of the intensity of the dif-

��A0 � �A1 cos�� A0�D � i�A1 sin��D, m � 0��A1 cos�� A0�D sinc�mD� � i�A1 sin��D sinc�mD�, m � 1, 2, . . .,

Fig. 1. Output complex wavefront immediately after the diffrac-tion grating.

Fig. 2. Diffraction efficiency of a phase grating as a function of theduty cycle and phase depth for the zero-order diffraction beam.

Fig. 3. Diffraction efficiency of phase grating as a function of theduty cycle for the non-zero-order diffraction beams with 0.5� phasedepth.

4224 APPLIED OPTICS � Vol. 45, No. 18 � 20 June 2006

fracted wavefront to that of the incident wavefront:

��U��2

�U0��2. (3)

Assume the intensity of the incident wavefront|U0��|2 is unit. Then, for the zero diffraction order�m � 0�,

�m�0 � A02�1 � D�2 � A1

2D2 � 2A0A1D�1 � D�cos��.(4)

For nonzero diffraction orders �m � 0�,

�m�0 � �A02 � A1

2 � 2A0A1 cos��D2 sinc2�mD�.(5)

B. Wavefront Phase

The diffracted wavefront phase function may also beachieved from Eq. (2). The phase function � is de-fined as the ratio of the imaginary part to the real

part of the complex wavefront U��:

tan� � �Im�U��Re�U��

. (6)

For any single order of diffraction, the phase � canbe interpreted as the phase of the plane wave at thegrating.

Substituting Eq. (2) into Eq. (6) leads to

m � 0 : tan� �m�0

�DA1 sin��

A0�1 � D� � A1D cos��, (7)

m � 1, 2, . . . : tan� �m�0

�A1 sin��sinc�mD�

��A0 � A1 cos��sinc�mD�. (8)

The phase can be obtained by taking the arctan-gent of these equations, and the unit of � is radians.The wavefront phase W in waves can be obtained by

W �

2�. (9)

Notice that the sinc�mD� functions are left in boththe numerator and denominator of Eq. (8). They needto preserve the sign information for the phase un-wrapping process. The phase unwrapping processuses the sign information of the real and imaginaryparts of the complex wavefront U��, which allowscalculation of phase from the 0 to 2� period. The newphase values are then evaluated point by point. Anyinteger multiple of 2� can be added or subtractedfrom the phase value to make the phase functioncontinuous.

Fig. 4. Diffraction wavefront phase of phase grating as a function ofthe phase depth and duty cycle for the zero-order diffraction beam.

Fig. 5. Wavefront phases versus phase depth for first- and second-order diffraction beams for the phase grating.


Fig. 6. Wavefront phase versus duty cycle for nonzero diffraction orders at 0.5� phase depth from phase grating.

Fig. 7. Analytical result for the wavefront phase sensitivity ofphase grating to phase depth variation for the zero-order beam.

Fig. 8. Analytical result for the wavefront phase sensitivity ofphase grating to the duty cycle for the zero-order beam.


C. Phase Sensitivity to the Duty Cycle

We have shown that the diffracted wavefront phasecan be expressed as a function of the duty cycle andphase depth. By taking the first derivative of Eq. (7)and Eq. (8) with respect to either the duty cycle orphase depth, the phase deviations due to the varia-tions of these two parameters can be determined.These deviations are defined as the wavefront sensi-tivity functions. Equations (10) and (11) show thewavefront sensitivity functions with respect to thegrating duty cycle:

m � 0 :� m�0

�D

�1

1 � �tan� �m�0�2

� tan� �m�0

�D

�A0A1 sin �

A12D2 � A0

2�1 � D�2 � 2A0A1D�1 � D�cos �,

(10)

m � 1, 2, . . . :� m�0

�D

�1

1 � �tan� �m�0�2

� tan� �m�0

�D

��, for sinc�mD� � 0,0, otherwise. (11)

Notice that duty-cycle errors produce wavefrontphase errors only in the zero-order diffraction. Thediffraction wavefront phase is not sensitive to CGHduty-cycle errors at nonzero diffraction orders. Forthe nonzero orders, diffraction efficiency is zero whensinc�mD� equals to zero, and the sensitivities to theduty cycle go to infinity at those points. Otherwise, inEq. (8), sinc�mD� in both numerator and denominatorcan be cancelled, and the wavefront phase is not sen-sitive to the duty cycle.

