Over the past couple of decades the interest in usingphase modulating spatial light modulators (SLMs)has grown rapidly.1–3 Irrespective of the application,it is critical that the SLM be characterized such thatthe phase modulation is determined as a function ofthe steerable parameter of a pixel in the SLM. Thephase response depends on the wavelength used andmay vary significantly from device to device as well asdrift slowly with time. Therefore a simple and reli-able method for obtaining the phase modulation char-acteristic should be of interest to many potentialusers of phase modulating SLMs.
Two different techniques that can be usedwhen characterizing a phase modulating SLM areinterferometer-based methods and diffractive phaseretrieval methods.4–6 Interferometer methods are wellproved, and the results obtained are nonambiguousand fairly accurate. However, the interferometer setupis inherently very sensitive to mechanical vibrations,air turbulence, and other factors that may inadver-
tently change the difference in optical path lengthsbetween the reference and the measurement arms ofthe interferometer. Also, the interferometer setup re-quires a number of additional optical components. De-pending on the situation (e.g., a setup using a low-costSLM for dynamic diffractive beam steering), the char-acterization equipment might significantly increasethe total cost and space requirements of the system.Further, since the SLM has to be placed in one of thearms of the interferometer, it is awkward to performcharacterization without removing the SLM from itsoperating position.
Alternatively, when a phase retrieval algorithm isused, dedicated patterns are placed onto the SLM,and the diffractive far-field intensity is measured,either in a few positions or over a large area. Thephase modulation of the SLM is then obtained byfitting the simulated far field of the SLM to the mea-sured data, often with the aid of a separately deter-mined amplitude modulation characteristic of theSLM. Of course, this approach requires the theoret-ical or numerical model of the SLM to be fairly accu-rate and the measured data to be relatively correctand free of noise, which is of less importance in in-terferometer measurements. On the other hand, theadvantages are a much lower sensitivity to environ-mental disturbances and a much less critical demandfor a temporally coherent light source.
Although popular phase retrieval algorithms areknown to work for almost any far-field intensity dis-tribution, a property used to advantage in the designof diffractive optical elements, they are also notorious
The authors are with the Photonics Laboratory, Department ofMicrotechnology and Nanoscience, Chalmers University of Technol-ogy, SE-412 96 Göteborg, Sweden. E-mail addresses are, for D.Engström, [email protected]; G. Milewski, [email protected]; J. Bengtsson, [email protected]; S. Galt, [email protected].
Received 1 March 2006; accepted 27 April 2006; posted 22 June2006 (Doc. ID 68587).
for their ambiguity in the retrieved phase. To reducethis type of uncertainty, it is wise to use simple pat-terns on the SLM with the number of unknownparameters as low as possible but still yielding suffi-ciently different phase response values. In this paperwe report such a low-parameter-number diffractivephase-determination method suited for SLMs. Otherresearch groups have also worked with characteriza-tion methods that can be said to belong to this cate-gory. Zhang et al.7 and McClain et al.8,9 both used asfitting data the ratio between the zeroth- and the first-order intensities of binary grating patterns shown onthe SLM. An even further simplified method that usedsolely the intensity in the zeroth order as fitting datawas presented by Delaye and Roosen.10 Unfortunately,the zeroth-order intensity of an SLM is also much in-fluenced by factors other than the modulation, perhapsmost notably the dead space, or gap, between the pix-els. The influence of this frame area surrounding eachpixel strongly depends on the physical realization ofthe SLM: whether it is transmissive or reflective,whether it uses nematic liquid crystals, ferroelectricliquid crystals, micromirrors or multiple quantumwells, etc. These artifacts may strongly influence anyphase modulation characterization that uses thezeroth-order intensity as a fitting parameter. This maybe a major reason why the referenced diffraction-basedmethods typically yield less accurate results than dointerferometer-based measurements. The accuracyalso decreases if there is strong amplitude modulationaccompanying the phase modulation. Also, restrictingthe gratings structures to be binary makes it moredifficult to create gratings and structures that yielddistinctly different intensity-versus-modulation char-acteristics. This leads to an increased ambiguity in thedetermination of the phase modulation, such that themodulation characteristics curve tends to have multi-ple possible branches, and the analysis gives no clearindication which branch is the correct one. Of course,physical plausibility and previous experience may helpto disqualify some of the mathematically feasiblebranches.
