Measurement of focal length of lens using phase shifting Lau phase interferometry

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1 Optics Communications 248 (2005) Measurement of focal length of lens using phase shifting Lau phase interferometry C.J. Tay a, M. Thakur a, *, L. Chen a, C. Shakher b a Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore , Singapore b National Institute of Technology, Hamirpur, India Received 20 August 2004; received in revised form 10 December 2004; accepted 16 December 2004 Abstract In this paper, we proposed a phase shifting Lau phase interferometry for the measurement of the focal length of a lens. The fundamentals of this application are discussed and the basis of the phase map interpretation is outlined. A three step phase-shifting algorithm is used to depict the phase maps of a lens-like test object and an application for measuring the focal length of a lens is reported. A phase map obtained with a coherent light source is also shown and the merits of using incoherent light over coherent light source are also discussed. Experimental results are presented. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Lau phase interferometry; Phase shifting; Focal length 1. Introduction The focal length is one of the most important parameters to evaluate the performance of a lens/ optical system. A number of methods of precision measurement of focal length includes Talbot interferometry, shearing interferometry and moiré technique and other interferometric approach [1 5] * Corresponding author. Tel.: ; fax: addresses: thakur_madhuri@hotmail.com, mpemt@ nus.edu.sg (M. Thakur). have been already investigated. DeBoo and Sasian [6] discussed a precise focal length measurement technique based on reflective Fresnel-zone hologram. In 1948, Ernst Lau described a diffraction experiment, whereby the object was illuminated by an extended white light source. The object consisted of two coarse gratings, positioned behind each other at a distance of a few centimeters and colored fringes appear at infinity. Several authors have studied LauÕs effect in great detail. Jahns and Lohman [7] explained the field properties behind the second grating using diffraction theory /$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi: /j.optcom

2 340 C.J. Tay et al. / Optics Communications 248 (2005) and self-imaging (Talbot effect) approach. Gori [8] and Sudol and Thompson [9,10] based their analysis on the coherence theory. In their analysis the Lau fringes were interpreted as a result of matching between the period of the second grating and the period of the complex degree of spatial coherence of the light field incident on the grating. Bartelt and Jahns [11] demonstrated the formation of Lau phase interferometric fringes when a phase object was placed at a finite distance. Swanson and Leith [12] used the grating imaging approach to explain the Lau effect. They showed in detail that the Lau fringes represent only one family of the many interference patterns produced by a double-grating interferometer. Studies of Lau effect for imaging of periodic structures in spatially incoherent radiation were reported by Aristov et al. [13]. An explanation of the Lau effect given by Patorski [14] is based on the concept of incoherent superposition of multiple self-imaging. He considered the case of incoherent illumination of two diffraction gratings separated by a finite distance along the direction of illumination. In 1986, Patorski [15] further suggested modification to Lau phase interferometer using the concept of multiple incoherent superposition of Talbot interferometry. He discussed the process of fringe formation when the grating separation distance was infinitely large. This process corresponds to illuminating the second grating with a multiple of mutually incoherent quasi-plane wavefronts. Thakur et al. [16 19] investigated Lau phase interferometry for the measurement of temperature and temperature profile of gaseous flame by using linear and circular gratings and focal length of lenses. The wide availability of digital image processing equipment prompted a number of studies to investigate different alternative of automatic interferogram analysis. Modern techniques have reached a point at which they can provide useful results, particularly with phase-stepping methods. By introducing a discrete shift in the position of the fringes, we can calculate a phase map in a relatively simple way. The phase-stepping technique is a simplification of the irradiance integration method [20] and requires a minimum of three unknowns in the interference equation. There are many algorithms used to extract the phase, and a certain combination of them can minimize phase step miscalculation and nonlinearity error [21,22]. In this paper, a phase shifting white light Lau phase interferometry is used to measure the focal length of a lens. This is carried out by using an in-plane shift of one of the interferometer gratings to produce the required phase steps. The Lau phenomena is a simple although useful and inexpensive interferometer. The advantage of using white-light is that it reduces coherent noise in the interferogram and leads to high signal-to-noise ratio. In addition, white-light interference bands have a high degree of stability. This procedure is applied to map a partial derivative of phase. The measurement of focal length using the phase map slope is also discussed in this paper. 2. Theory An experimental arrangement is shown in Fig. 1. It consists of three identical linear gratings G 0, G 1 and G 2. The first grating G 0 can be considered as multiple mutually incoherent sources. Light from grating G 0 and collimating lens L C illuminates grating G 1 as many mutually incoherent plane wavefront beams. The self-image of the second grating would appear at a distance ÔZÕ, given as [15] Z ¼ kd2 k ; ð1þ where ÔdÕ is the pitch of the second grating, ÔkÕ is an integer and ÔkÕ is the wavelength of the light source. The grating G 1 can be represented mathematically by a Fourier-series [23] G 1 ðx 1 ; y 1 ; t 1 Þ¼A 1 þ cos 2pðx 1 t 1 Þ ; ð2þ d where t 1 represents the position of grating G 1 and A is an amplitude of the grating. The intensity distribution of grating G 1 immediately behind test object O(x,y) is given as I 1 ðx; yþ ¼I s Að1 þ cosð/ðx; yþþ2pt 1 =dþþ; ð3þ where I s is the intensity of the light source and /(x,y) is the phase of the object O(x,y). The intensity distribution is not only modified by the spatial

