Automated collimation testing by incorporating theFourier transform method in Talbot interferometry

Jitendra Dhanotia and Shashi Prakash*Photonics Laboratory, Department of Electronics & Instrumentation Engineering, Institute ofEngineering & Technology, Devi Ahilya University, Khandwa Road, Indore 452017, India

*Corresponding author: sprakash_davv@rediffmail.com

Received 26 October 2010; revised 30 December 2010; accepted 15 February 2011;posted 15 February 2011 (Doc. ID 137003); published 30 March 2011

In this paper, we report an automated technique for collimation testing by incorporating Fourier fringeanalysis of the recorded interferograms in Talbot interferometry. The triangular profile of Talbotinterferometric fringes has been recorded using a CCD and computer system. The interferograms cor-responding to the in-focus, at-focus, and out-of-focus positions of the collimating lens have been recorded.Direct phasemeasurement using the Fourier transformmethod has been used for detection of collimationpositions. Good accuracy and precision in measurement have been achieved. © 2011 Optical Society ofAmericaOCIS codes: 070.6760, 120.1680, 070.2465, 100.2650.

1. Introduction

Talbot interferometry has been used for testing ofbeam collimation because it is inexpensive, simple,and offers good measurement characteristics interms of accuracy and precision. Because of these ad-vantages, Talbot interferometry has been used for awide range of other applications, also [1].

The use of Talbot interferometry for collimationtesting was first reported by Silva [2]. The author de-fined the accuracy obtainable in the collimation testas the smallest detectable deviation from the perfectcollimation position. Later, collimation testing usingdifferent kinds of gratings were reported, with theprimary aim being to get improved accuracy in colli-mation testing [3–7]. In recent years, efforts havebeen made to undertake quantitative measurementsand to improve the measurement characteristics byautomating the collimation detection mechanism.Toward this, Bera et al. [8] reported collimation test-ing using a double grating system based on Talbotinterferometry. In the setup, two photodetectors wereplaced behind the detector grating and the phase

shift between the two signals of the photodetectorswas measured as the collimation position was ap-proached. In an exhaustive arrangement, the signalsat two laterally displaced positions on the imageplane were recorded using photodetectors and theirphase difference was evaluated electronically usingLissajous figures. The technique involved the use ofextra instrumentation, and the phase information atonly two points at the image plane was used to locatethe collimation position.

Collimation of an optical beam by incorporatinga phase shifting test procedure in Talbot interfero-metry [9] and Lau interferometry [10] has also beenreported. Here the phase along the whole of thewavefront was determined to extract exact informa-tion regarding collimation position. However, thetechnique required recording of either three or fourimages to extract the phase information, dependingon the magnitude of calibrated phase step needed.Also, phase plots obtained using the phase shiftingtechnique are very noisy. The phase shifting methodworks well with low frequency gratings, while, for thehigh frequency gratings, the method becomes moretedious. The technique requires a precision transla-tion stage, which is relatively costly.

0003-6935/11/101446-07$15.00/0© 2011 Optical Society of America

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In this paper, we report our studies undertaken to-ward collimation testing of an optical beam using theFourier transform method in Talbot interferometry.The technique needs only one interferogram to beanalyzed, making the method simple, effective, andadvantageous. The requirement of specialized trans-lating devices, as in the phase shifting technique, isalleviated.

2. Basic Principle

The Talbot effect is a near-field diffraction effect thathas been observed both with light and atom optics.When a plane parallel wavefront illuminates a grat-ing G1, it is observed to generate the self-image of thegrating at certain well-defined planes called “Talbotplanes.” These planes are located at a distance d,called the Talbot distance:

d ¼ 2p2mλ ; ð1Þ

where m, p, and λ represent an integer, the pitch ofthe grating, and the wavelength of the light used, re-spectively. The location of a self-image when a grat-ing G1 is illuminated by a plane wavefront is shownin Fig. 1. Here, G0

1 corresponds to the self-image ofgrating G1, located at the first Talbot plane.

