Digital phase-shifting point diffraction interferometer - OSA Publishing

March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS

1641

Digital phase-shifting point diffraction interferometer Vyas Akondi,* A. R. Jewel, and Brian Vohnsen Advanced Optical Imaging Group, School of Physics, University College Dublin, Ireland *Corresponding author: [email protected] Received November 21, 2013; revised February 11, 2014; accepted February 16, 2014; posted February 18, 2014 (Doc. ID 201769); published March 13, 2014 A digital phase-shifting (PS) point diffraction interferometer is demonstrated with a transmitting liquid crystal spatial light modulator. This novel wavefront sensor allows tunability in the choice of pinhole size and eliminates the need for mechanically moving parts to achieve PS. It is shown that this wavefront sensor is capable of sensing Zernike aberrations introduced with a deformable mirror. The results obtained are compared with those of a commercial Hartmann–Shack wavefront sensor. © 2014 Optical Society of America OCIS codes: (010.7350) Wave-front sensing; (120.5050) Phase measurement; (010.1080) Active or adaptive optics; (280.4788) Optical sensing and sensors; (350.1260) Astronomical optics; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/OL.39.001641

Hartmann–Shack (HS) wavefront sensors are widely used in adaptive optics applications [1–4]. The sensitivity and dynamic range of this sensor is primarily limited by the number of subapertures and the focal distance of the microlenses, respectively [5], which motivated the development of pyramid [6], curvature [7,8] and signal-based [9] wavefront sensors. Previously known common path interferometers like the point diffraction interferometer (PDI) [10–13], which are less sensitive to environmental changes, were mainly used in applications involving optical quality testing in lenses and fluid flow characterization. It is a challenging task to build more robust phaseshifting (PS) interferometers with the PDI. Previous studies showed that the use of a liquid crystal (LC) in PS-PDI could make it both robust and effective [14]. In this Letter, results from implementing a digital PS-PDI based on a transmitting LC spatial light modulator (SLM) are presented. The use of the SLM for making a pyramid wavefront sensor showed great tunability earlier [15]. As the reference pinhole size increases in the PS-PDI, the wavefront estimation error increases [16]. The current technology allows generation of a pinhole in the range of tens of micrometers with the SLM. In addition to allowing tunability of the pinhole size, the novel digital PS-PDI eliminates problems with making customized pinholes [17]. The schematic diagram of the proposed digital PS-PDI is shown in Fig. 1(a), similar to its conventional counterpart, with the difference that the pinholes are tailored using the SLM. Figure 1(a) shows the interference fringes recorded by the camera when a defocus aberration with a phase magnitude of 2.32 μm is introduced in the aperture plane. The intensity, I
x0 ; y0 , at the detector plane can be expressed as I
x0 ; y0   jFTFT
P
x; y:eiϕ
x;y :M
X; Y j2 ;

Here, L and l are the lengths of the sides of the two square pinholes in the mask function, h defines the separation of the pinholes, and c is a constant value that defines the degree of transmittance. θ1 is fixed at zero and θ2 is varied to achieve PS. P
x; y is the defined pupil function. FT represents the Fourier transform function. The optimal value of L depends largely on the spatial frequency of the aberration function, ϕ
x; y [18]. The use of the SLM allows a tunable “L” in closed-loop operation. The choice of “l” is made such that the reference pinhole is much smaller than the diffraction-limited spot size, and hence it could in general be fixed. The smaller the value of “l,” the more uniform is the reference wave [16], but at the cost of decreased light intensity. The values “h” and “c” can be tuned to gain in terms of fringe contrast. However, the separation between the pinholes cannot be indiscriminately increased due to the finite size of the point spread function in the plane of the SLM. Three intensity measurements, I 1 , I 2 , and I 3 corresponding to phase shifts of 0, π∕2, and π, respectively, help in estimating the wrapped wavefront phase, ϕ
x; y, as described by Hariharan [19]:
ϕ
x; y  tan−1

 2I 2
x; y − I 3
x; y − I 1
x; y : I 3
x; y − I 1
x; y

(3)

Using a simple and quick phase unwrapping algorithm, the wavefront phase is estimated [20]. It is convenient to represent the wavefront in terms of the standard Zernike polynomials since they form a complete,

(1)

where M
X; Y  is the pinhole mask function [see Fig. 1(b)], defined as follows: 8 iθ1 e ; if − L∕2 ≤ X ≤ L∕2; > > > > > and − L∕2 ≤ Y ≤ L∕2 > < iθ (2) M
X; Y   e 2 ; if − l∕2 ≤ X ≤ l∕2; > > > > and − h − l∕2 ≤ Y ≤ −h  l∕2 > > : c; otherwise: 0146-9592/14/061641-04$15.00/0

Fig. 1. (a) Schematic of the digital PS-PDI. (b) Pinhole mask function. © 2014 Optical Society of America