3 C.J. Tay et al. / Optics Communications 248 (2005) Fig. 1. Experimental arrangement. frequency of the grating but also by the test lens which is placed at one of the self-image planes of grating G 1. The modified image of grating G 1 overlaps grating G 2 and a moiré fringe pattern is observed on the image plane. Grating G 2 is mathematically represented by G 2 ðx 2 ; y 2 ; t 2 Þ¼A 1 þ cos 2pððx 2 DÞ t 2 Þ ; ð4þ d where t 2 represents the position of grating G 2, D = kf T /d (where f T is the focal length of the test lenses specified by the manufacture and d is the pitch of the grating) and intensity distribution for grating G 2 is given as I 2 ðx; yþ ¼Að1 þ cosð/ðx D; yþþ2pt 1 =dþþ: ð5þ It is assumed that when no object is located at the self-image plane of grating G 1, the dark fringes of the self-image of grating G 1 and the bright fringes of grating G 2 would overlap when t 2 t 1 = k Æ d (k = 0,±1,±2,...) and t 2 t 1 represents the magnitude of the gratings relative (in-plane) shift. Lau interferometric fringes are generated when the intensity field in Eqs. (3) and (5) overlapped each other and the intensity distribution at the image plane is given by Iðx; yþ ¼I 1 ðx; yþi 2 ðx; yþ; ð6þ Iðx; yþ¼bðx; yþð1 þ cosð/ðx; yþ /ðx D; yþþdðt 21 ÞÞÞ; ð7þ where B(x,y)=I s A 2 and dðt 21 Þ¼ 2pðt 2 t 1 Þ. Now d intensity I(x, y) consists of multiple sheared images, where the shear is given by D and / (x D,y) can be expressed as a series /ðx D; yþ ¼ X1 1 o k /ðx; yþ ðdþ k : ð8þ k! ox k k¼0 If the phase of the object varies slowly, Eq. (8) can be approximated as o/ðx; yþ /ðx D; yþ/ðx; yþþ ðdþþ 1 o 2 /ðx; yþ D 2 ox 2 ox 2 ð9þ then o/ðx; yþ Iðx; yþ ¼B 1 þ cos D þ dðt 21 Þ : ð10þ ox If a lens is considered as a phase object then /ðx; yþ ¼ p kf T ðx 2 þ y 2 Þ, where f T is the focal length of the test lens. Consequently from Eq. (10) 2p D Iðx; yþ ¼B 1 þ cos x þ dðt 21 Þ : ð11þ k f T Eq. (11) shows that for a test lens, it gives an equidistance fringe pattern whose period is P = kf T /D.Itis