If another identical grating, G2, is placed at theTalbot plane corresponding to the grating G1, thecharacteristic moiré pattern is formed. This setuphas been named the Talbot interferometer and hasbeen used extensively for applications in science andengineering. The interferometer provides the slopeinformation of the phase object under investigation,and the intensity profile of the fringes at the imageplane is triangular and can be expanded in theFourier series as [11]

Iyðx; yÞ ¼14þ 2

π2X∞n¼0

cos½πð2nþ 1Þ2d=ðp2=λÞ�ð2nþ 1Þ2

× cos�2πð2nþ 1Þ

�χpþ yθ

pþΦðx; yÞd

pþ sp

��;

ð2Þ

where χ=p and yθ=p are the phase shifts related tothe relative translation and rotation, respectively,of the Ronchi gratings, χ is the offset of the gratinglines in the x direction, and θ is the relative angle be-tween the gratings. Φðx; yÞ is the deflection angle oflight, and s=p is an additional phase shift caused bythe displacement of one grating in the x direction.

Pfeifer et al. [12], while presenting phase shiftingmoiré deflectometry, presented that, because of thefactor 1=ð2nþ 1Þ2, the higher harmonic terms contri-bute little to the entire intensity, and, hence, higher-order terms may be neglected. So the term corre-sponding to n ¼ 0 may be considered a dominantterm. Under paraxial approximation and Fresneldiffraction, the above equation can be written as

Iðx; yÞ ≈ 14

þ 2

π2 cosðlπÞ cos�2π

�χpþ yθ

pþΦðx; yÞd

pþ sp

��:

ð3Þ

To determine the deflection angle and/or phase var-iation caused due to decollimation of the collimator,direct phase determination using Fourier fringe ana-lysis has been incorporated. In this case, Eq. (3) canbe written in more general form as

Iðx; yÞ ¼ Aþ B cos½ψðx; yÞ þ αðx; yÞ�; ð4Þ

where

A ¼ 14; B ¼ 2

π2 cosðlπÞ;

ψðx; yÞ ¼ 2π�Φðx; yÞd

pþ yθ

p

�; αðx; yÞ ¼ 2π

�χpþ sp

�:

Since there is no displacement of any of the gratingsin the x direction, α ¼ 0 and, so,

Iðx; yÞ ¼ Aþ B cos½ψðx; yÞ�: ð5Þ

Undertaking the Fourier transform of Eq. (5) yields

Iðu; vÞ ¼ Aðu; vÞ þ Cðu; vÞ þ C�ðu; vÞ; ð6Þ

where Cðu; vÞ is the Fourier transform of Cðx; yÞ,which may be defined as

Cðx; yÞ ¼ 12Bðx; yÞ exp½jψðx; yÞ�; ð7Þ

and C�ðu; vÞ is the complex conjugate of Cðu; vÞ. Ofthe three terms in Eq. (6), the third term correspond-ing to the first-order sidelobe is selected and isshifted to the center. The centrally shifted lobe is fil-tered out using a bandpass filter. By taking the in-verse Fourier transform, the desired phase, ψðx; yÞ,may be obtained. Cðx; yÞ may be represented as

Fig. 1. Schematic of the experimental arrangement for testingthe collimation of an optical beam using Talbot interferometryand the Fourier transform method.

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Cðx; yÞ ¼ Re½Cðx; yÞ� þ Im½Cðx; yÞ�; ð8Þand ψðx; yÞ can be evaluated back as

ψðx; yÞ ¼ tan−1

�ImðCðx; yÞÞReðCðx; yÞÞ

�: ð9Þ

The deflection angle Φðx; yÞ may be determinedusing the relation [11]

Φðx; yÞ ¼ p2πdψðx; yÞ:

The phase we obtain from Eq. (9) is the wrappedphase because the arctangent is defined only overthe limited range from −π=2 to π=2. To extend therange of phase measurements to 0 to 2π, we adoptthe phase unwrapping process such that two conse-cutive pixel positions are checked; if the phase valuesvary by more than �π, we add or subtract 2π to orfrom the second pixel phase value.

The deflection angleΦðx; yÞ is related to ψðx; yÞ andis effectively the measure of the slope of the rays inthe direction perpendicular to the grating lines. Itsvariation with respect to the x axis can be used asa measure for checking the detection of collimationposition. For the in-focus position of the collimator,the incident beam is diverging, representing the po-sitive value of the deflection angle and, hence, a po-sitive value of the slope in the phase map. For theout-of-focus position of the collimator, the beam con-verges, which corresponds to a negative value of thedeflection angle of the beam and, hence, the slope ofthe phase map is negative. The plot of phase map forthe at-focus position of the collimator corresponds toa zero deflection angle and, hence, a zero value of theslope of the phase map. Thus, the slope of the phasemap represents the type of beam emerging out fromthe collimator.