1642

OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014

orthogonal basis over the unit circle. Least squares fitting-based Zernike modal decomposition is performed to estimate the Zernike coefficients corresponding to the reconstructed phase [8]. Here, an 8-bit transmitting SLM device with 1024 × 768 pixels, from Holoeye (Model: LC 2012, pixel pitch  36 μm and pixel fill factor  58%), was used. The intensity transmission of the SLM, placed in between two crossed polarizers (input polarizer at 45° with the horizontal) was measured using a power meter by varying the uniform gray scale. Due to the nonlinear behavior of the SLM, the measured power did not follow the expected curve of the normalized theoretical power. To achieve linearization, a ninth degree polynomial was fitted on the measured power, and by applying inversion, the gray scale values were modified. This linearization procedure allowed good agreement with the theoretical power. The schematic of the experimental setup to demonstrate the digital PDI is shown in Fig. 2. A 5 mW He–Ne laser (632.8 nm) is used as light source. The HS wavefront sensor is placed in the conjugate plane of the deformable mirror (DM, with 140 actuators from Boston Micromachines), which is used to generate aberrations in a controlled manner. The SLM is placed in the focal plane of a 500 mm lens that is located at the back focal plane of the DM. Beyond the SLM, the beam size is reduced on to the CCD camera with the help of a 300 mm lens to capture the interference fringes. The results of wavefront sensing with the SLM-based PDI are shown in Fig. 3. The larger pinhole was generated using 3 × 3 SLM pixels and the smaller pinhole using 2 × 2 SLM pixels, with a separation of seven pixels between them. Hence, L  108 μm, l  72 μm, and h  252 μm. The first 15 Zernike terms alone were considered for wavefront representation here. Also, tip-tilt terms were neglected since they arise primarily due to the PS approach and alignment errors. The root mean square (RMS) wavefront error between the estimated wavefront ˆ and the HS measured wavefront (ϕ) was evaluated as (ϕ) follows: s P ˆ 2 ij jϕij − ϕij j RMS  : N ×M

(4)

Fig. 2. Experimental setup to demonstrate the digital PDI. All lenses are achromatic doublets and the focal lengths are shown in millimeters.

Fig. 3. Comparison of the wavefronts reconstructed using a digital PDI with that measured by the HS while sensing (a) primary defocus, (b) primary astigmatism, (c) coma, and (d) secondary astigmatism. The peak-to-valley of the HS measured and the PDI reconstructed wavefronts is shown above individual wavefronts.

Here, N × M represents the number of CCD pixels used to capture the pupil plane, which is also the dimension of ˆ The predicted values of the RMS the matrices, ϕ and ϕ. wavefront error are 152.65, 94.14, 104.34, and 234.63 nm for defocus, astigmatism, coma, and secondary astigmatism, respectively. The low accuracy is attributed to inaccurate generation of the reference wavefront, noncommon path tilt aberrations, the presence of noise in the obtained fringes, and resultant inaccuracies in the phase unwrapping algorithm. The RMS wavefront error values obtained using the novel digital PDI are close to those obtained using a digital pyramid wavefront sensor [15]. A closed-loop operation could lead to lower RMS wavefront error in the case of the digital PS PDI case as well. Figure 4 shows the plot of the estimated defocus coefficient while sensing defocus aberration of increasing magnitude. It was seen that as the HS measured defocus coefficient increases (approximately) beyond 0.69 μm (corresponding to the HS measured peak-to-valley of 2.32 μm), the estimated defocus coefficient using the digital PDI saturates. Simulations were performed to test the working of the novel digital PS-PDI. Defocus (Z 20 ), astigmatism (Z 22 ), coma (Z 31 ), and secondary astigmatism (Z 42 ) were simulated with the peak-to-valley equal to that measured by the HS in Fig. 3. The intensities I 1 , I 2 , and I 3 corresponding to phase shifts θ2  0, π∕2, π, respectively, at the detector plane were evaluated by using Eq. (1). In order to closely match the experimental situation, λ  632.8 nm, L  108 μm, l  72 μm, and h  252 μm were chosen. The wrapped phase was estimated using Eq. (3). The RMS wavefront error was then calculated after phase

March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS

1643

Fig. 4. As the defocus magnitude is increased, the accuracy of measuring the defocus coefficient, a4 , saturates beyond a certain level.