D. Phase Sensitivity to Phase Depth

Similarly, wavefront phase sensitivities to phasedepth at different diffraction orders are given as

m � 0 :� m�0

��

�1

1 � �tan� �m�0�2

� tan� �m�0

��

�A1

2D2 � A0A1D�1 � D�cos �

A12D2 � A0

2�1 � D�2 � 2A0A1D�1 � D�cos �,

(12)

m � 1, 2, . . . :� m�0

��

�1

1 � �tan� �m�0�2

� tan� �m�0

��

�A1

2 � A0A1 cos �

A12 � A0

2 � 2A0A1 cos �. (13)

The wavefront sensitivity functions [Eqs. (10)–(13)]provide a means of calculating the phase changes inthe wavefront that result from duty-cycle or phasedepth variations. They can be used to identify holo-gram structures that are the most or the least sensi-tive to duty-cycle and groove depth fabricationuncertainties. The information may also be used toestimate error budgets for the applications of CGHs.8

The sensitivity functions can be evaluated directlyto give the wavefront error due to variations in theduty cycle or phase depth:

�WD �1

2�

�

�D �D �1

2�

�

�D��DD �D, (14)

�W� ��

��

�

��

� ��, (15)

where �D is the duty-cycle variation across the grat-ing; �WD is the phase variation in waves due to duty-cycle variation, �� is the phase depth variation inradians across the grating, and �W� is the phasevariation in waves due to phase depth variation.

As long as the duty-cycle and phase depth varia-tions vary over spatial scales that are large comparedto the grating spacing, we can use the functions aboveto determine the coupling between fabrication errorsand system performance.

3. Numerical Simulations

A linear phase grating with an index of refraction of1.5 is analyzed with the diffraction model derived inthe Section 2. Phase gratings are typically made byetching the grating patterns onto a bare glass sub-strate, and modulate only the phase function of theincident wavefront. For this analysis, the grating isused in reflection, so the groove depth is half of thephase depth �. If the grating works in transmission,then the groove depth is equal to the phase depth �.An ideal reflection is assumed in our study, soA0 � A1 � 1.

Fig. 9. Experimental setup for diffraction wavefronts measure-ments of phase grating.


Fig. 10. Design layout of the sample phase grating.

Fig. 11. Diffraction wavefront phase sensitivities to phase depth for the zero-order beam. Experimental data (vertical bar) versustheoretical results (solid line).


A. Diffraction Efficiency

Diffraction efficiencies for the phase grating are calcu-lated using Eqs. (4) and (5). Figure 2 shows the diffractionefficiency as a function of both the duty cycle and phasedepth for the zero-order beam. The minimum value ofdiffraction efficiency occurs at the point where the phasedepth is ��2 and the duty cycle is 50%. The distribu-tion is symmetric about this point.

Diffraction efficiencies for non-zero-order diffrac-tion fields �m � 0� vary periodically with changes inthe grating duty cycle. Figure 3 shows the diffractionefficiencies of different orders with the phase depth of��2. Moreover, the number of times that maximumefficiency values reach for each diffraction beam isequal to the order number of that beam.

B. Wavefront Phase

Wavefront phase distributions are obtained fromEqs. (7) and (8). Figure 4 shows the diffracted wave-

front phase as a function of the phase depth and theduty cycle for the zero-order beam. The phase curvesat all duty cycles are continuous as a function ofphase depth, with the exception of the 50% duty cycle.An anomalous phase discontinuity is observed alongthe 50% duty-cycle line at the ��2 phase depth point.It implies a phase reversal in the diffracted wavefrontwhen the phase depth reaches the ��2 point. Thediscontinuity also indicates a high sensitivity of thewavefront phase function in the zero diffraction orderto groove depth fabrication errors for gratings with a50% duty cycle and ��2 phase depth. Further discus-sions of this discontinuity are given in Section 5,using the graphical complex field representation.

