Polarization sensitive phase-shifting Mirau interferometry using a

Letter

Vol. 40, No. 19 / October 1 2015 / Optics Letters

4567

Polarization sensitive phase-shifting Mirau interferometry using a liquid crystal variable retarder PETR BOUCHAL,1,2,* RADEK Cˇ ELECHOVSKÝ,3

AND

ˇ BOUCHAL3 ZDENEK

1

Institute of Physical Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic Central European Institute of Technology, Brno University of Technology, Technická 10, 616 00 Brno, Czech Republic 3 Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic *Corresponding author: [email protected] 2

Received 13 August 2015; revised 11 September 2015; accepted 13 September 2015; posted 15 September 2015 (Doc. ID 247859); published 1 October 2015

We present all-optical motionless arrangement for polarization sensitive phase-shifting (P-S) interferometry, where the phase shifts are accurately implemented by a liquid crystal variable retarder (LCVR). The LCVR is used as a polarization selective device capable of introducing a computercontrolled phase retardance between signal and reference waves with orthogonal linear polarizations. The proposed optical P-S is deployed in a polarization adapted commonpath Mirau interferometer. Application of the method to a setup using the Michelson interference objective or Linnik interference module is also outlined. The accuracy of the quantitative phase reconstruction is examined theoretically, and a possibility to reduce the optical path difference error below 1/200 wavelength is demonstrated experimentally. Benefits and application potential of the polarization P-S interferometry supported by versatile liquid crystal devices are also discussed. © 2015 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.5050) Phase measurement; (230.3720) Liquid-crystal devices. http://dx.doi.org/10.1364/OL.40.004567

Interferometry and holography are historically well-established areas of optics with many applications in metrology and industry. In recent years, the significance of these techniques has been further enhanced because of extensive use in optical and digital microscopy. The P-S interferometry represents a widely used method for surface characterization that relies on the digital processing of phase-shifted interference patterns [1]. In experiments, the P-S is most often performed by a mechanical motion using a piezoelectric transducer. However, piezoelectric materials exhibit nonlinearity, hysteresis, and response to temperature resulting in P-S errors that must be reduced by special algorithms [2]. Methods enabling optical P-S have also been developed. The phase grating displayed on a spatial light modulator (SLM) was used in a Michelson-type P-S interferometer [3]. An achromatic P-S was advantageously realized by the 0146-9592/15/194567-04$15/0$15.00 © 2015 Optical Society of America

geometric Pancharatnam phase. In this case, the required phase retardance was achieved using a rotating half-wave plate [4], or a rotating polarizer [5], in a circularly polarized beam. The P-S is of fundamental importance also in digital holography, where the holograms recorded in off-axis or in-line geometries are numerically reconstructed. In the preferred in-line holography, an optimal space-bandwidth product is achieved [6], but at least three phase-shifted holograms are needed for a faithful image reconstruction. Recently, digital holography has been successfully deployed in advanced imaging techniques of optical microscopy. In the experiments, an exceptionally stable common-path recording configuration has been of particular interest. The stability of the common-path systems is inherently ensured by the identical optical path of interfering waves, but implementation of the P-S is more challenging. To introduce a phase retardance between these waves, the phases must be controlled selectively. In the available techniques, the Fourier filtering is usually combined with light-shaping technologies providing a precise phase modulation. In a point diffraction interferometer, a SLM was used to realize the three-step P-S, enabling a robust wavefront sensing [7]. In the spatial light interference microscopy (SLIM) [8], instantaneous spatial light interference microscopy (iSLIM) [9], or modified Gabor holography [10], a reference wave is created by filtering the spatial spectrum of the specimen. The phase shifts are then imposed on this wave by the SLM placed at the Fourier plane. In the Fresnel incoherent correlation holography (FINCH) [11–13], the SLM is used as a diffractive beam splitter that also provides a polarization sensitive phase modulation of incoming light waves. Spherical waves emanating from each point of incoherently illuminated or fluorescent object are divided by the SLM to create the signal and reference waves. Their wavefronts are shaped independently by a phase modulation, and, simultaneously, the required phase retardance is applied. The SLM-based P-S was also used in a coherent imaging realized by the common-path lensless holographic microscopy [14]. Unfortunately, in the spatial filtering techniques, the P-S is applied to the light passing through the sample so that an independent reference wave is not provided. In holographic microscopy, the systems operating with the

