Abstract

A parallel two-step spatial carrier phase-shifting common-path interferometer with a Ronchi grating placed outside the Fourier plane is proposed in this paper for quantitative phase imaging. Two phase-shifted interferograms with spatial carrier can be captured simultaneously using the proposed interferometer. The dc term can be eliminated by subtracting the two phase-shifted interferograms, and the phase of a specimen can be reconstructed through Fourier transform. The validity and stability of the interferometer proposed are experimentally demonstrated via the measurement of a phase plate.

© 2013 OSA

1. Introduction

Digital imaging interferometry (DII) is a noninvasive, high precision and full field of view technique, which can be used for quantitative phase imaging of surface profile or transparent specimen [1,2]. It includes the recording of interferograms using CCD camera and the extraction of the phase of a specimen using numerical reconstruction methods on a computer. However, there are unwanted dc and twin image terms in the interferograms. It is therefore of great significance to find ways and means to eliminate them, and so, much work has been done on this particular aspect. For example, it is suggested in Refs. [3] and [4] to eliminate these terms by imposing a large tilt between the object and reference beams to introduce a spatial carrier into the interferograms in off-axis DII. The phase can then be reconstructed from a single interferogram, which makes real-time measurement possible. However, the space-bandwidth of a CCD camera cannot be fully utilized by just doing so, and the spatial details of a specimen cannot be completely recorded, either. In order to make full use of the space-bandwidth of a CCD camera, on-axis DII [5,6] is proposed as a possible solution, but the time-sequent phase-shifting operation required makes real-time measurement impossible. What is more, the phase noise may increase as the surroundings of a system may vary from frame to frame. Parallel phase-shifting methods are therefore introduced into on-axis DIIs to record the phase-shifting interferograms in a single camera shot [7,8]. The field of view of a CCD camera cannot be efficiently utilized because at least three interferograms have to be captured. As a matter of fact, the utilization of the field of view of a CCD camera decreases as the number of interferograms recorded in one shot increases.

In order to further improve the utilization of the field of view of a CCD camera, a two-step on-axis DII is then proposed for recording only two (rather than three) on-axis interferograms [911]. But, this technique can be applied only after the reference field is pre-measured off-line. Parallel two-step slightly off-axis DII [1113] is also proposed for elimination of dc terms by subtracting the two parallel phase-shifted interferograms recorded in one shot. It provides an intermediate solution between the traditional off-axis and on-axis DII. But, this technique still needs adjustment of optical elements to generate a tilt between two beams.

C. Meneses-Fabian et al. [14] introduced carrier fringes into interferograms by placing a grating outside the Fourier plane in a two-aperture common-path interferometer in comparison with other interferometries. It does not need to tilt elements, while system complexity is reduced. And especially, a common-path configuration can be provided to improve its stability in variety of surroundings. However, even after doing so, the space-bandwidth of a CCD camera is still limited by the unwanted dc and twin image terms.

Therefore, a parallel two-step spatial carrier phase-shifting common-path interferometer with a Ronchi grating placed outside Fourier plane is proposed in this paper for quantitative phase imaging.

2. Experimental setup

As shown in Fig. 1 , the experimental setup consists of a nonpolarizing He-Ne laser with wavelength λ, polarizer P, collimator & expander CE, quarter wave plates QWL and QER, aperture Ap, Lenses L1 and L2 with focal length f, Ronchi grating G with period d, polarizing filter array PLA and CCD camera. It is a standard 4f optical image system with two transforming lenses L1 and L2. The light coming from a He-Ne laser is linearly polarized by polarizer P through an angle of 45° with respect to the horizontal axis, and then gets collimated and expanded by the collimator & expander (CE). In the input plane, the aperture with sides B and b is split into two identical apertures, with size B/2 × b each, to support the reference and object beams. Specimen S is placed before the object aperture. Quarter-wave plates QWR and QWL are 0° and 90° oriented with respect to horizontal axis, and placed before the two apertures to transform the polarized beam into orthogonal circular polarized object and reference beams. Ronchi grating G is placed outside the Fourier plane with displacement Δf, and acts as a spatial filter to diffract the incident light into the +1, 0 and −1 orders. While aperture side B is selected usingB = 2λf/d [8,14] with respect to grating period d, focal length f and wavelength λ, the superposition of diffraction orders can cause interferences to produce two interferograms in the output plane. Polarizing filter array (PLA) consists of two polarizing filters positioned in front of the CCD camera to perform polarization phase shifting. The two polarizing filters are placed with an angle α/2 between. Two interferograms with phase shift α modulated by PLA can be then obtained using a CCD camera in one shot.