3. Experimental Arrangement

The schematic of the experimental arrangement forchecking the collimation of a laser beam is shown inFig. 1. The light from a 15mW He–Ne laser is inci-dent on a microscopic objective of magnification

40× and is used to focus the beam at the focal pointof the collimating lens. A pinhole of 5 μm diameter isplaced at the focus of the microscope objective toserve as a spatial filter. The precision achromaticdoublet lens PAC088, supplied by Newport Corpora-tion, USA, having a focal length of 250mm, has beenused as collimating lens LC. The beam from the col-limating lens LC impinges on the grating G1, havinga 0:2mm pitch. This results in the formation of theself-image of the grating G1 at certain well-definedplanes, called “self-imaging planes.” Grating G2, ofa period the same as that of grating G1, is placedat the first self-image plane of grating G1. The twogratings are oriented such that they make equaland opposite angles with respect to y axis (gratinglines). This results in introduction of tilt errors inthe interfering wavefronts. The shearing interfero-gram was imaged on the phase plate of CCD camerausing lenses L1 and L2. Aperture A is placed at thefocal plane of L1 to block the directly transmittedlight from reaching the image plane. The CCD has1392 × 1040 pixels, with each pixel sized 4:65 μm×4:65 μm. For image acquisition, Artray 150PIII soft-ware supplied by Artray Co., Japan was used andthe results were displayed online via the computermonitor.

Initially, the collimating lens LC was mounted onthe precision translating stage to translate it alongthe optical axis, making it possible to introduce aknown amount of decollimation. The gratings G1 andG2 were placed in such a way that the grating lineswere parallel to each other. At the image plane, uni-form illumination was obtained. Then the gratingsG1 and G2 were rotated in the clockwise and antic-lockwise directions, so that the lines of the gratingsmake small but equal and opposite angles with thevertical. This introduces error corresponding to tiltof the wavefront. When the incident optical beamis collimated, equidistance parallel fringes, perpendi-cular to the grating lines, were observed at the imageplane. To achieve this condition, the collimating lenswas adjusted in such a manner that the fringes werealigned parallel to the set reference line. These hor-izontal fringes were indicative of the collimationposition as shown in Fig. 2(b), and the position of

Fig. 2. (Color online) Fringe pattern recorded (when grating lines are inclined with equal and opposite angles to each other) using a CCDcamera at the (a) in-focus, (b) at-focus, and (c) out-of-focus positions of the collimating lens of focal length 250mm.

1448 APPLIED OPTICS / Vol. 50, No. 10 / 1 April 2011

the collimating lens was recorded using an attachedmicrometer screw. Also, upon moving the collimatinglens corresponding to the in-focus or out-of-focus po-sition, the fringes get inclined. This is indicative ofthe setting of the decollimation position, as shownin Figs. 2(a) and 2(c). This may correspond to a coarsesetting of the collimation position by visual inspec-tion. For collimation testing using Fourier fringeanalysis, a series of interferograms near the collima-tion position were recorded and analyzed using thealgorithm in Fig. 3. This may be treated as a fine set-ting for checking the collimation position.

4. Results

The corrected collimating lens of focal length of250mm was used in the experiment. For determin-ing the collimation position, the interferograms have

been recorded using a CCD camera with a framegrabber card and stored in the computer memory.The interferograms were analyzed using the Fouriertransform method. The MATLAB software has beenused to implement the algorithm.

Figures 2(a)–2(c) show the recorded interferogramsat the in-focus, at-focus, and out-of-focus positions, re-spectively. It is quite evident from these figures that,in the case of the at-focus position, the fringes are par-allel to the horizontal direction and, for the in-focusposition, they are oriented in the anticlockwise direc-tion. For the out-of-focus position, the fringes areoriented in the clockwise direction.