The method presented in this paper is, as men-tioned, also diffraction based but avoids sampling thezeroth order and utilizes three-level gratings, ratherthan binary (two-level) gratings, to span a larger por-tion of the modulation space. It samples a single po-sition in the far field, is ideally capable of handlingany arbitrarily strong amplitude modulation, andcorrectly determines the phase modulation withinthe full 2� rad range.
2. Diffraction-Based SLM Characterization Method
To collect information on the phase response of theSLM, we measure a single, fixed diffraction order (byusing a single photodetector in a fixed position in thefar field) for differently shaped gratings on the SLM.Here, a “grating” is a certain setting of the pixels inthe SLM that varies in one dimension (say, in the xdirection) and that repeats after a certain number ofpixels d, the grating period. Each setting of a pixelcorresponds to a certain phase modulation � of the
light traversing the pixel; the determination of � as afunction of the pixel setting is the goal of the charac-terization method. The pixel setting, denoted L, is theuser-controlled parameter that one changes in orderto, e.g., change the voltage over the pixel electrodes,in case the SLM is a liquid-crystal device. The grat-ings used in this work consist of three different seg-ments in each period, characterized by three differentpixel setting values L1, L2, and L3. One period of sucha grating is schematically illustrated in Fig. 1. Theoptical field modulation of each segment is given bythe amplitude modulations A1, A2, and A3 and thephase modulations �1, �2, and �3. Since only relativephases are of interest, such a grating involves in effectonly two unknown phase values, ��1 � �1 � �3 and��2 � �2 � �3. The geometry of the grating is definedby d, a1, and a2, where a1 and a2 are the transitionpoints, within the grating period, between the differentpixel settings.
To summarize our phase retrieval method, it con-sists of two independent sets of measurements, fol-lowed by a numerical analysis of the collected data.The first set of measurements is the determination ofthe amplitude modulation A(L) by a conventionalmethod; see Subsection 4.B. In the second set of mea-surements, the SLM gratings described above areused, and the intensity, IM, in the first diffractionorder in the far field is measured. The gratings useddiffer in their geometry, i.e., in the values a1 and a2,as well as in their settings L1 and L2. The third pixelsetting, L3, is the reference level and is therefore keptfixed. In the following the steps of the method aredescribed in more detail.
A. Phase Retrieval Algorithm
First, in the next subsection we derive the relationbetween the phase and the amplitude modulation ofthe three-level grating and the far-field intensity inthe first diffraction order. Then we show how suchmeasured intensities and corresponding grating pa-rameters may be used to retrieve the two unknownphase values ��1 and ��2.
1. Far-Field Intensity From an Amplitude andPhase Modulated GratingBy use of the parameters describing the three-levelgrating illustrated in Fig. 1, the electric field in the
Fig. 1. Example of one period of a grating consisting of threelevels with different pixel settings. Note that the staircase shape ofthe grating is only a way to illustrate the difference in phasebetween the levels; in reality the values of A1, A2, A3, �1, �2, and �3,are unknown at the beginning of the characterization.
�1 diffraction order in the far-field of the three-levelgrating can be calculated with the Fraunhofer ap-proximation, neglecting constant prefactors, as11
E�1 ��0
d
A�x�exp�i��x��exp�i 2�xd �dx
� A1 exp�i�1��0
a1
exp�i 2�xd �dx
� A2 exp�i�2��a1
a2
exp�i 2�xd �dx
� A3 exp�i�3��a2
d
exp�i 2�xd �dx
� T�C1Ar,1 exp�i��1� � C2Ar,2 exp�i��2� � C3�,(1)
where
T �d
i2�A3 exp�i�3�; (2)
k � 2��� is the wavenumber; � is the vacuum wave-length; the complex constants
C1 � exp�i2�a1�d� � 1,
C2 � exp�i2�a2�d� � exp�i2�a1�d�,
C3 � 1 � exp�i2�a2�d�, (3)
are given solely by the geometry of the grating, i.e.are independent of the amplitude and phase modu-lation of the SLM; and
Ar,1 � A1�A3, Ar,2 � A2�A3, (4)
are the amplitude modulation ratios. Thus the theo-retical intensity in the �1 diffraction order is given by
IT � �E�1�2 � �T�C1Ar,1 exp�i��1� � C2Ar,2
exp�i��2� � C3��2. (5)
2. Phase Modulation from Far-FieldIntensity ValuesTo retrieve the unknown phase values, ��1 and ��2,from Eq. (5), numerical data fitting has to be used. Tocollect a sufficient amount of data, the intensity in thefirst diffraction order is measured for a series of Ngratings. All gratings use the same three pixel set-tings, i.e., the same amplitude and phase modulationvalues, and the same period d, but differ in theirgeometries; i.e., they have different values of a1 anda2 (and thus C1, C2, and C3). This yields a correspond-ing series of measured intensities IM
�1�, IM�2�, . . . , IM
�N�.