 

Fig. 1 (a) Experimental setup for two-step spatial carrier phase-shifting common-path interferometer; (b) Polarization state of beams in the input aperture; (c) Polarization state of polarizing filter array(PLA). At the right side of grating G, the blue, red and green beams indicate the diffraction orders +1, 0 and −1, respectively; P, Polarizer; CE, Collimator & expander; QWR, QWL, Quarter wave plates; Ap, Aperture; S, Specimen; G, Ronchi grating; L1, L2, Lenses.

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2.1 Generation of spatial carrier

A carrier frequency is analyzed once again in this paper from another view point, although it has been analyzed using near-field diffraction theory in Ref. [14]. For simplicity, as shown in Fig. 2 , only the +1, 0 and −1 orders are considered in our analysis.

 

Fig. 2 Schematic for diffraction of grating. Orders +1, 0 and −1 are indicated with blue, red and green; Δf, displacement of grating; Δx, distance between adjacent orders in the Fourier plane; θ, diffraction angle of ±1st orders.

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In a Fourier plane, if the separation distance between adjacent orders is Δx, carrier frequency f0 in the interferograms introduced by order ±1 can be given by

f0=Δx/λf.

If Δf >>Δx, the expression can be obtained by combining the grating equation and the geometry relation in Fig. 2 as shown below:

sinθ=λd=tanθ=ΔxΔf.

From Eqs. (1) and (2), carrier frequency f0 can be given by

f0=Δffd.

From Eq. (3), the carrier frequency is directly proportional to displacement Δf and inversely proportional to grating period d and focal length f, which agrees well with the results reported in Ref. [14].

2.2 Phase reconstruction method

If the two identical apertures in the input plane are illuminated using a uniform plane wave, and a specimen with phase distribution φ(x,y) is placed in the object aperture, two interferograms with carrier frequency f0 and phase shift α can be obtained in the output plane, and intensity distributions [15] of two interferograms can be expressed as

I1(x,y)=a(x,y)+c(x,y)exp(i2πf0x)+c*(x,y)exp(i2πf0x),
I2(x,y)=a(x,y)+c(x,y)exp(iα)exp(i2πf0x)+c*(x,y)exp(iα)exp(i2πf0x)
and
c(x,y)=Aexp[iφ(x,y)],
where a(x,y) and A are the variations in background, and the local contrast of fringe; * denotes the complex conjugation. It can be seen from Eqs. (4a) and (4b) that dc term a(x,y) can be easily eliminated by subtracting I2 from I1.

To reduce the error resulting from the mismatch in carrier frequency during phase reconstruction [13], a digital reference wave can also be used as shown below:

Rr=exp(i2πf0x)=exp(i2πΔffdx).

The spatial carrier of real image term can be eliminated using Rr(I1-I2), and the frequency spectrum of real image can also be moved to the center of the spectrum map. The real image term can then be obtained using a low pass filter in the frequency domain. The complex amplitude of real image term can be calculated through inverse Fourier transform [15] as shown below:

c(x,y)=c(x,y)[1exp(iα)]=IFT{FT{Rr(I1I2)}LF},
where LF is a low pass filter, FT and IFT denote Fourier transform and inverse Fourier transform. For 1-exp(iα) is only a constant term, it does not influence the phase distribution of a measured specimen. So, the phase distribution of a measured specimen can be expressed as
φ(x,y)=Im[c(x,y)]Re[c(x,y)]φC,
where Im() and Re() mean extractions of image and real part, and φC is a constant value.

It can be seen from the presentation above that most common noise and background are removed using the interferometer proposed. Moreover, the proposed interferometer can also achieve the time resolution of off-axis DII because the information gained for each observation of a specimen can be obtained in a single shot. The spatial bandwidth of a CCD camera can also be utilized more efficiently, and the information gained from the specimen can be maximized. The rate of measurement is determined not by the rate at which the dynamic processes change, but by the true frame rate of the CCD camera used. So, it is believed that this interferometer proposed can be used to provide a powerful tool for measuring moving objects and dynamic processes.

3. Experimental results

To prove the effectiveness of the interferometer proposed, three experiments are made under normal ordinary laboratory conditions as shown in Fig. 1. The experimental setup is established using a collimated and expanded He-Ne laser with wavelength λ = 632.8nm, two lenses with focal length f = 250mm and diameter 25.4mm, a Ronchi grating with period d = 50μm. The width of the aperture can be calculated using B = 2λf/d = 6.328mm, and the measurement width of Specimens is limited by aperture width B/2 = 3.164mm. A PLA with two transmission angles at 0 and π/2 is used to perform phase-shifting, and a CCD camera with 1600 × 1200 pixels and pixel size 4.4um × 4.4um is used to record the interferograms.

