2788 OPTICS LETTERS / Vol. 33, No. 23 / December 1, 2008

One-shot phase-shifting interferometry:five, seven, and nine interferograms

Gustavo Rodriguez-Zurita,* Noel-Ivan Toto-Arellano, Cruz Meneses-Fabian, and José F. Vázquez-CastilloFacultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Apartado Postal 1152,

Puebla PUE 72000, México*Corresponding author: gzurita@fcfm.buap.mx

Received August 25, 2008; revised October 10, 2008; accepted October 15, 2008;posted October 21, 2008 (Doc. ID 100539); published November 21, 2008

To extract phase distributions, which evolve in time using phase-shifting interferometry, the simultaneouscapture of several interferograms with a prescribed shift has to be done. Previous interferometric systemsaimed to fulfill such a task were reported to get only four interferograms. It is pointed out that more thanfour suitable interferograms can be obtained with an interferometer that uses two windows in the objectplane, a phase grid as a pupil, and modulation of polarization for each diffraction orders in the image plane.Experimental results for five, seven, and nine interferograms are given. © 2008 Optical Society of America

OCIS codes: 050.2770, 050.5080, 070.6110, 120.3180, 120.2650, 260.5430.

In phase-shifting interferometry using n interfero-grams, n−1 shifts must be carried out. To extract op-tical phase distributions that evolve in time, the cap-ture of the n-shifted interferograms with one shot isdesirable. Some approaches to perform this task havebeen already demonstrated [1–3], although only forn=4 to our knowledge. Among these systems, the oneusing two windows in the object plane of a 4f systemwith a phase-grating in the Fourier plane and modu-lation of polarization [two-windows phase-gratinginterferometer (TWPGI)] is a very simple possibility[3]. In this Letter, the capability of a TWPGI to cap-ture more than four interferograms in one shot isdemonstrated with the introduction of a phase grid inplace of the grating. To test a TWPGI for more thanfour interferograms, the case of n= �N+1� interfero-grams has been chosen. This method reduces errorsin phase calculations when noisy interferograms areinvolved [4]. Experimental results for n=5, 7, 9 inter-ferograms are shown.

The setup is depicted in Fig. 1. A combination of aquarter-wave plate Q and a linear polarizing filter Pgenerates linearly polarized light at an azimuthangle of 45°. Retardation plates (QL and QR) with thesame retardation �� and with mutually orthogonalfast axes are placed in front of each rectangular win-dow �A ,B� to generate left and right nearly circularpolarized light. The windows are symmetrically dis-placed from the origin along a line at 45° with respectto the x axis. A phase grid is placed at the system’spupil. The phase grid was carefully constructed bysuperposition of two commercially available phasegratings with their respective grating vectors at 90°.Diffraction orders appear in the image plane, forminga rectangular array. Around each order, an interfer-ence pattern appears owing to the optical fields asso-ciated with each window when proper matching con-ditions are met. Polarizing filters in the image planecentered on each diffraction order are the compo-nents that finally achieve the desired shift �i, i

=1. . .n [3], as described later (Fig. 1). Each polarizing

0146-9592/08/232788-3/$15.00 ©

filter transmission axis must be adjusted at theproper angle �i. The calculation of �i for the case ofnonexact quarter-wave retardation has been consid-ered [3].

Object and image planes are described by �x ,y� co-ordinates. A periodic phase-only transmittanceG2��u /�f� , �v�f�� is placed in the frequency plane�� ,��. Then �=u /�f and �=v /�f are the frequency co-ordinates scaled to the wavelength � and the focallength f. In the plane �u ,v�, the period of G2 is de-noted by d (the same in both axis directions) and thusits spatial frequency by �=1/d. Two neighboring dif-fraction orders have a distance of X0��f /d in the im-age plane. Then, � ·u=X0 ·�. Taking the rulings ofone grating along the � direction and the rulings ofthe second grating along the � direction, the resultingcentered phase grid can be written as

G2��,�� = �q=−�

�

Jq�2Ag�ei2·qX0� �r=−�

�

Jr�2Ag�ei2·rX0�,

�1�

with 2Ag as the grating’s phase amplitude; Jq is theBessel function of the first kind of integer order q [5].