Since we have neglected constant prefactors (whichare generally lost in the measurements), IT and IM
values are comparable only after a normalizationwith the respective value for, say, the sum of all thevalues in the series. The phase retrieval problem canthus be formulated as finding the minimum of thefunction f given by
f���1, ��2� � n�1
N IM�n�
n�1N IM
�n�
�IT
�n����1, ��2�n�1
N IT�n����1, ��2��
2
, (6)
which is a least-squares regression.Because of the strongly nonlinear relation between
the phase values ��1 and ��2 and the intensity in thefirst diffraction order, IT, it is not obvious that it ispossible to retrieve the phase modulation from themeasured intensities. Fortunately, our simulationshave shown that if IM and IT had been directly com-parable, i.e., without the need to normalize them tothe sum of all the values in the series, the correctsolution alone would always result from the minimi-zation provided that N were sufficiently large. How-ever, the normalization results in a loss of information,which in turn results in an optimization task withmore than one solution, as will be shown.
The numerical implementation of finding the mini-mum of f is done most easily by calculating f ���1, ��2�for the entire variable space ���1, ��2� with a suffi-ciently fine discretization. The solution is then the���1, ��2� pair yielding the smallest function value.Although this scheme works fine, it is much less timeconsuming to use a more intelligent optimization al-gorithm. Therefore we have implemented our solverin MATLAB, using the fminsearch function. Thisfunction utilizes the Nelder–Mead simplex algo-rithm,12 which finds the local minimum closest to agiven starting guess. To facilitate finding the globalminimum, we use a grid of 19 19 evenly spreadstarting guesses. Typically, the multiple searches re-sult in two to four different solution pairs ���1, ��2�,all of which are stored for further analysis.
B. Coupled Measurements: Reduction to OneFree Parameter
In the previous subsection the set of measurementsand the following minimization gave only the desiredrelation between phase modulation and pixel settingfor two specific values of the pixel settings, L1 and L2,and even then multiple solution pairs appeared for��1 and ��2. The situation can be resolved by per-forming what we refer to as coupled measurements.We again perform measurements with the same Ngrating geometries, but the pixel setting for level 1,L1, is changed to a new value. For instance, in ourcase the possible settings are the integer numbersL1 � �0, 1, . . . , 255�, representing digital values sentby the user to the control unit of the SLM. We suc-cessively performed the set of N measurements for
each chosen value of L1, where L1 took on the valuesL1 � �0, 10, . . . , 250�. In contrast, the setting of level2, L2, should be fixed (say L2 � 130) not only duringone measurement of the N different grating geome-tries (as is L1) but also for each different value of L1,i.e., during the full series of measurements. Likewise,of course, the reference level, level 3, is always keptfixed. The most natural choice is perhaps L3 � 0,which was adopted in our case. One of the benefits ofusing a fixed level L2 is that it becomes very simple topick the correct solution pair ���1, ��2� from the asmany as four solution pairs that typically result foreach value of L1: since ��2 should be the same irre-spective of L1, the correct solution is the pair that hasa value for ��2 that also appears among the multiplesolutions for (almost) all other L1 values. Conse-quently, a false solution has a value for ��2 that ismissing in most of the solutions for other L1 values.