4568

Vol. 40, No. 19 / October 1 2015 / Optics Letters

reference wave unaffected by the specimen are highly desirable, and their design is a challenge. In this Letter, the modified Mirau interference objective (MO) is utilized in the polarization adapted setup, operating with a motionless all-optical P-S. The P-S is based on a liquid crystal light modulation, and, advantageously, imposed on an unaffected reference wave by the LCVR. In the experiment, incoming linearly polarized light is divided into reference and signal waves having orthogonal linear polarizations at the output of the interferometer. Because of birefringence properties of liquid crystals, the LCVR is operated as a polarization sensitive device introducing a required phase retardance between the output waves. The computer-controlled phase changes are extremely accurate and can be created in the operation time of several milliseconds. The proposed motionless P-S benefits from the polarization MO recently invented in [15,16]. The P-S is achieved by a polarization adjustment of the experiment preserving geometric configuration of the standard Mirau interferometry using the piezoelectric transducer. With a simple polarization modification, the LCVR can be deployed also in the setup, using the Michelson interference objective or the Linnik interference module. The proposed polarization sensitive Mirau interferometer using the motionless all-optical P-S is shown in Fig. 1. In the illumination part, a rotating diffuser RD is placed into the laser beam to reduce the spatial coherence. A static diffuser SD and a polarizer P support Köhler illumination of the sample by linearly polarized light. The illumination beam is deflected by a beam splitter BS toward the MO. The MO is supplemented by

Fig. 1. Schema of the polarization sensitive P-S Mirau interferometer: RD, rotating diffuser; SD, static diffuser; P, linear polarizer; MO, Mirau interference objective; QWP1 and QWP2, quarter-wave plates; BS, beam splitter; LCVR, liquid crystal variable retarder; A, analyzer; TL, tube lens; CCD, charge coupled device.

Letter a pair of quarter-wave plates, QWP1 and QWP2, placed into the optical paths of the reference and signal waves, respectively. The fast axis of the QWP1 coincides with the polarization direction of the input beam, while the fast axis of the QWP2 is rotated by 45°. After double passage of the reference wave through the QWP1, the polarization direction is retained. Since the polarization direction of the signal wave is rotated by 90° after the double passage through the QWP1 and QWP2 and the reflection from the sample, the waves with orthogonal linear polarizations are obtained behind the MO. The orthogonally polarized signal and reference waves then pass through the polarization selective LCVR. The P-S is enabled by anisotropic nematic liquid crystal molecules arranged in uniaxial birefringent layers in the LCVR. With no voltage applied, the liquid crystal molecules lie parallel to the glass substrates, and maximal retardation between orthogonally polarized waves is achieved. As voltage increases, the orientation of the molecules is changed causing a reduction in the effective birefringence, and, hence, the phase retardance. In this way, the phase shift between the signal and reference waves can be precisely adjusted. By analyzer A, the orthogonal linear polarizations are projected into the same direction to enable interference of the signal and reference waves. By setting analyzer orientation, the intensity of the interfering waves can be adjusted to achieve the highest contrast of the interference pattern. The accuracy of the proposed method is affected by several factors, including P-S errors, intensity fluctuations, system vibrations, detector nonlinearities, and quantization errors. To assess the feasibility of the polarization P-S based on the liquid crystal technology, an ideal anti-vibration system free from detection and data acquisition errors was considered, and only inaccuracies caused by the LCVR were included in the theoretical analysis. In this case, the deviations of the reconstructed phase from the precise values are directly related to the P-S errors associated with the LCVR accuracy and the throughput stability. A coherent superposition of the signal wave U S
AS expiΦS  and the reference wave U R j
AR expiΦR iφj  results in the interference patterns, whose intensity detected by the CCD is given as I j
jU S  U R j2 . Here AS , AR and ΦS , ΦR are the amplitudes and the phases of the signal and reference waves, respectively, and φj denotes the constant phase shifts imposed on the reference wave by the LCVR. Applying the four-step P-S with φj
j −2π∕2, j
1; 2; 3; 4, the phase retardance between the signal and reference waves, ΔΦ
ΦS − ΦR , can be quantitatively reconstructed using the formula ΔΦ
arctanI 1 − I 3 ∕I 2 − I 4 . In interference experiments with the LCVR, the phase shifts φj are provided with a limited accuracy, and the amplitude of the reference wave AR may slightly vary, when the driving voltage is changed. The operation errors of the LCVR determine the accuracy of the phase reconstruction, and their examination is of particular importance for an assessment of the LCVR applicability in the polarization sensitive P-S. The effects of the LCVR accuracy were tested both numerically and experimentally. In the simulation, the phase reconstruction was examined for the phase retardance with a linear dependence on the x-coordinate. The recording and reconstruction of phase-shifted interferograms were repeated 500×, while randomly varying amplitude and phase shift of the reference wave to simulate the LCVR errors. The maximal amplitude deviation was set to 0.05AR , while the maximal