The interferograms were processed using the FFT(fast Fourier transform) algorithm shown in Fig. 3.Direct phase determination of the recorded interfer-ograms was processed using this FFT algorithm. Itwas applied on each of the recorded interferograms.Figures 4(a)–4(c) show the Fourier spectra for the in-terferograms corresponding to the in-focus, at-focus,and out-of-focus positions. The central dot with max-imum intensity shows the zero order, while the dotsimmediately on the two sides of the central spot showthe �1-order spectra. In the case of the at-focus posi-tion [Fig. 2(b)], the fringes are parallel to the horizon-tal reference line; hence, the Fourier spectra havedots aligned in the vertical direction, as shown inFig. 4(b). In the cases of the in-focus and out-of-focus[Figs. 2(a) and 2(c)] positions, as the fringes get in-clined with respect to the horizontal reference line,the corresponding Fourier spectra get inclinedwith respect to the vertical direction, as shown inFigs. 4(a) and 4(c). The Fourier spectra in each two-dimensional FFT (2D-FFT) are sufficiently sepa-rated; the �1-order spectra can be easily separatedand shifted at the center by replacing the zero orderusing an appropriate filtering scheme. The 2D-FFTfringe orientation tan−1ðx=yÞ has been calculatedfrom the values of coordinates x and y of the first-order spectra. The centrally shifted þ1 order is fil-tered out, its inverse FFT (IFFT) calculated, andthe wrapped phase of each interferogram obtained.To obtain a reliable phase map, the field correspond-ing to the wrapped phase map is scanned and 2π isadded or subtracted every time an edge is detected.

The wrapped and unwrapped phase map has beenevaluated for in-focus, at-focus, and out-of-focuspositions. The value ψðx; yÞ, the phase of the recordedinterferograms corresponding to different positions ofthe collimator was determined. The variation ofphase with respect to the x and y axes is plotted.Figures 5(a)–5(c) represent the unwrapped phasemaps of the interferograms corresponding to in-focus,at-focus, and out-of-focus positions of the collimatinglens, respectively. Thephasemapshaveaparticularlydistinct orientation in each case. It is quite evidentfrom these figures that the slope of the phase mapalong the x axis is positive for the in-focus positionof the collimating lens. The slope of the phase mapalong the x axis decreases as we move toward theat-focus position and it becomes zero at the at-focus

Fig. 3. Flow chart for the Fourier transform algorithm used toprocess fringes.

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position. This is the collimation position of the colli-mator. The slope of the phase map corresponding tothis value of Δf was 6:8 × 10−6 rad, approximately.As the collimating lens is further moved away fromthe collimation position, the slope of the phase mapsalong the x axis becomes negative. In the present con-dition, the phase depends not only on the defocusingerror, but also on the tilt errors in the interferingwavefronts. The tilt is introduced between the two in-terfering wavefronts in a plane perpendicular to thedirection of shear.Tilt hasbeen introducedby rotationof the gratings in their own plane, but in opposite di-rections, so that the gratings make equal but verysmall angles with the vertical. In the case of the at-focus position, the defocusing error is zero and, thus,only the contribution because of the tilt (along the yaxis) exists. The plot of phase ψ with respect to thex domain, for a fixed value of y (the plot of ψ alongthe same row for different values of the columnvector)gives a straight line, as anticipated and shown inFig. 6(b).

In the case of the in-focus position, the defocusingterm appears and the contribution is because of bothterms, i.e., defocus and tilt. But in this case, the con-tribution due to the defocusing term is in the positivedirection. The plot of phase ψ with respect to the xdomain for a fixed value of y gives a positive slope,as anticipated.

The reasoning for out-of-focus position remains thesame; the only change is that the defocusing errorhas a negative value. The three-dimensional phasewith respect to the x and y axes is plotted in Fig. 5(c).The plot of phase ψ with respect to the x domain for afixed value of y has a negative slope, as shown inFig. 6(c).

5. Discussion

Collimation testing using Talbot interferometry withdifferent types of grating has been proposed fromtime to time. Researchers have put in concentratedefforts to achieve higher accuracy in collimation test-ing. Accordingly, several configurations have beenproposed in the literature to achieve improved accu-racy in setting the collimation position. Two distinctmethodologies for detecting the collimation positionwhile using Talbot interferometry for collimationtesting have been proposed. These may be classifiedas static and dynamic detection mechanisms. In astatic mechanism, the grating or any other elementused in the setup is not disturbed as the collimationposition is approached, while in a dynamic mode, thegratings are displaced or translated so as to get dis-tinct information about the location of the collima-tion position of the collimating lens. Prakash et al.[9] reported an automated method for collimationtesting of coherent beams using a temporal phaseshifting technique in Talbot interferometry. The

Fig. 4. Fourier spectrum of recorded images at the (a) in-focus, (b) at-focus, and (c) out-of-focus positions.