Once the true value for ��2 has been established, apostoptimization is performed in which the describedminimization problem for f ���1, ��2� is solved onceagain, but this time using the found value for ��2.Since this is now only a one-parameter determinationof ��1 as a function of L1 � �0, 10, . . . , 250� (theresult of which is the desired phase response of theSLM), it is performed very quickly. Because a reliablevalue for ��2 is used (an average over several goodsolutions), this improves the precision in extractingthe correct ��1 values. This allows the determinationof ��1 as a function of L1 even if there are singularvalues of L1 for which the original (two-parameter)minimization completely failed, i.e., when none of themultiple solutions was the correct one. For the ma-jority of L1 values, for which the original optimizationincluded the correct solution among its multiple so-lutions, the postoptimization still generally gives aslight improvement of the corresponding ��1 values,since we use a better value for ��2.
Finally, one may of course ask why binary (two-level) gratings are not used rather than three-levelgratings. It is true that a binary grating has only oneunknown, ��1, the phase of the nonreference level,which at first glance should be possible to determinedirectly in a one-dimensional optimization procedure.Unfortunately, irrespective of their geometries, bi-nary gratings are too similar in their phase responseto be useful; for instance, for a given value of ��1, anytwo gratings differ in their first-order intensity val-ues only by a ��1-independent factor. This makes thefunction given in Eq. (6) independent of the test val-ues of ��1 and thus clearly not useful for obtainingthe phase response of the SLM.
3. Simulated Performance of the CharacterizationMethod
To investigate the numerical stability and perfor-mance of the described method, a plausible amplitudeand phase modulation were assumed; see Fig. 2.Based on these, the intensity IT in the first diffractionorder was calculated from Eq. (5) for different set-tings of L1 and L2. These intensity values were then
also used as the measured values IM�n� to retrieve the
phase modulation with the method described in theprevious section. Of course, in the absence of simu-lated noise the obtained phase response ��1�L1�should be equal to the assumed values, provided thatthe original minimization finds essentially all thecandidates for ��2 so that the correct one can besingled out before the postoptimization. As is seen inFig. 2, the assumed phase modulation basically in-creases linearly above a threshold setting value, witha total phase interval of 540°. The amplitude modu-lation was chosen to allow inspection of the behaviorwhen the three levels have different values of ampli-tude modulation and particularly when one valuebecomes relatively small compared with the othertwo.
As is seen from the results obtained before thepostoptimization (see Fig. 3), even given ideal datathe algorithm yields multiple solutions. However, theconstant level of ��2 is easily determined; the figureshows the ad hoc chosen boundaries within which
Fig. 2. Assumed (a) amplitude and (b) phase modulation used tosimulate the performance of the diffraction-based characterizationmethod. In (a), the range of possible noisy values, with a noiseamplitude of na � 0.10, is indicated with gray, and one specificexample is also plotted (the irregular curve).
Fig. 3. Results from the original optimization; shown are themultiple solutions for ��1 (pluses) and ��2 (squares) for each valueof L1 and the chosen boundaries (dashed lines) defining the rangeused to find the actual value of ��2.
those ��2 values are located that are used to calculatethe true ��2 value (obtained as the average). Forthese ideal data, the reason why these ��2 values arenot perfectly equal (though hardly noticeable fromthe plot) is that the minimization was interruptedbefore the problem had settled completely. Finally,the result from the postoptimization is shown in Fig.4. Shown are the original pairs of ���1, ��2� that hada ��2 between the boundaries given in Fig. 3 (plusesand squares), the average value of these ��2 values(dashed line), the postoptimized data (circles), andthe correct phase modulation (solid lines).
Note that the original optimization failed to yieldthe correct solution for a few values of L1 �L1 � 130,L1 � 170, and L1 � 240). This can occur in exceptionalcases when the found phase pair ���1, ��2� is sensi-tive to the starting guess. As expected, it also occursfor certain choices of L1, L2, and L3 for which theintended three-level grating degenerates to a binarygrating. This is the case when two phase levels coin-cide, i.e., for ��L1� � m2� � ��L2� rad or ��L1� �m2� � ��L3� rad, where m is an integer. This is alsothe case when the amplitude modulation is zero for acertain pixel value. As is seen in Fig. 4, the postop-timization manages to overcome this problem anddelivers the correct solution for all pixel settings. Ofcourse, at the setting for which the amplitude mod-ulation is exactly zero, there is no field after the pixel,and thus the phase modulation is not defined in thiscase.