Letter ambiguity of the adjusted phase 0.05 rad was used. Both values were found experimentally as typical accuracy limits of the LCVR. The phase recovered by the four-step technique was unwrapped using the Goldstein algorithm [17] and compared with the precise phase profile ΔΦ. The standard deviation σ was evaluated for 500 realizations. In Fig. 2, the standard deviation at different x-positions of the phase profile is illustrated with the maxima reaching approximately σ
0.02 rad. This means that the inaccuracies of the LCVR allow us to obtain a surface profile of the test object with an accuracy better than λ∕300, where λ is the wavelength of the used light. The feasibility of the polarization sensitive P-S Mirau interferometry was tested experimentally using the setup shown in Fig. 1. To ensure conditions of a high-precision measurement, the system was built at the optical breadboard with a passive vibration isolation and placed on an active damped optical table. To suppress the effects of air turbulence and increase the thermal stability, the Mirau interferometer was operated in an enclosure system with door assemblies. As a light source, a He–Ne laser (632 nm, 15 mW) was used. For spatial coherence reduction of the laser beam, the rotating diffuser RD was utilized, operating with the electric stepper motor Precistep RDM66200. The polarization version of the MO has been patented [15], but it is still commercially unavailable. Therefore, the standard MO (Nikon 10×, NA
0.3) was used to perform the polarization modification. To avoid disassembling of the MO, the QWP1 included to compensate the optical path of the reference wave was not used in the realized experiment. To minimize the optical path difference, λ∕4 polymer retarder film with a thickness of 75 μm was used as the QWP2 and placed into the optical path of the signal wave. The polarization sensitive P-S was carried out by the LCVR (Thorlabs LCC1223-A). The LCVR is based on nematic liquid crystal technology providing a high retardance uniformity and stability with a low wavefront distortion. The interferograms were formed by the tube lens (Nikon CFI60) and captured at the CCD (uEye UI-164x or Retiga 4000R). Data acquisition and processing were controlled by the computer through the LabVIEW interface. System adjusting and checking of the LCVR and polarization components were performed using the polarimeter (Thorlabs, PAX5710VIS-T). Experimental testing of the system was performed in two steps devoted to the P-S reproducibility and the achievable accuracy of the phase recovery. For testing, a tilted plane mirror introducing a linear phase retardance was used as a sample. In each measurement, four interferograms I j were recorded to apply the P-S procedure. In Fig. 3(a), the normalized intensity of the interferogram I 1 obtained for φ1
−π∕2 is shown. The wrapped phase ΔΦ and its 3D representation after unwrapping are illustrated in Figs. 3(b) and 3(c), respectively. To obtain

Fig. 2. Standard deviation σ of the recovered phase obtained by the numerical simulation of random amplitude and P-S errors of the LCVR.