4 342 C.J. Tay et al. / Optics Communications 248 (2005) apparent that the phase of the fringe is shifted by an in-plane translation of one of the gratings. Specifically, a grating displacement of magnitude t 2 t 1 results in a fringe shift of magnitude ( t 2 t 1 P)/d. In order to incorporate the phase shifting technique in the Lau fringe analysis, the Lau phase interferometric fringe intensity distribution in Eq. (11) rearranged in the form Iðx; yþ ¼Bðx; yþ½1 þ cos ðuðx; yþþdðt 21 ÞÞŠ; ð12þ In Eq. (12), d(t 21 ) is the phase shift. The intensity distribution at each point (x,y) varies as a sinusoidal function of the introduced phase shift d(t 21 ). If the pitch of the grating is ÔdÕ and one of gratings is moved by a distance t 2 t 1 =d/k (k =1,2,...) this would represent phase shift of d(t 21 ) = 2p/k for each fringe. To retrieve the phase u(x, y), additional phase shift d(t 21 ) in the steps of p/2 are required. In this study, three phase shifted Lau interferometric patterns with a phase shift of 0, p/2, p are recorded. The following equations represent the corresponding phase shifted intensity distributions [14]: I 1 ðx; yþ ¼aðx; yþþbðx; yþ cos uðx; yþ; ð13þ I 2 ðx; yþ ¼aðx; yþþbðx; yþ cos½uðx; yþþp=2š; ð14þ I 3 ðx; yþ ¼aðx; yþþbðx; yþ cos½uðx; yþþpš: ð15þ Solving the above equations simultaneously, the phase u(x, y) at each point (x, y) on the object can be obtained by uðx; yþ ¼arctan I 1ðx; yþ 2I 2 ðx; yþþi 3 ðx; yþ : I 1 ðx; yþ I 3 ðx; yþ ð16þ The phase u(x, y) obtained from Eq. (16) resulted with the principal value of phase u(x,y) in region between p to +p regardless of the actual value of the phase. Phase unwrapping is carried out in order to remove phase ambiguities by adding 2p or subtracting 2p from individual pixel until the phase difference between the adjacent pixel is less then p. The phase shifting technique can also be used to map the partial derivative D[o/(x, y)/ox] of the object phase. For example, when a lens of focal distance f T is under study, it follows that the partial derivative D[o/(x,y)/ox]=(2p/k)(D/f T )x, and the corresponding phase map depicts a plane whose slope is proportional to 1/f T. 3. Experimental arrangement An experimental arrangement is shown in Fig. 1. A white light source is used to illuminate grating G 0. The incoherently illuminated grating G 0 and collimating lens L C of focal length 100 mm provides a multiple mutually incoherent plane wavefront illumination system. The pitch of the grating used in the experiment is 0.5 mm. The angular separation between the beams is matched with the diffraction angle of grating G 1. A test lens (L T ) is placed at the self-image of grating G 1. If there is no test object at the self-image plane of grating G 1 a uniform intensity distribution is recorded in the observation plane. Its brightness depends on the relative in-plane displacement of the bright fringes of the self-image ðg 0 1Þ of grating G 1 and G 2. In particular if the bright fringe of G 0 1 and that of G 2 coincide, then the intensity reaches a maximum value. If G 0 1 and G 2 are laterally shifted by half a period relative to each other, no light emerging from G 1 would pass through G 2. If a test object is properly located between G 1 and G 2 additional phase shifts would occur at grating G 2. This would result in axial displacement of G 0 1. The resulting fringe pattern is particularly simple when a lens is used as a test object. The phase shifted Lau phase interferometric fringes are recorded using a CCD camera and a real time processor. For each test object under study, three images are recorded successively. Each subsequent image is recorded with a phase difference of p/2. The phase difference is introduced by a prescribed in-plane displacement of grating G 2 which is mounted on a linear translation stage. A linear displacement of 125 lm on the translation stage would translate into a phase step of p/2. 4. Results and discussion In this investigation a three step PSI method and subsequent phase unwrapping operation are used to retrieve a derivative phase map.

5 C.J. Tay et al. / Optics Communications 248 (2005) Fig. 2. (a) White-light Lau phase interferometric fringes. (b) Wrapped phase-map. (c) Unwrapped phase map. (d) 3-D plot of a test lens (100 mm focal length). Fig. 3. Lateral view of the phase maps for a lens with focal length 100 mm. Fig. 2(a) shows a fringe pattern of a test lens with focal length f T = 100 mm. Figs. 2(b) and (c), shows, respectively, a wrapped and the corresponding unwrapped phase maps and Fig. 2(d) shows the corresponding 3-D phase map. As expected, in Fig. 2(a), parallel equidistance fringes