Fig. 6. Simulations: the RMS wavefront error drops with increasing SNR while sensing primary defocus aberration with a peak-to-valley of 1.87 μm.

unwrapping [20] and least squares fitting with 15 Zernike terms. Figure 5 shows the simulation results for defocus, astigmatism, coma, and secondary astigmatism. The RMS wavefront error estimated through simulations is nearly 2 orders of magnitude lower than λ. It was shown earlier that an in-line configuration can reduce coherent noise by low-pass filtering or lateral averaging of the interference pattern [21]. Here, a median filter was used to nullify the high spatial frequency content caused by dust and inhomogeneities in the interference pattern. The inaccuracies in generating a proper reference wavefront and the presence of extraneous fringes are other error sources. Since the wavefront phase is retrieved from the difference of the interference patterns [see Eq. (3)], even a small amount of detector

readout noise or scatter noise could affect the accuracy of wavefront sensing. To examine this, uniform random noise was added to the simulated intensity while sensing primary defocus aberration with a peak-to-valley of 1.87 μm. The results of the estimated RMS wavefront error are shown in Fig. 6. The error bars correspond to six repetitions of the RMS wavefront error estimation. The RMS wavefront error drops exponentially with increasing SNR with a decay constant of approximately 0.59. It has to be noted that the finite pixel size of the SLM used does not permit an ideal reference wavefront for generating the interferograms. The difference between an ideal point source reference wavefront (ϕ) and the SLM generated reference wavefront (ϕSLM ) was estimated. Under the experimental conditions described in this Letter, the maximum phase difference along the cross section of ϕ − ϕSLM is nearly λ∕5. However, simulations show that using an SLM with smaller pixel pitch allows reduction of this number further, and hence decreasing the resultant RMS wavefront error while sensing different aberrations. In addition, a smaller pixel pitch offers greater tunability. An overestimation of the wavefront phase could result in a runaway process in closed-loop wavefront sensing. Nevertheless, underestimation of the wavefront magnitude is not much of a problem in closed-loop wavefront sensing applications. The sensitivity of the digital PDI could further be improved by increasing the focal length of the lens in front of the SLM. The use of a longer focal length lens increases the area occupied by the aberrated point spread function on the SLM and hence allows greater tunability in the values of “L” and “l.” However, similar results as shown in Fig. 3 were obtained while using a lens of focal length 1000 mm and by setting L  216 μm, l  144 μm, and h  504 μm. In summary, a novel digital PDI has been demonstrated. The use of the LC SLM allows tunability in the pinhole size. In addition, the implementation of PS does not require moving parts. The use of advanced phase unwrapping methods could further improve the accuracy of wavefront sensing.

Fig. 5. Simulations: results of wavefront reconstruction using the digital PS-PDI. Here, L  108 μm, l  72 μm, and h  252 μm. The estimated RMS wavefront error while sensing (a) primary defocus (Z 20 ), (b) primary astigmatism (Z 22 ), (c) primary coma (Z 31 ), and (d) secondary astigmatism (Z 42 ) aberrations is 3.43, 2.86, 6.41, and 6.59 nm, respectively.

This work was supported by Science Foundation Ireland (grants: 07/SK/B1239a and 08/IN.1/B2053) and University College Dublin (seed funding: SF665).

1644

OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014

References 1. R. V. Shack and B. C. Platt, J. Opt. Soc. Am. 61, 656 (1971). 2. V. Akondi, M. B. Roopashree, and B. R. Prasad, in Topics in Adaptive Optics, R. K. Tyson, ed. (InTech, 2012), pp. 167–196. 3. D. Rativa, R. de Araujo, A. Gomes, and B. Vohnsen, Opt. Express 17, 22047 (2009). 4. J. Liang, B. Grimm, S. Goelz, and J. Bille, J. Opt. Soc. Am. A 11, 1949 (1994). 5. V. Akondi and B. Vohnsen, Ophthalmic Physiol. Opt. 33, 434 (2013). 6. R. Ragazzoni, J. Mod. Opt. 43, 289 (1996). 7. F. Roddier, Appl. Opt. 27, 1223 (1988). 8. R. M. Basavaraju, V. Akondi, S. J. Weddell, and R. P. Budihal, Opt. Commun. 312, 23 (2014). 9. A. R. Jewel, V. Akondi, and B. Vohnsen, JEOS RP 8, 13073 (2013). 10. V. Linnik, Proc. Acad. Sci. USSR 1, 208 (1933).

11. J. E. Millerd, S. J. Martinek, N. J. Brock, J. B. Hayes, and J. C. Wyant, Proc. SPIE 5380, 422 (2004). 12. M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, Opt. Lett. 31, 3597 (2006). 13. Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, Opt. Lett. 37, 3927 (2012). 14. C. Mercer and K. Creath, Opt. Lett. 19, 916 (1994). 15. V. Akondi, S. Castillo, and B. Vohnsen, Opt. Express 21, 18261 (2013). 16. R. Neal and J. Wyant, Appl. Opt. 45, 3463 (2006). 17. X. Jia, T. Xing, J. Xu, W. Lin, and Z. Liao, Proc. SPIE 8683, 86832F (2013). 18. C. Mercer and K. Creath, Appl. Opt. 35, 1633 (1996). 19. P. Hariharan, Appl. Opt. 26, 2506 (1987). 20. M. Herraez, D. Burton, M. Lalor, and D. Clegg, Appl. Opt. 35, 5847 (1996). 21. P. Gao, I. Harder, V. Nercissian, K. Mantel, and B. Yao, Opt. Lett. 35, 712 (2010).