Vol. 40, No. 19 / October 1 2015 / Optics Letters

4569

Fig. 3. Experimental evaluation of the P-S reproducibility performed with a tilted plane mirror: (a) demonstration interferogram I 1 , (b) wrapped phase, (c) unwrapped phase, (d) standard deviation σ A related to the average phase ΔΦA (realization of 99 measurements).

statistical data, the measurement was repeated 99×. From the phase reconstruction carried out in the individual measurements, an average phase ΔΦA was calculated, and, subsequently, used to evaluate the standard deviation σ A overall realizations. The distribution of σ A in the plane x; y, where the average linear phase ΔΦA was evaluated, is shown in Fig. 3(d). The one-dimensional profile of σ A along the direction shown is also added. The standard deviation σ A evaluated by the average phase ΔΦA gives information about the reproducibility of the phase shifts provided by the LCVR. Examining the maxima of σ A , it can be estimated that the phase shifts are reproducibly adjusted with the accuracy better than λ∕250. The absolute accuracy of the proposed technique was assessed using a single phase reconstruction randomly selected from a set of measurements performed. The phase errors at the individual points of the x; y plane were evaluated with respect to an ideal planar surface, whose inclination was determined by fitting the experimental data [Fig. 4(a)]. Profiles of the phase errors in the selected sections are shown in Fig. 4(b). The standard deviation was evaluated by processing the phase errors over all pixels of the measured phase surface. Its value was determined as σ
0.028 rad, providing the surface profile accuracy of λ∕225 with the peak-to-valley given by λ∕43. The determined accuracy is in good agreement with the numerical analysis predicting the theoretical precision of λ∕300. A slight discrepancy is caused by previously mentioned effects, which were not included in the simulations. In further experiment, an array of microlenses with the parameters provided by the producer was used as a sample (Thorlabs, MLA300-14AR). Because of the antireflection coatings applied to the microlenses, the reflectivity of the

4570

Vol. 40, No. 19 / October 1 2015 / Optics Letters

Fig. 4. Experimental evaluation of the absolute measurement accuracy: (a) phase errors at individual points of the measured planar surface and (b) phase errors in indicated sections.

measured surface was less than 1% so that the interferograms were recorded under low light conditions. Although the captured interference patterns were noisy, the phase unwrapping was successful, even if filtering of the records for the noise reduction [18] was not applied. The experimental results shown in Fig. 5 were compared with the numerical simulation carried out using the parameters given in the Thorlabs catalog. In the measured array, the square plano–convex microlenses with the theoretical focal length f
18.6 mm, the radius of curvature r
8.6 mm and the pitch p
300 μm were used. By means of r and p, the maximal height of the spherical cap along the diagonal of the array was calculated, h
2.616 μm. The maximal height h is given by the sum of the spherical cap heights h1 and h2 evaluated along Sections A and C shown in Fig. 5(a), h
h1  h2 . For the theoretical parameters, both heights take the same value, h1
h2
1.308 μm. The surface topography obtained by the measurement [Figs. 5(a) and 5(b)] was evaluated in Sections A, B [Fig. 5(c)] and C, D [Fig. 5(d)] taken across the valleys and the tops of the microlenses, respectively.

Fig. 5. Phase reconstruction of the array of microlenses (Thorlabs, MLA 300-14 AR) carried out under low light conditions: (a) unwrapped phase with color representation of the phase levels, (b) 3D illustration of the unwrapped phase, (c) height profiles across the valleys of the microlenses (Sections A and B), (d) height profiles across the tops of the microlenses (Sections C and D).