Sensitivity to Noise and Measurement Errors. It iscritical that small deviations in the input data (mea-surements error due to, e.g., noise and SLM imper-fections) lead to only a small error in the retrievedphase. To investigate the stability of the phase re-trieval method, the ideal input data were disturbed.The ideal amplitude and phase modulation shown inFig. 2 was still used to calculate the ideal intensities
in the first orders, IM,ideal�n�, but random noise was
then added as
IM�n� � IM,ideal
�n��1 � N1�.
Also, noise was added to the amplitude modulationvalues used as the measured data, as
AM � ��Aideal�2�1 � N2�.
Here Aideal is the amplitude modulation shown in Fig.2(a) and N1 and N2 are random numbers uniformlydistributed in ��na, na�, where na is the noiseamplitude.
As an illustration, a noisy amplitude modulationis shown in Fig. 2(a) for the case when na � 0.10. InFig. 5 the optimization results before the postoptimi-zation are shown for the same case. As expected, theadded noise leads to a slight variation of the retrievedvalue of ��2 from one L1 value to the next, althoughideally ��2 should always be the same. The postopti-mized results are shown in Fig. 6. As can be seen, theobtained phase modulation differs slightly from theideal, and this difference tends to be larger whenthe amplitude modulation is low, i.e., for L1 220.
The simulations were repeated for a noise amplitudena between 0 and 0.20. For each value of na, the phasemodulation was determined for 50 cases of randomnoise. The resulting average standard deviation aswell as the standard deviation of the standard devi-ations between the ideal and the determined phasemodulation is plotted in Fig. 7 for each na. Since anamplitude modulation close to zero had a rather det-rimental effect on the determined phase, the stan-dard deviations in the phase were calculated for all L1values with an amplitude modulation above 0.1(dashed line) and above 0.2 (dotted line).
From the simulations it was also possible to con-clude that the choice of the constant value ��2 has an
Fig. 4. Final postoptimized phase modulation response of ��1
(circles). Also shown are those phase pairs ���1, ��2� in Fig. 3having a ��2 value between the two boundaries (pluses andsquares), the averaged values of ��2 (dashed line), and the correctphase modulation (solid lines).
Fig. 5. Multiple solutions for ��1 (pluses) and ��2 (squares) andthe boundaries for identifying the correct solution for ��2 (dashedlines) from the original optimization in the presence of noise witha noise amplitude na � 0.10.
almost negligible influence compared with the errorinduced by the noise. In all, the simulations showedthat the presented phase retrieval method is fairlyinsensitive to the disturbed input data. Thus, themethod should yield a resulting phase modulationthat is close to the correct one, even when using mea-sured input data containing small stochastic errors.
4. Measurements
To verify our method experimentally, it was used tocharacterize a commercial SLM (HoloEye PhotonicsAG13). The SLM is a reflective liquid-crystal-on-silicon
device based on a twisted nematic liquid-crystal cell.The number of pixels is 1024 768, and the SLM iscontrollable as an additional computer monitor viathe digital video interface (DVI) on a personal com-puter. The setting, L, of each individual pixel is con-trolled by an 8 bit value, thus yielding a total rangeL � �0, 1, . . . , 255�.
In the setup the amplitude and phase modulationresponse of the SLM could be altered by changingthe polarization state of the light incident on thepolarization-sensitive SLM. The polarization statecould be arbitrarily set by using a polarizer followed bya quarter-wave plate before the SLM. The behavior ofthe phase retrieval method was investigated for threedifferent sets of phase and amplitude modulation re-sponses of the SLM. Therefore three different inputpolarization states were chosen such that the obtainedamplitude modulation, as a function of the pixel set-ting L1, differed significantly for the three cases. AHe–Ne laser emitting green light at a wavelength of543.5 nm was used as the light source for the coher-ent measurements.
It should also be noted that, when considering thephysical setup, we use the term “SLM” solely for theactive liquid-crystal device, whereas the modulationproperties include, as they should, the effects of theretarder, polarizers, and beam splitter, since they arean integral part of the device to be characterized.