The first experiment is made to verify the generation of spatial carrier. No specimen is placed in the object aperture, and an interferogram with a spatial carrier can be obtained in the output plane with displacement Δf = 100.34mm as shown in Fig. 3(a) . The carrier frequency can be obtained by counting the number of fringes. Figure 3(b) shows that there are 16 periods of fringes in a length of about 1.975mm, and so, the carrier frequency is about 8.101 lines/mm. The spectrum distribution in the Fourier plane is also recorded and shown in Fig. 3(c). It can be seen from Fig. 3(d) that the separation distance Δx is about 1.276mm. According to Eq. (1), the carrier frequency is about 8.066 lines/mm. In addition, according to Eq. (3), the carrier frequency can also be 8.027 lines/mm. It can be seen through comparison that all the experimental results coincide well with theoretical results, which proves the validity of our analysis, although the experimental results suffer from such perturbations as the fabrication error of grating, the nonuniformity of illumination light, and the nonlinearity of a CCD camera.

 

Fig. 3 Experimental results without specimen: (a) 2D intensity distribution recorded in the image plane and (b) 1D profile of (a) along the red line; (c) 2D and (d) 1D spectrum distribution recorded in the Fourier plane.

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A phase plate of about 580.22nm high is used as the specimen for the second experiment, and the height of the phase plate is measured using BRUKER Atomic Force Microscopy (AFM). The phase plate was etched in BK7 glass with a refractive index of 1.5168. The Ronchi grating is placed outside the Fourier plane with displacement Δf = 112.43mm. The two interferograms with carrier frequency f0 = Δf/(fd) = 8.994 lines/mm and phase shift of π are obtained in one shot as shown in Fig. 4(a) . The optical path differences (OPD) distribution of the specimen can be retrieved as shown in Fig. 4(b) from a single interferogram using the method proposed in Ref. [14], and as shown in Fig. 4(c) from two interferograms using the interferometer proposed in this paper. Figures 4(b) and 4(c) show that the proposed interferometer can be used to obtain a higher quality, and especially finer details about the phase distribution compared to what can be achieved using the method in Ref. [14]. The 1D profile underlined by a dash line in Fig. 4(c) also obtained as shown in Fig. 4(d). Figure 4(d) shows that the difference in height of the specimen is approximately 567.41nm, which agrees well with the results obtained using AFM. In addition, the standard deviations for the upper and lower sections of OPD are 12.540nm and 24.348nm respectively, which also indicates that the noise is low and approximately constant. It should be also noted that due to the unitary magnification of the system, the lateral resolution of the setup is limited by the pixel size of a CCD camera. Our next job is to achieve the diffraction-limited resolution using a microscopic objective with high magnification [4,1113].

 

Fig. 4 Experimental results for a phase plate: (a) interferograms with phase shift π; (b) reconstructed OPD distribution using the method reported in Ref. [14]; (c) reconstructed OPD distribution using the proposed method; (d) profile along the dash line marked in (b).

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To evaluate the stability of the experimental setup, ninety measurements are carried out on the phase plate with a time interval of one minute. As shown in Fig. 5 , the OPD fluctuation at a randomly selected point in the OPD map is highlighted with a blue rectangular line, and the standard deviation for the OPD fluctuation in height is 4.6nm, i.e., the setup has long stability. In addition, to assess the repeatability of the experimental setup, the difference between two successive measurements Hti-Hti-1 is also investigated using the standard deviation, and the measurements are shown using a red circular line in Fig. 5. The average standard deviation is 3.3nm, which means the setup has a fairly high repeatability. Both the high stability and repeatability of the experimental setup are attributed to the common-path configuration of the experimental setup.

 

Fig. 5 Stability and repeatability of the proposed method. Hti denotes the ith measurement and RMS denotes the standard deviate.

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4. Conclusions

A common-path interferometer with a Ronchi grating placed outside the Fourier plane is proposed to capture two phase-shifted interferograms with spatial carrier simultaneously. The dc term can be eliminated by subtracting the two phase-shifted interferograms, and the phase of a specimen can be reconstructed through Fourier transform. Experimental results indicate that the proposed method can obtain finer details of the specimen with long stability and high repeatability.

Acknowledgments

This work is supported by National Natural Science Foundation of China (60908026, 61102004), Major National Scientific Instrument and Equipment Development Project of China (No. 2011YQ040136) and Fundamental Research Funds for the Central Universities.