Fig. 1. (Color online) Setup. A, B, windows; G2, phasegrid; tf, transmitted image; Qj, retarders; P, linear polar-izer; xq=x− �q+ 1

2 �X0, yr=y− �r+ 12 �X0, coordinates about dif-

fraction order qr. Polarization filter arrays at the rightshowing different angle �n for each transmission axis of po-

larizing filter Pi.

2008 Optical Society of America

December 1, 2008 / Vol. 33, No. 23 / OPTICS LETTERS 2789

The frequencies along each axes directions are takenof the same value. The Fourier transform of thephase grid becomes

G̃2�x,y� = �q=−�

q=�

�r=−�

r=�

Jq�2Ag�Jr�2Ag�

��x − qX0,y − rX0�, �2�

which consists of pointlike diffraction orders distrib-uted in the image plane on the nodes of a lattice witha period given by X0. A convenient window pair for agrating interferometer implies an amplitude trans-mittance given by

t�2�x,y� = J� L · w�x −x0

2,y −

y0

2 � + J� Rw��x +x0

2,y +

y0

2 � ,

�3�

where the polarizing vectors are defined as

J� L = � 1

ei���, J� R = � 1

e−i��� , �4�

and x0 and y0 give the mutual separations betweenthe centers of each window along the coordinate axis.One rectangular aperture can be written as w�x ,y�=rect�x /a� · rect�y /b� whereas the second one asw��x ,y�=w�x ,y�expi��x ,y�. A relative phase be-tween the windows is described with the function��x ,y�, and a represents the side length of each win-dow. The image t�f�x ,y� formed by the system consistsbasically of replications of each window at distancesX0, that is, the convolution of t�2�x ,y� with the pointspread function of the system, defined by the inverseFourier transform of ̃2�� ,��= 1

2IG2�� ,�� with I beingthe unit matrix 22. This results into the following:

t�f�x,y� =1

2I · t�2�x,y� � I−1G̃2�� − �0,� − �0�, �5�

where �0 and �0 denote a possible translation of thegrid along the axis directions. Assuming x0=y0, by in-voking the condition of matching first-neighboring or-ders �X0=x0� the image t�2�x ,y�* �ei2��0x+�0y�G̃2�x ,y��can be described by

�q=−�

�

�r=−�

�

J� LJqJr + J� RJq+1Jr+1ei2��0+�0�·x0

· e�i��x−�q+1/2�x0,y−�r+1/2�x0��ei2�q�0+r�0�x0, �6�

where some inessential constants are dropped. By se-lecting the diffraction term of order qr, after placinga linear polarizing filter with transmission axis atthe angle � ,J�

L, its irradiance results proportional to(using position coordinates xq, yr with respect to or-der qr, see Fig. 1)

A��,��� · ��JqJr�2 + �Jq+1Jr+1�2

+ 2JqJrJq+1Jr+1 · cos����,��� − ��xq,yr���, �7�

where �0=�0=0 without loss of generality with

J�L = �cos � − sin �

sin � cos ��, J� L� = J�

LJ� L, J� R� = J�LJ� R,

�8�

and [4]

A��,��� = 1 + sin�2�� · cos����,

���,��� = ArcTan�sin����

cot�2��

1 + tan��� · cos����+ cos����� .

�9�

Each pattern is shifted by an amount �. For thecase of exact quarter-wave retardation, A�� , /2�=1and ��� , /2�=2�. Otherwise, these quantities mustbe evaluated with Eq. (9). The Fourier spectrum ofthe grid in our tests behaves as sketched in Fig. 2,where two equal phase gratings are shown with theirrespective +4th diffraction order assumed negative[Fig. 2(a)]. Thus, the −4th diffraction order resultsare also negative. A phase grid is formed with thegratings at 90° and the resulting Fourier spectrumforms a rectangular reticule [Fig. 2(b)]. Owing to the phase difference between orders, there are orderspointing out toward the reader (circles) or away(crosses). Because the window are displaced, twoFourier spectra become shifted from the origin diago-nally and in opposite directions [Fig. 2(c)]. Similarrows and columns are encircled within the dottedlines. Under our matching condition, the order qrsuperimposes with the order �q−1��r−1�. Thus, someorders are in phase (dots with dots or crosses withcrosses, but only one symbol is depicted) and othersout of phase (dot with cross). Then only one symbolmeans positive contrast, while both symbols meancontrast reversal [Fig. 2(c)].