A. Interferometer-Based Phase Measurements
A Michelson interferometer, see Fig. 8, was used toobtain a reliable measure of the phase modulationfor reference. To reduce the phase error due to drift,turbulence, and mechanical instability, the area ofthe SLM was divided into two halves. In one half,all the pixels were set to L � 0, while in the otherhalf all the pixels were successively set to L ��0, 10, . . . , 250�. For each setting the CCD camerathus simultaneously recorded the fringe patterns for
Fig. 8. (Color online) Michelson interferometer setup. The beamis expanded by two lenses �L1 and L2) before a desired polarizationstate is generated by a polarizer �P1� and a quarter-wave plate(QWP). The beam is divided by a nonpolarizing beam splitter cube(BS). The SLM is placed in the measurement arm, while the ref-erence arm only holds a flat mirror (M). The outgoing light from theSLM and the mirror is made linearly polarized with a polarizer �P2�before the interference pattern is captured by a CCD camera.
Fig. 6. Results from a complete optimization in the presence ofnoise with na � 0.10. Those phase pairs ���1, ��2� in Fig. 3 thatwere used for the determination of ��2 are marked as pluses ���1�and squares ���2�; also indicated are the average value of ��2
(dashed line), the postoptimized ��1 values (circles), and the idealphase modulation (solid lines).
Fig. 7. Simulated standard deviation of the obtained phase mod-ulation values from the correct ones, in the presence of noise.Shown are the simulation results and corresponding fitted lines forthe cases where only values corresponding to an amplitude mod-ulation above 0.1 (circles and dashed line, respectively) and 0.2(squares and dotted line, respectively) were used. The standarddeviation within each value of na is also indicated.
both the reference setting L � 0 and one of the otherpixel settings, as shown in Fig. 9. Consequently, thephase modulation of the pixel setting was obtainedsimply by calculating the phase shift between the twopatterns.
B. Amplitude Measurements
The amplitude modulation was measured by settingall pixels of the SLM to the same value and measur-ing the intensity in the zeroth diffraction order. Thesetup used is illustrated in Fig. 10. The common pixelsetting took on the values L � �0, 10, . . . , 250�, andthe zeroth-order intensity was measured as a func-tion of the pixel setting. The amplitude modulation isthen simply the square root of the measured inten-
sity. In Fig. 11, the normalized amplitude modulationis shown for the three input polarization states.
C. Phase Measurements Based on Diffraction into theFirst Diffraction Order
When realizing the gratings on the SLM, we used agrating period d of 48 pixels. To ensure that enoughdata were collected, 19 different grating geometrieswere used for each set of three pixel values L1, L2,and L3, where the two fixed settings were kept atL2 � 150 and L3 � 0 while the actual characterizationwas made for 26 pixel settings such that L1 ��0, 10, . . . , 250�. The grating geometry is describedby the size of the grating segments having the pixelsettings L1, L2, and L3, which are shown for all 19cases in Table 1.
As can be seen, the smallest building block of anygrating is four pixels. The reason is that we wantedthe transition areas, where the LC material gradu-ally adjusts to different pixel settings, to be small incomparison with the features of the gratings. Mea-surements showed that with a building block sizesmaller than four pixels the phase modulation de-pended slightly on the size of the building block.However, when the building block was four pixels orlarger, the measured phase modulation was not af-fected by the building block size. We observed thesame behavior when the schemes in Refs. 7 and 10were used. Obviously, as the SLM technology im-proves further, the transition areas should be definedmore sharply, and thus gratings with smaller fea-tures might also be used.
1. Laser-Based Diffractive Phase MeasurementsThe setup is identical to the one used for amplitudecharacterization, as shown in Fig. 10, only now thedetector in the far-field plane from the SLM is posi-tioned at the �1 diffraction order. The output fromthe characterization method for the case with the
Fig. 9. Picture captured by the CCD camera during the inter-ferometer measurements. The upper and lower halves of the pic-ture show the fringe pattern for the pixel settings L � 0 andL 0, respectively. The somewhat curved fringes are caused by theslightly spherical backplane of the SLM.
Fig. 10. (Color online) Setup used both in the amplitude charac-terization and in the diffraction-based phase characterization. Itdiffers from the setup in Fig. 8 in that the flat mirror is removedand that a lens �L3� is used to transform the output from the deviceto its far-field intensity distribution, which is then magnified witha second lens �L4� onto a plane where a moveable detector (D) ispositioned in either the zeroth or first diffraction order.