References and links

1. D. T. Lin and D. S. Wan, “Profile measurement of an aspheric cylindrical surface from retroreflection,” Appl. Opt. 30(22), 3200–3204 (1991). [CrossRef]   [PubMed]  

2. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999). [CrossRef]   [PubMed]  

3. M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21(3), 367–377 (2004). [CrossRef]   [PubMed]  

4. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006). [CrossRef]   [PubMed]  

5. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef]   [PubMed]  

6. C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun. 282(15), 3063–3068 (2009). [CrossRef]  

7. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000). [CrossRef]  

8. B. Hao, M. Shan, M. Diao, Z. Zhong, and H. Ma, “Common-path interferometer with a tri-window,” Opt. Lett. 37(15), 3213–3215 (2012). [CrossRef]   [PubMed]  

9. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006). [CrossRef]   [PubMed]  

10. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47(19), D183–D189 (2008). [CrossRef]   [PubMed]  

11. P. Gao, B. Yao, I. Harder, J. Min, R. Guo, J. Zheng, and T. Ye, “Parallel two-step phase-shifting digital holograph microscopy based on a grating pair,” J. Opt. Soc. Am. A 28(3), 434–440 (2011). [CrossRef]   [PubMed]  

12. N. T. Shaked, Y. Zhu, M. T. Rinehart, and A. Wax, “Two-step-only phase-shifting interferometry with optimized detector bandwidth for microscopy of live cells,” Opt. Express 17(18), 15585–15591 (2009). [CrossRef]   [PubMed]  

13. P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, and K. Mantel, “Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beamsplitters,” Opt. Express 19(3), 1930–1935 (2011). [CrossRef]   [PubMed]  

14. C. Meneses-Fabian and G. Rodriguez-Zurita, “Carrier fringes in the two-aperture common-path interferometer,” Opt. Lett. 36(5), 642–644 (2011). [CrossRef]   [PubMed]  

15. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]  

References

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  • |

  1. D. T. Lin and D. S. Wan, “Profile measurement of an aspheric cylindrical surface from retroreflection,” Appl. Opt.30(22), 3200–3204 (1991).
    [CrossRef] [PubMed]
  2. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt.38(34), 6994–7001 (1999).
    [CrossRef] [PubMed]
  3. M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004).
    [CrossRef] [PubMed]
  4. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett.31(6), 775–777 (2006).
    [CrossRef] [PubMed]
  5. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997).
    [CrossRef] [PubMed]
  6. C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
    [CrossRef]
  7. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng.39(4), 960–966 (2000).
    [CrossRef]
  8. B. Hao, M. Shan, M. Diao, Z. Zhong, and H. Ma, “Common-path interferometer with a tri-window,” Opt. Lett.37(15), 3213–3215 (2012).
    [CrossRef] [PubMed]
  9. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett.31(10), 1414–1416 (2006).
    [CrossRef] [PubMed]
  10. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt.47(19), D183–D189 (2008).
    [CrossRef] [PubMed]
  11. P. Gao, B. Yao, I. Harder, J. Min, R. Guo, J. Zheng, and T. Ye, “Parallel two-step phase-shifting digital holograph microscopy based on a grating pair,” J. Opt. Soc. Am. A28(3), 434–440 (2011).
    [CrossRef] [PubMed]
  12. N. T. Shaked, Y. Zhu, M. T. Rinehart, and A. Wax, “Two-step-only phase-shifting interferometry with optimized detector bandwidth for microscopy of live cells,” Opt. Express17(18), 15585–15591 (2009).
    [CrossRef] [PubMed]
  13. P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, and K. Mantel, “Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beamsplitters,” Opt. Express19(3), 1930–1935 (2011).
    [CrossRef] [PubMed]
  14. C. Meneses-Fabian and G. Rodriguez-Zurita, “Carrier fringes in the two-aperture common-path interferometer,” Opt. Lett.36(5), 642–644 (2011).
    [CrossRef] [PubMed]
  15. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
    [CrossRef]

2012 (1)

2011 (3)

2009 (2)

N. T. Shaked, Y. Zhu, M. T. Rinehart, and A. Wax, “Two-step-only phase-shifting interferometry with optimized detector bandwidth for microscopy of live cells,” Opt. Express17(18), 15585–15591 (2009).
[CrossRef] [PubMed]

C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
[CrossRef]

2008 (1)

2006 (2)

2004 (1)

2000 (1)

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng.39(4), 960–966 (2000).
[CrossRef]

1999 (1)

1997 (1)

1991 (1)

1982 (1)

Awatsuji, Y.

Blu, T.

Cai, L. Z.

Cuche, E.