Phase functions for non-zero-order beams �m � 0� areshown in both Figs. 5 and 6. Figure 5 gives the rela-tionship between the wavefront phase and the phasedepth for all non zero orders. It is shown that thewavefront phase varies linearly with the phase

Fig. 12. Diffraction wavefront phase deviations per 1% duty-cycle variation for the zero-order beam. Experimental data (vertical bar)versus theoretical results (solid curve).


depth. It is also obvious that the dependence of wave-front phase on variations in phase depth errors isconstant.

Figure 6 shows the relationship between the wave-front phase and duty cycle for non-zero-order beams.Notice that the phase functions exhibit ��2 disconti-nuities for diffraction orders higher than 1. A ��2phase discontinuity occurs at a 50% duty-cycle pointfor the second-order diffraction beam. Two ��2 phasesteps occur for the third-order diffraction, at 33.3%and 66.6% duty-cycle points. These phase jumps arethe direct results of the sign changes of the sinc�mD�function in Eq. (8). The alteration between positiveand negative values of the sinc function causes thecalculated phase values to change by � rad. Thenumber of discontinuities that occur when the grat-ing duty cycle varies from 0% to 100% is equal to thediffraction order number minus one. Figure 6 alsoimplies that deviations in the grating duty cycle have

no effect on the wavefront phase for non-zero-orderbeams.

C. Wavefront Phase Sensitivity

In Subsections 2.C and 2.D, the analytical solutionsof diffraction wavefront phase sensitivities to duty-cycle and phase depth variations are derived. Basedon these equations, the diffracted wavefront phasesensitivities can be represented in a graphicalformat.

Figure 7 shows the wavefront phase sensitivity tothe phase depth for the zero-order beam. Gratingswith various duty cycles are shown. A 0.1� phasedepth error will produce a �0.075� wavefront phaseerror for the phase grating with a 30% duty cycle anda nominal phase depth of 0.5�. The same phase deptherror produces a �0.03� wavefront phase error forthe phase grating with a 30% duty cycle and a nom-inal phase depth of 0.1�. It is also realized that phase

Fig. 13. Wavefront phase sensitivity functions per phase depth variations for the first-order diffraction wavefront. Experimental data(vertical bars) versus theoretical data (solid curve).

Fig. 14. Wavefront phase sensitivity functions per 1% duty-cycle variations for the first-order diffraction wavefront of phase grating.Experimental data (vertical bars) versus theoretical data (solid curve).


Fig. 15. Interferogram indicating diffraction efficiency distribution and wavefront phase as a function of the duty cycle at differentdiffraction orders. The top chart shows the corresponding theoretical values.


gratings with a 50% duty cycle are extremely sensi-tive to the phase depth variations at a 0.5� nominalphase depth.

Figure 8 illustrates the wavefront phase changesthat result from 1% deviations in the duty cycle. Forinstance, a 1% variation will result in �0.004 wavesof phase error in the zero-order beam for phase grat-ing with 30% duty cycle and 0.6 waves nominal phasedepth.

Using the wavefront sensitivity functions, wave-front phase and efficiency errors can be estimatedwith knowledge of the uncertainties from the CGHfabrication processes. The wavefront sensitivity func-tions can also be used for guidance for the design ofCGHs to reduce or eliminate wavefront errors pro-duced by fabrication uncertainties.

4. Validation of Model

To verify the wavefront prediction by the model, twosample phase gratings were fabricated. A phase-shifting Fizeau interferometer was used to measurethe diffraction wavefronts produced by the samplegratings, as shown in Fig. 9. The sample grating wastreated as a flat test plate, and a transmission flat

with ��20 accuracy �� 632.8 nm� was used as thereference. The distance between the test grating andreference flat was minimized during the experimentsin order to reduce air turbulence in the test beampath.

The first sample phase grating was designed for areference wavelength of 0.633 �m. The sample grat-ing is divided into a 5 � 11 array, where each cell inthe array contains a linear grating with the specificduty cycle and phase depth, as shown in Fig. 10.Along the rows of the grating array, the duty cyclevaries in 2% increments from 44% to 60%. Duty-cyclegratings of 0% and 100% are placed at the first andthe last cell of each row in the grating array. Groovedepths or half-phase depths of the columns are 0.22�,0.24�, 0.25�, 0.27�, and 0.29�.