6 344 C.J. Tay et al. / Optics Communications 248 (2005) Table 1 Comparison of results with specifications Phase plane slope of test lens Phase plane slope (f 100 = 100 mm) Measured value (mm) ManufacturerÕs specifications (mm) 4.81 ± ± Relative discrepancy (%) are observed, and consequently, a plane surface is apparent in the phase map (Fig. 2(d)). Since speckle noise is not a concern in an incoherent light source the resulting phase maps are of remarkable good quality. Fig. 3 shows a lateral view of a phase map of two lenses using white light Lau phase interferometry. The nominal effective focal lengths are f 50 = 50 mm, and f 100 = 100 mm. The test lens with a focal length of 50 mm is not an achromatic doublet lens. It can be seen from Fig. 3 that due to aberrations the graph is non-linear outside the paraxial region of the lens. The slopes of the planes were determined from data points resulting in m 50 = 4.81 ± 0.02 and m 100 = 2.41 ± Note that (m 100 /m 50 (f 50 /f 100 ). As stated above, the slope corresponding to the phase map of a test lens of focal distance f T is proportional to 1/f T. If the method is calibrated using a lens of known focal length, then it is possible to determine the focal length of lenses on the basis of slope measurement of the phase map planes. A lens of focal length 50 mm was measured using the purposed method and the results (see Table 1) agree well with that specified by the manufacturer and the discrepancy is about 0.2%. The order of the accuracy is similar to that measured by Thakur and Shakher [19]. In this measurement the main advantage of evaluating the focal length from the slope is that very little mathematical calculation is required. The Lau phase interferometry experiment is also conducted by using a lens which is not achromatic. Figs. 4(a) and (b) shows, respectively, a wrapped phase and the corresponding 3-D phase map of aspheric lens having a focal length of 50 mm by using the Lau phase interferometry. It can be seen that phase plane remains flat at the mid section however towards the corner, a curved plane is observed. This is due to the aperture of the lens is also not in the paraxial region and the lens can have aberrations. In this study it is note worthy that the grating displacement required to produce the phase steps are of the same order as the grating pitch. Since the proposed method employs a course grating, it is relatively easy to achieve sufficiently accurate grating translations. 5. Conclusions Fig. 4. (a) Wrapped phase map for a non-achromatic lens. (b) 3-D plot of a test lens. The application of Phase shifting Lau phase interferometry for the measurement of the focal length of the lens was studied. The required phase step is achieved by an in-plane translation of an interferometer grating. As a white light

7 C.J. Tay et al. / Optics Communications 248 (2005) source is used no filtering is required to obtain high quality maps. Phase shifting techniques are being increasingly used with all interferometers. Incorporation of phase shifting in white light Lau phase interferometry has resulted in the removal of noise, increase in accuracy and almost real time operation. By introducing discrete shifts in the position of the fringes we can calculate the phase map in a relatively simple way. It is shown that a phase map corresponding to a test lens is a plane surface whose slope is inversely proportional to the focal length. Thus the focal length of a lens can be determined through slope measurement. The present study is carried out on a simple lens. However this technique can be used to interpret qualitatively phase objects of more complicated shapes. Acknowledgements The authors acknowledge the financial supports provided by the National University of Singapore under research Project R References [1] L.M. Bernardo, O.D.D. Soares, Appl. Opt. 27 (1988) 296. [2] Y. Nakano, K. Murata, Appl. Opt. 24 (1985) [3] K. Mastuda, T.H. Barnes, B.F. Oreb, C.J.R. Sheppard, Appl. Opt. 38 (16) (1999) [4] M.D. Angelis, S.De. Nicol, P. Ferraro, A. Finizio, G. Pierattini, Opt. Commun. 136 (1997) 370. [5] S.De. Nicol, P. Ferraro, A. Finizio, G. Pierattini, Opt. Commun. 132 (1996) 432. [6] B. DeBoo, J. Sasian, Appl. Opt. 42 (19) (2003) [7] J. Jahns, A.W. Lohman, Opt. Commun. 28 (3) (1979) 263. [8] F. Gori, Opt. Commun. 31 (1) (1979) 4. [9] R.J. Sudol, B.J. Thompson, Opt. Commun. 31 (2) (1979) 105. [10] R.J. Sudol, B.J. Thompson, Appl. Opt. 10 (5) (1971) [11] H.O. Bartelt, J. Jahns, Opt. Commun. 30 (3) (1979) 268. [12] G.J. Swanson, E.N. Leith, J. Opt. Soc. Am. 10 (4) (1982) 552. [13] V.V. Aristov, A.I. Erko, V.V. Martynov, Opt. Commun. 53 (3) (1985) 159. [14] K. Patorski, Opt. Acta 30 (1983) 745. [15] K. Patorski, Appl. Opt. 25 (14) (1986) [16] M. Thakur, A.L. Vyas, C. Shakher, Opt. Laser Eng. 36 (4) (2001) 373. [17] M. Thakur, A.L. Vyas, C. Shakher, Appl. Opt. 41 (4) (2002) 654. [18] M. Thakur, S.K. Angra, C. Shakher, Opt. Eng. 42 (1) (2003) 86. [19] M. Thakur, C. Shakher, Appl. Opt. 41 (10) (2002) [20] S.E. Bialkowski, Appl. Opt. 32 (18) (1993) [21] J.E. Greivenkamp, J.H. Bruning, in: D. Malacara (Ed.), Optical Shop Testing, Wiley, New York, [22] K. Creath, Temporal Phase Measurement Methods, Bristol and Philadelphia, [23] L. Angel, M. Tebaldi, R. Henao, Opt. Commun. 164 (4) (1999) 247.