Dasari, R. R.

Depeursinge, C.

Diao, M.

Dong, G. Y.

Encarnacion-Gutierrez, M.-C.

C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
[CrossRef]

Feld, M. S.

Gao, P.

Guo, R.

Hao, B.

Harder, I.

Hettwer, A.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng.39(4), 960–966 (2000).
[CrossRef]

Ikeda, T.

Ina, H.

Kaneko, A.

Kobayashi, S.

Koyama, T.

Kranz, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng.39(4), 960–966 (2000).
[CrossRef]

Kubota, T.

Liebling, M.

Lin, D. T.

Ma, H.

Mantel, K.

Marquet, P.

Matoba, O.

Meneses-Fabian, C.

C. Meneses-Fabian and G. Rodriguez-Zurita, “Carrier fringes in the two-aperture common-path interferometer,” Opt. Lett.36(5), 642–644 (2011).
[CrossRef] [PubMed]

C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
[CrossRef]

Meng, X. F.

Min, J.

Nercissian, V.

Nishio, K.

Popescu, G.

Rinehart, M. T.

Rodriguez-Zurita, G.

C. Meneses-Fabian and G. Rodriguez-Zurita, “Carrier fringes in the two-aperture common-path interferometer,” Opt. Lett.36(5), 642–644 (2011).
[CrossRef] [PubMed]

C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
[CrossRef]

Schwider, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng.39(4), 960–966 (2000).
[CrossRef]

Shaked, N. T.

Shan, M.

Shen, X. X.

Tahara, T.

Takeda, M.

Toto-Arellano, N. I.

C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
[CrossRef]

Unser, M.

Ura, S.

Wan, D. S.

Wang, Y. R.

Wax, A.

Xu, X. F.

Yamaguchi, I.

Yang, X. L.

Yao, B.

Ye, T.

Zhang, T.

Zheng, J.

Zhong, Z.

Zhu, Y.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. Meneses-Fabian, G. Rodriguez-Zurita, M.-C. Encarnacion-Gutierrez, and N. I. Toto-Arellano, “Phase-shifting interferometry with four interferograms using linear polarization modulation and a Ronchi grating displaced by only a small unknown amount,” Opt. Commun.282(15), 3063–3068 (2009).
[CrossRef]

Opt. Eng. (1)

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng.39(4), 960–966 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

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Figures (5)

Fig. 1
Fig. 1

(a) Experimental setup for two-step spatial carrier phase-shifting common-path interferometer; (b) Polarization state of beams in the input aperture; (c) Polarization state of polarizing filter array(PLA). At the right side of grating G, the blue, red and green beams indicate the diffraction orders +1, 0 and −1, respectively; P, Polarizer; CE, Collimator & expander; QWR, QWL, Quarter wave plates; Ap, Aperture; S, Specimen; G, Ronchi grating; L1, L2, Lenses.

Fig. 2
Fig. 2

Schematic for diffraction of grating. Orders +1, 0 and −1 are indicated with blue, red and green; Δf, displacement of grating; Δx, distance between adjacent orders in the Fourier plane; θ, diffraction angle of ±1st orders.

Fig. 3
Fig. 3

Experimental results without specimen: (a) 2D intensity distribution recorded in the image plane and (b) 1D profile of (a) along the red line; (c) 2D and (d) 1D spectrum distribution recorded in the Fourier plane.

Fig. 4
Fig. 4

Experimental results for a phase plate: (a) interferograms with phase shift π; (b) reconstructed OPD distribution using the method reported in Ref. [14]; (c) reconstructed OPD distribution using the proposed method; (d) profile along the dash line marked in (b).

Fig. 5
Fig. 5

Stability and repeatability of the proposed method. Hti denotes the ith measurement and RMS denotes the standard deviate.

Equations (9)

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f 0 =Δx/λf.
sinθ= λ d =tanθ= Δx Δf .
f 0 = Δf fd .
I 1 ( x,y )=a( x,y )+c( x,y )exp( i2π f 0 x )+c*( x,y )exp( i2π f 0 x ),
I 2 ( x,y )=a( x,y )+c( x,y )exp( iα )exp( i2π f 0 x )+c*( x,y )exp( iα )exp( i2π f 0 x )
c( x,y )=Aexp[ iφ( x,y ) ],
Rr=exp( i2π f 0 x )=exp( i2π Δf fd x ).
c ( x,y )=c( x,y )[ 1exp( iα ) ]=IFT{ FT{ Rr( I 1 I 2 ) }LF },
φ( x,y )= Im[ c ( x,y ) ] Re[ c ( x,y ) ] φ C ,

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