For the zero-order measurement, the samplegrating was aligned parallel to the reference surface.Optical interferometric measurements provide infor-mation on the wavefront phase of the test beam rel-ative to the reference wavefront. In order to avoidunnecessary complications or confusion when com-paring the experimental results with the theoreticaldata, phase deviations between adjacent grating cellswere used in our analysis instead of the absolutephase values. The phase deviation functions in Sec-tion 5 represent wavefront phase changes as a resultof variations in the grating duty cycle or the phasedepth.

Linear gratings with the same duty cycle on thesample grating array were measured together to de-termine wavefront phase sensitivities as a function ofphase depth variations. Measured wavefront phasesensitivities to the phase depth for gratings with var-ious duty cycles are shown in Fig. 11. Theoreticalwavefront phase sensitivities are overlaid on theexperimental results. The error bars of the experi-mental data provide information on the noise andconfidence level of our results. A large standard de-viation of the data implies a high noise level or a largeamount of random errors in our measurements. Forthe phase sensitivities to phase depth analysis, theexperimental results agree well with the theoreticaldata in most cases.

Gratings with the same groove depth were mea-

Fig. 16. Graphical representation of the complex diffractionwavefront produced by a phase grating with 40% duty cycle for thezero-order beam.

Fig. 17. Graphical representation of the complexdiffraction fields at five different diffraction orders fora phase grating with a 40% duty cycle.


sured together in order to determine wavefront phasesensitivities as a function of grating duty-cycle vari-ations. Wavefront phase deviations per 1% duty-cyclevariations for gratings with various phase depths aregiven in Fig. 12. Although the measured data did notagree perfectly with the theoretical predication, gen-eral behaviors hypothesized by our analytical modelswere clearly observed. And the measured wavefrontsensitivities to the duty cycle are within the theoret-ical scale. Random errors, such as the substrate non-flatness and measurement errors, may result in thedeviations of the measured data from the theoreticalcalculations.

To measure the first diffraction order wavefront,the sample grating was tilted with respect to thereference flat, so that the first-order beam was

seen by the interferometer [Fig. 9(b)]. Our analyticalmodel predicts constant sensitivity to phase depthvariations for all non-zero-order diffraction beams.This agrees with the measured data, shown inFig. 13.

The measured wavefront sensitivities to gratingduty-cycle variations are shown in Fig. 14. Accordingto our analytical model, the diffracted wavefrontphases are not sensitive to grating duty-cycle varia-tions. The measured data agree well with this theo-retical predication.

The second grating sample was fabricated withduty cycles ranging from 0% to 100%, in 10% incre-ments and with a constant groove depth. As shown inFig. 3, the diffraction efficiency for nonzero beamsvaries periodically with changes in the grating dutycycle, and the number of times that maximum effi-ciencies can be obtained for each diffraction order isequal to the order number. Also recall Fig. 6 showsthat phase discontinuity occurs when sinc�mD� � 0.These predictions can be verified directly with thehelp of interferograms obtained from the samplegrating, as shown in Fig. 15. Parallel fringes seenacross the vertical direction (y direction) of all inter-ferograms indicate the tilt of the sample gratingin the vertical direction. The tilt of the sample gratingin the horizontal direction was eliminated. Observedintensity levels along the x direction of the inter-ferograms correspond directly to the diffraction effi-ciencies of gratings with various duty cycles at eachdiffraction order. A normalized intensity distributionplot based on the theoretical diffraction efficiency isabove each interferogram. The phase reversal effectsbetween different grating duty cycles are also in-cluded in the intensity plots. Discontinuities of thefringes in the interferograms correspond to the dis-continuities of the phase function, or phase reversal.As seen from Fig. 15, observed intensity distributionfunctions agree perfectly with the theoretical predic-tions. Periodicity of the diffraction efficiency func-tions for the non-zero-order beams produced by thesample gratings can be clearly identified. Phase re-versals, or ��2 phase discontinuities, are also ob-served in the interferograms.