Fig. 5. Unwrapped phase map for a fringe patterns corresponding to the (a) in-focus (Δf ¼ −20 μm), (b) at-focus, and (c) out-of-focus(Δf ¼ 15 μm) positions of the collimating lens of focal length 250mm.

1450 APPLIED OPTICS / Vol. 50, No. 10 / 1 April 2011

authors reported the relative comparison of the accu-racy achievable in various collimation checking testprocedures. To date, the dynamic detection mech-anism has been reported to offer higher accuracy(6 μm) than the static detection mechanism. How-ever, the dynamic detection mechanism involves theneed for extra equipment (such as a precision trans-lation stage) and the effort to translate the grating.Hence, the static detection mechanism is a preferredmode if ease of operation is required. However, withthe use of a static detection mode, the best accuracyobtainable reported to date has been with the use ofcircular gratings in a Talbot interferometer. The ac-curacy reported for a lens of focal length 250mmhas been 24 μm when the distance between thetwo gratings is 1750mm [7]. This makes the systembulky, too.

Toward improving the accuracy in the staticdetection mechanism, we incorporate the Fouriertransform method of analysis for direct phase mea-surement. The accuracy obtainable in setting collima-tion has been obtained to be 5 μm. In fact, it is possibleto improve the accuracy achievable below this value.In our case, we could not improve it below this valuedue to the unavailability of a translation stage withresolution better than this value. However, the ulti-mate accuracy obtainable with the use of the tech-nique shall be determined by diffraction effects.

We compare the performance of the technique withrespect to the case when temporal phase shifting hasbeen used for checking collimation in Talbot inter-ferometry. We find that phase plots obtained in thepresent case are much less noisy than those obtainedin the case of phase shifting. As the collimation posi-tion is approached, the phase plot obtained usingtemporal phase shifting becomes very noisy. This hasbeen reported to be the main limitation of the tech-nique [9]. It has been reported by the authors that, asthe collimation position is approached, the contribu-tion due to higher-order terms present in the re-corded interferogram play a role and it becomesdifficult to ascertain the exact at-focus position. Inour case, we could easily isolate the higher-order dif-fraction spots using digital filters. Hence, in compar-ison to earlier investigations, it may be concludedthat the proposed technique provides the better mea-surement characteristics.

However, there are several factors limiting the ac-curacy and sensitivity of measurement. These maybe because of the quality of the optics used, imper-fections in the gratings, grating tilt angle, etc. Wehave taken precautions to minimize these errors.There may be errors due to quantization errors indigitizing the data, source instabilities, and detectornonlinearity.

6. Conclusions

The applicability of collimation testing using theFourier transform method in Talbot interferometryhas been tested. The interferograms correspondingto the in-focus, at-focus, and out-of-focus positionsof the collimating lens have been recorded. The inter-ferograms are then analyzed using the Fourier trans-form method. The technique involves undertakingFFT of the recorded data, design of suitable digitalfilters to select the desired frequency spectrum, andshifting it to the center. In the final step, the compu-tation of the phase information using a phase un-wrapping algorithm has been undertaken. The phaseplot provides an accurate and quantitative measurefor detection of the collimation position of the colli-mating lens. Marked improvement in the accuracyhas been achieved. The technique proposed is simpleand economical.

The authors are grateful to the reviewers for pro-viding valuable suggestions. Also we are indebtedto Dr. D. S. Mehta, Associate Professor, InstrumentDesign Development Centre, Indian Institute ofTechnology Delhi, New Delhi, for helping us withtheproblemsolving inMATLABenvironment. Finan-cial support from the University Grants Commission(UGC), New Delhi in terms of research project grant33-395/2007(SR) is gratefully acknowledged.

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Fig. 6. (Color online) Variation of phase with respect to the x axis for the fringe patterns corresponding to the (a) in-focus (Δf ¼ −20 μm),(b) at-focus, and (c) out-of-focus (Δf ¼ 15 μm) positions of the collimating lens of focal length 250mm.

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