Letter In the statistical assessment of the experimental results, the microlenses available in the field of view of the MO were used. Evaluating the experimental data, the heights of the spherical cap in the Sections A, B and C, D were determined as h1
1.304  0.004 μm and h2
1.290  0.006 μm, respectively, and the pitch of the microlenses was measured as p
303.6  0.9 μm. By the best fitting of the experimental data, the radius of curvature of the microlenses r
8.68 mm was determined, which is also close to the value given by the producer. In this Letter, a polarization modification of the Mirau interferometer enabling all-optical motionless P-S implemented by the LCVR has been proposed and examined. The results obtained have proved a conceptual correctness and feasibility of the proposed method and its applicability to quantitative high-resolution phase measurements. It is important to note that the proposed experiment goes beyond the demonstrated applications. It represents a promising polarization technique for selective control and independent shaping of the signal wave and the unaffected reference wave in the common-path interferometry. Utilizing more versatile liquid crystal devices, such as liquid crystal lenses or SLMs, the proposed configuration can open new horizons for innovative experiments of optical and digital microscopy operating with adaptive elements [19] or special vortex beams [20,21]. Funding. Central European Institute of Technology (CZ.1.05/1.1.00/02.0068); Grant Agency of the Czech Republic (15-14612S); IGA Palacky University (PrF 2015 002); POSTUP II, European Social Fund in the Czech Republic (CZ.1.07/ 2.3.00/30.0041).

REFERENCES 1. I. Yamaguchi and T. Zhang, Opt. Lett. 22, 1268 (1997). 2. Ch. Ch. Jaing, Y. L. Shie, Ch. J. Tang, Y. Y. Liou, Ch. M. Chang, and Ch. R. Yang, Opt. Rev. 16, 170 (2009). 3. Y. Bitou, Opt. Lett. 28, 1576 (2003). 4. P. Hariharan, K. G. Larkin, and M. Roy, J. Mod. Opt. 41, 663 (1994). 5. S. S. Helen, M. P. Kothiyal, and R. S. Sirohy, Opt. Commun. 154, 249 (1998). 6. L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, Opt. Express 13, 2444 (2005). 7. V. Akondi, A. R. Jewel, and B. Vohnsen, Opt. Lett. 39, 1641 (2014). 8. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, Opt. Express 19, 1016 (2011). 9. H. Ding and G. Popescu, Opt. Express 18, 1569 (2010). 10. V. Micó, J. García, Z. Zalevsky, and B. Javidi, Opt. Lett. 34, 1492 (2009). 11. J. Rosen and G. Brooker, Opt. Lett. 32, 912 (2007). 12. G. Brooker, N. Siegel, V. Wang, and J. Rosen, Opt. Express 19, 5047 (2011). 13. P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, Opt. Express 19, 15603 (2011). 14. V. Micó and J. García, Opt. Lett. 35, 3919 (2010). 15. J. Schmit and P. Hariharan, “Polarization Mirau interference microscope,” U.S. patent 8,072,610 B1 (December 6, 2011). 16. A. M. Tapilouw, L. Ch. Chen, Y. J. Jen, S. T. Lin, and S. L. Yeh, Opt. Lett. 38, 2502 (2013). 17. R. M. Goldstein, H. A. Zebken, and C. L. Werner, Radio Sci. 23, 713 (1988). 18. J. C. Estrada, M. Servin, and J. A. Quiroga, Opt. Express19, 5126 (2011). 19. T. Shirai, T. H. Barnes, and T. G. Haskell, Opt. Lett. 25, 773 (2000). 20. Ch. Maurer, S. Bernet, and M. R. Marte, J. Opt. A 11, 094023 (2009). 21. P. Bouchal and Z. Bouchal, Opt. Lett. 37, 2949 (2012).