To demonstrate the use of the several interfero-grams, we choose the symmetrical N+1 phase stepsalgorithms for data processing in the cases of N=4, 6,8. A constant phase shift value of 2 /N is employedin using these techniques. In these systems, the N

Fig. 2. (a) One-dimensional spectra of identical phasegratings to be crossed to construct a grid. (b) Correspondingimage plane for the phase grid. (c) Two shifted Fourierspectra are superimposed according to the windows dis-

placement A–B.

2790 OPTICS LETTERS / Vol. 33, No. 23 / December 1, 2008

+1 interferogram results from a shift of 360°. Thephase formula for N shifts can be seen in [4]. Figure3 shows the polarizing filters employed. For the caseof five interferograms (Schwieder–Hariharan algo-rithm), only three linear polarizing filters have to beplaced. This is because one filter serves instead oftwo as long as it covers two patterns with a 180°phase shift in between [Fig. 3(b)]. Considering the re-tarders at disposal, according to Eq. (9) it can beshown that �1=0°, �2=46.577°, and �3=92.989°, eachstep of � being of 90°. For the case of symmetricalseven, each step of � is 60°, so �i are �1=0°, �2=30.800°, �3=62.330°, and �4=92.989°. For sym-metrical nine, �1=0°, �2=92.989°, �3=22.975°, �4=46.577°, and �5=157.903°. In this case, each step of� is 45°. The corresponding results and calculatedphases are shown in Fig. 4. The object was an oil droprunning down over a microscope slide. Each inter-ferogram was the subject of the same scaling processfrom 0 to 255 and the same low-pass filtering prior tophase calculation.

In conclusion, this system is able to obtain n= �N+1� interferograms with only one shot �n�16�. Testswith 2 /N phase shifts were presented, but other ap-proaches using different phase shifts could be at-tained using linear polarizers with their transmis-sion axes at the proper angle before detection. Thephase shifts of due to the grid spectra allow the useof a number of polarizing filters that is less than thenumber of interferograms, simplifying the filter ar-ray. Other configurations for the window positionsthat are different from the one reported in this Letter

Fig. 3. Polarizing filter arrays for several cases. (a) Twelveinterference patterns detected with a polarizing filter at �=35° covering all of them. (b) N=4, symmetrical five. (c)N=6, symmetrical seven. (d) N=8, symmetrical nine.

are also possible. The accuracy in measurements isthe one typical of phase shifting. Some trade-offs ap-pear while placing several images over the same de-tector field, but for low-frequency interferograms(with respect to the inverse of the pixel spacing) theinfluence of these factors seems to be rather small ifnoticeable. The interferometer could be used for ob-jects with no changes of polarization.

Partial support from Consejo Nacional de Ciencia yTecnologiá (CONACYT) (grants 90497 and 165912) isgreatly appreciated.

References

1. B. Barrientos-García, A. J. Moore, C. Pérez-López, L.Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).

2. M. Novak, J. Millerd, N. Brock, M. North-Morris, J.Hayes, and J. Wyant, Appl. Opt. 44, 6861 (2005).

3. G. Rodriguez-Zurita, C. Meneses-Fabian, N. I. Toto-Arellano, J. Vázquez-Castillo, and C. Robledo-Sánchez,Opt. Express 16, 7806 (2008).

4. D. Malacara, M. Servin, and Z. Malacara, inInterferogram Analysis for Optical Testing (MarcelDekker, 1998).

5. N. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, Opt. Express 16,19330 (2008).

Fig. 4. Flow of oil drops on glass. Phase-shifted interfero-grams and unwrapped phases. Upper two rows, two ex-amples of five 90° phase shifts. Center rows, seven 60°phase shifts. Lower rows, nine 45° phase shifts. Referencesquare is for scale dimensions.