Fig. 11. Measured normalized amplitude modulation as a func-tion of pixel setting for the three different modulation character-istics of the SLM, corresponding to the three different inputpolarization states on the SLM.
most strongly varying amplitude modulation (inputpolarization 1, as shown in Fig. 11) is shown in Fig.12. The figure shows those original ���1, ��2� pairswhose ��2 values were used to find the averagedvalue of ��2 (dashed line). The postoptimized results,representing the final output from the method, arealso shown.
The results from our method were compared withthose obtained both from interferometer measure-ments and the diffraction-based methods describedby Zhang et al.7 and Delaye and Roosen.10 In Figs.13–15 these comparisons are shown for input polar-ization states 1–3. As can be seen in the figures, thethree diffraction-based methods typically agree wellwith the interferometer-based method. However, it isalso seen that, when they disagree slightly, themethod described in this work generally performsbetter than the other two diffraction-based methods.This is particularly true for phase modulation valuesclose to multiples of 180°, an example of which isshown clearly in Fig. 15. The standard deviation ofthe phase modulation from its interferometer-basedvalue, for all 26 pixel settings, is shown in Table 2 forthe diffraction-based methods for each of the threeinput polarization states.
It is evident from Figs. 13–15 and Table 2 that thediffraction-based method described in this work andthe interferometer-based method agree very well.The remaining small discrepancy can be attributedpartly to the noise and the limited accuracy and res-olution inherent in both the diffraction-based andinterferometer-based methods. Also, SLM-specific
imperfections, such as those causing the slightlycurved fringes in Fig. 9, may have an influence. Tothis category also belongs any error caused by the factthat we did not measure with both methods simulta-neously. Since the SLM is slightly sensitive to smallvariations in the environment, its modulation perfor-mance is not perfectly identical at any two differentpoints in time.
Making further measurements (without compari-son to other diffraction-based methods) for a largenumber of different input polarization states, wefound that the described method works accurately, asreferenced to the interferometer-based method, evenfor cases where the amplitude modulation varies asmuch as between 0.2 and 1.0. Also, we tested howcritical the choice of the fixed pixel settings L2 and L3is. It was found that these settings could be chosenquite freely, provided the corresponding phase mod-ulation values were not too close to one another (i.e.,��2 should not be an integer multiple of 360°), sincethen the three-level grating in effect has only twodistinct levels. In the unlucky event that a disadvan-tageous L2 is chosen, the measurements should atleast give enough information to make it possible tochoose a more suitable L2 setting and then repeat thecharacterization procedure.
2. LED-Based Diffractive Phase MeasurementsWhen an interferometer is used, the requirement fora light source with high temporal coherence makes itnecessary to use a laser. However, diffraction alsooccurs with less temporally coherent light sources,e.g., for a white-light source filtered through a mono-
Fig. 12. Retrieved phase modulation values from the two stagesof the characterization method, for input polarization state 1. Fig. 13. Resulting phase modulation for input polarization state 1.
Table 1. Parameters of the 19 Different Grating Geometries Used (pixels)
chromator. To verify that a light source with a lowtemporal coherence is sufficient for the characteriza-tion method described in this work, a setup utilizinga light-emitting diode (LED) was used; see Fig. 16.The idea is to image the LED via the SLM. When theSLM is activated, multiple images of the LED appearin the far field, centered at the locations of the dif-fraction orders, with the power contained in a certainimage being equivalent to the intensity in the corre-sponding diffraction order.
To obtain a usable far-field pattern, i.e., to avoidadjacent diffraction orders’ overlapping, the effectiveemitting area of the LED had to be decreased. First amicroscope objective lens was used to reduce thetransverse scale 10 times, and then a pinhole (aper-ture 25 �m) was placed in the image plane to furtherreduce the effective area. To increase the measurablerange (i.e., increase the number of illuminated pixelson the CCD camera), the far-field pattern was mag-nified 10 times with a microscope objective lens be-fore the central part (containing the diffraction orders�2, �1, 0, 1, and 2) was captured with a CCD camera.