5. Graphical Representation of Diffraction Fields

A. Representation of Diffraction in the Complex Plane

Diffraction fields may be represented by plottingthe real and the imaginary parts of the diffractionwavefronts in a complex coordinates. This graphicalrepresentation provides an intuitive view of the dif-fracted wavefront phase function. Both diffraction ef-ficiency and wavefront phase values may be easilyobtained from the plot. Figure 16 gives a demonstra-tion of this graphical representation. The circle in theplot contains the complete solutions of a phase grat-ing with a 40% duty cycle at the zero-order diffrac-tion. Each point in the circle corresponds to differentvalues of phase depth. A vector �r� pointing from theorigin of the complex coordinate to a point on thecircle corresponds to a solution of the diffraction fields

Fig. 18. Graphical representation of the complex diffraction fieldsin zero diffraction order produced by a phase grating with a 50%duty cycle.

Fig. 19. Graphical representation of the complex diffraction fieldsin zero diffraction order produced by a phase grating with 45% and55% duty cycles.


produced by the grating at a specific phase depth. Asthe grating phase depth increases, the tip of vectortravels along the circle in the counterclockwise direc-tion that is indicated by the arrow in the figure. Themagnitude of the vector �r� gives the amplitude of thediffraction field, while the angle between the vector�r� and the real axis gives the phase value � � of thediffraction field.

As an example, in Fig. 16 the vector r is pointing tothe � � 0.2� point on the circle. This point corre-sponds to the zero-order diffraction beam of a phasegrating with a 40% duty cycle and 0.2 waves ofphase depth. The magnitude of the vector is 0.82 andthe corresponding efficiency is � 0.822 � 67%. Theangle between the vector r and the real axis isapproximately 28°, which gives a wavefront phase � 28°�360° � 0.077. Figure 17 illustrates thegraphical representations of the complex diffractionfields for both zero and nonzero diffraction orders.Those graphical representations can be directly de-rived from Eq. (2).

B. Graphical Explanation of Phase Anomaly

One advantage of the complex field plot is that anom-alous phase discontinuities can be explained easily.Recall Fig. 4 where a phase discontinuity was ob-served for the zero-order diffraction beam along the50% duty-cycle line at a ��2 phase depth. The expla-nation of the phase discontinuity was not obviousfrom the derived equations. However, it may be ex-plained directly using the complex field representa-tion.

Figure 18 gives the graphical representation of thezero-order diffraction field at 50% duty cycle with thephase depth varied from zero to one wave. The zerophase depth point is located on the real axis one unitaway from the origin. As the phase depth increases,the solution point travels counterclockwise alongthe circle. For the phase depth of 0.45� and 0.55�, thewavefront phases are 81° and �81°, respectively. Thephase change is about 162°. When the vector �r�passes the origin, where the phase depth equals to��2, the angle of the vector varies abruptly from �90°to �90°, or the wavefront displays a � phase change.At this point, the diffraction efficiency of the gratingis zero. Therefore, the apparent discontinuity inphase is actually a continuous transition in the com-plex field.

Figure 19 shows how the wavefront phase changeswith respect to the duty cycle at the 0.5� phase depth.For the duty cycles of 45% and 55%, the wavefrontphases are 0° and 180°, respectively. For duty cyclesof less than 50%, the wavefront phases at the 0.5�phase depth are 0°. Likewise, the wavefront phasesare 180° for duty cycles greater than 50%. It meansthere is a discontinuity at the point where the dutycycle is 50%.

6. Conclusion

We have shown that the method of Fourier analysiscan be applied to determine wavefront phase as wellas amplitude and diffraction efficiency. The validityof this analysis is demonstrated by direct measure-ment of wavefront phase using an interferometer. Wenoted two interesting cases where the phase canjump abruptly by a half-wave. These jumps are pre-dicted by the Fourier model and they can be readilyexplained using a graphical depiction of how the dif-fracted wavefront behaves in the complex plane.

This work has been supported in part by the East-man Kodak Company. Special thanks go to SteveArnold at Diffraction International for making thegrating samples.

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3. C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Opt. Eng. 43,2534–3540 (2004).

4. S. Reichelt, C. Pruß, and H. J. Tiziani, “Specification and char-acterization of CGHs for interferometrical optical testing,” inProc. SPIE 4778, 206–216 (2002).

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