We used a surface mounted LED (Osram LT W5SG)with a central wavelength of 528 � 9 nm and a spec-tral full width at half-maximum (FWHM) of 30 nm,according to the data sheet. Note that there is a slightdifference in the light source wavelength between theLED and the laser � 530 and 543.5 nm, respectively),and thus the behavior of the quarter-wave plate differsslightly for the two light sources. Since we wanted touse the interferometer-based values for the phasemodulation as a rough reference, only input polariza-tion state 1 (with the quarter-wave plate parallel to thefirst polarizer, P1, which is equivalent to removingthe quarter-wave plate altogether) was analyzed. Thesame measurement and analysis scheme as used inthe laser-based diffraction characterization was fol-lowed. In contrast to that method, though, all inten-sities were measured by evaluating appropriatelychosen subareas of the captured CCD images.
The results obtained for the LED setup are shownin Fig. 17 for the three diffraction-based methods(that presented in this work and those by the Zhang7
and Delaye10 groups, respectively). As a reference,the results previously shown for the interferometer-based measurements using the He–Ne laser are also
Fig. 14. Resulting phase modulation for input polarization state 2.
Fig. 15. Resulting phase modulation for input polarization state 3.
Fig. 16. (Color online) LED-based setup used both in the ampli-tude characterization and in the diffraction-based phase charac-terization. A lens �L1� is used to collect the light emitted from thesurface-mounted LED. The polarization state is controlled by apolarizer �P1� and a quarter-wave plate (QWP). A microscope ob-jective �L2� is used to demagnify the LED 10 times, and an aperture(A) further reduces the effective emitting area of the LED. A thirdlens �L3� images the aperture plane via the beam splitter (BS) andthe SLM. The outgoing light is made linearly polarized with apolarizer �P2� before the far field is magnified with a microscopeobjective �L4�. The central diffraction orders are detected with aCCD camera.
Table 2. Standard Deviation of the Obtained Phase Modulation fromthe Reference Phase Modulationa
plotted. As for the laser-based measurements, thediffraction-based methods of Zhang et al.7 and Delayeand Roosen10 may yield unreliable results for certainvalues of L1, a phenomenon which is again avoidedwith the presented method. Note that the plottedinterferometer results may not necessarily representthe exactly correct solution, since these were obtainedfor a slightly different wavelength, and the operationof the liquid-crystal-based SLM is generally wave-length dependent.
5. Conclusions
We have presented a diffraction-based method forphase modulation characterization of general SLMswith more than two phase levels. The SLM is set toproduce a large number of three-level gratings, forwhich the corresponding first diffraction order in-tensities are used to deduce the phase characteris-tics of the device. A comparison with a Michelsoninterferometer-based method showed that the re-sults of the two methods are very similar. Thepresented method is also able, like interferometer-based methods, to characterize cases where theamplitude modulation of the SLM is strongly de-pendent on the pixel setting and where the mini-mum amplitude transmission is low, correspondingto an intensity transmission through the pixel ofless than 5% of its maximum value. The high predic-tion accuracy of the method, for all values of thephase modulation and for very general amplitudemodulation characteristics, implies that the majordrawbacks of previously used diffraction-based meth-ods can be avoided and that the method may beconsidered an alternative to an interferometric char-acterization.
Compared with an interferometer-based method,the major advantages of the presented method are itsinsensitivity to drift, turbulence, and mechanical in-stability and a simple optical setup that generally doesnot include any optical components in addition to thoseneeded for the intended SLM application, thus alsopermitting straightforward in situ characterization ofthe SLM. In particular, a setup using a transmissiveSLM rather than a reflective one would be very simple.Further, since the temporal coherence of the lightsource is much less critical, it is possible to character-ize the SLM by using an LED as the light source, whichwas also demonstrated. The obvious drawback, com-pared with previously presented diffraction-basedcharacterization methods, is that a larger numberof measurements are required and that the analysisof the measured data is more complicated. For thisreason the methods in Refs. 7 and 10 may some-times be more convenient for a quick estimation ofthe phase modulation.
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Fig. 17. Obtained phase modulation characteristics for input po-larization state 1. Note that the results from the diffraction-basedmethods (circles, pluses, and squares) were obtained with an LED�� 530 nm� as a light source, while the interferometer measure-ments (dotted curve) were achieved with a laser �� � 543.5 nm�.
