Abstract

Phase-only liquid crystal spatial light modulator has wide ranging applications that require accurate phase retardance. The phase calibration of the spatial light modulator is therefore of vital importance. Available self-referenced calibration methods face the challenges of high time consumption, low efficiency, and low stability against the conditions. A self-referenced multiple-beam interferometric method is proposed to derive the global grayscale-phase response. As is presented theoretically and experimentally, the proposed method reduces the measuring time and improves the calibration efficiency by encoding multiple fringes in a single hologram. Results also show that the method is equally accurate when compared with traditional two-beam interferometric method, whereas providing a greater robustness against measuring errors since the standard deviation is only 56% of that of the traditional method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
    [Crossref]
  2. W. Osten, A. Faridian, P. Gao, K. Körner, D. Naik, G. Pedrini, A. K. Singh, M. Takeda, and M. Wilke, “Recent advances in digital holography,” Appl. Opt. 53(27), G44–G63 (2014).
    [Crossref]
  3. A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
    [Crossref]
  4. L. Hu, L. Xuan, Y. Liu, Z. Cao, D. Li, and Q. Mu, “Phase-only liquid-crystal spatial light modulator for wave-front correction with high precision,” Opt. Express 12(26), 6403–6409 (2004).
    [Crossref]
  5. P. M. Prieto, E. J. Fernández, S. Manzanera, and P. Artal, “Adaptive optics with a programmable phase modulator: applications in the human eye,” Opt. Express 12(17), 4059–4071 (2004).
    [Crossref]
  6. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
    [Crossref]
  7. W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
    [Crossref]
  8. T. Baumbach, W. Osten, C. von Kopylow, and W. Jüptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45(5), 925–934 (2006).
    [Crossref]
  9. S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. 52(12), 2610–2618 (2013).
    [Crossref]
  10. C. Kohler, X. Schwab, and W. Osten, “Optimally tuned spatial light modulators for digital holography,” Appl. Opt. 45(5), 960–967 (2006).
    [Crossref]
  11. A. Bergeron, J. Gauvin, F. Gagnon, D. Gingras, H. H. Arsenault, and M. Doucet, “Phase calibration and applications of a liquid-crystal spatial light modulator,” Appl. Opt. 34(23), 5133–5139 (1995).
    [Crossref]
  12. J. L. M. Fuentes, E. J. Fernández, P. M. Prieto, and P. Artal, “Interferometric method for phase calibration in liquid crystal spatial light modulators using a self-generated diffraction-grating,” Opt. Express 24(13), 14159–14171 (2016).
    [Crossref]
  13. Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
    [Crossref]
  14. X. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. 43(35), 6400–6406 (2004).
    [Crossref]
  15. H. Zhang, J. Zhang, and L. Wu, “Evaluation of phase-only liquid crystal spatial light modulator for phase modulation performance using a Twyman–Green interferometer,” Meas. Sci. Technol. 18(6), 1724–1728 (2007).
    [Crossref]
  16. S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
    [Crossref]
  17. Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33(9), 3018–3022 (1994).
    [Crossref]
  18. O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
    [Crossref]
  19. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45(8), 1688–1703 (2006).
    [Crossref]
  20. F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices,” Opt. Express 22(12), 15064–15074 (2014).
    [Crossref]
  21. R. Li and L. Cao, “Progress in phase calibration for liquid crystal spatial light modulators,” Appl. Sci. 9(10), 2012 (2019).
    [Crossref]
  22. J. B. Bentley, J. A. Davis, J. Albero, and I. Moreno, “Self-interferometric technique for visualization of phase patterns encoded onto a liquid-crystal display,” Appl. Opt. 45(30), 7791–7794 (2006).
    [Crossref]
  23. https://holoeye.com/gaea-4k-phase-only-spatial-light-modulator/

2019 (1)

R. Li and L. Cao, “Progress in phase calibration for liquid crystal spatial light modulators,” Appl. Sci. 9(10), 2012 (2019).
[Crossref]

2018 (1)

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

2016 (3)

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

J. L. M. Fuentes, E. J. Fernández, P. M. Prieto, and P. Artal, “Interferometric method for phase calibration in liquid crystal spatial light modulators using a self-generated diffraction-grating,” Opt. Express 24(13), 14159–14171 (2016).
[Crossref]

2014 (3)

2013 (2)

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
[Crossref]

S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. 52(12), 2610–2618 (2013).
[Crossref]

2011 (1)

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

2009 (1)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

2007 (1)

H. Zhang, J. Zhang, and L. Wu, “Evaluation of phase-only liquid crystal spatial light modulator for phase modulation performance using a Twyman–Green interferometer,” Meas. Sci. Technol. 18(6), 1724–1728 (2007).
[Crossref]

2006 (4)

2004 (3)

1995 (1)

1994 (1)

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33(9), 3018–3022 (1994).
[Crossref]

Albero, J.

Alfalou, A.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Arsenault, H. H.

Artal, P.

Baumbach, T.

Beléndez, A.

Bentley, J. B.

Bergeron, A.

Bhattacharya, K.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
[Crossref]

Brosseau, C.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Cao, L.

R. Li and L. Cao, “Progress in phase calibration for liquid crystal spatial light modulators,” Appl. Sci. 9(10), 2012 (2019).
[Crossref]

Cao, Z.

Carbonell-Leal, M.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Chen, W.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

Chen, X.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

Chipman, R. A.

Cohn, R. W.

Davis, J. A.

Doñate-Buendía, C.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Doucet, M.

Dudley, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Faridian, A.

Fernández, E. J.

Fernández-Alonso, M.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Forbes, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Francés, J.

Fuentes, J. L. M.

Gagnon, F.

Gallego, S.

Gao, P.

Gauvin, J.

Gingras, D.

Hazra, L.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
[Crossref]

Hu, L.

Javidi, B.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

Jüptner, W.

Kohler, C.

Körner, K.

Lancis, J.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Li, D.

Li, R.

R. Li and L. Cao, “Progress in phase calibration for liquid crystal spatial light modulators,” Appl. Sci. 9(10), 2012 (2019).
[Crossref]

Liu, Y.

Lu, G.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33(9), 3018–3022 (1994).
[Crossref]

Manzanera, S.

Márquez, A.

Martínez, F. J.

Martínez-León, L.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

McLaren, M.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Mendoza-Yero, O.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Mínguez-Vega, G.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Moreno, I.

Mu, Q.

Mukhopadhyay, S.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
[Crossref]

Naik, D.

Ortuño, M.

Osten, W.

Pascual, I.

Pedrini, G.

Pérez-Vizcaíno, J.

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Prieto, P. M.

Reichelt, S.

Sarkar, S.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
[Crossref]

Schwab, X.

Singh, A. K.

Takeda, M.

von Kopylow, C.

Weiner, A. M.

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Wilke, M.

Wolfe, J. E.

Wu, L.

H. Zhang, J. Zhang, and L. Wu, “Evaluation of phase-only liquid crystal spatial light modulator for phase modulation performance using a Twyman–Green interferometer,” Meas. Sci. Technol. 18(6), 1724–1728 (2007).
[Crossref]

Xiao, Z.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

Xuan, L.

Xun, X.

Yu, F. T. S.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33(9), 3018–3022 (1994).
[Crossref]

Zhang, H.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

H. Zhang, J. Zhang, and L. Wu, “Evaluation of phase-only liquid crystal spatial light modulator for phase modulation performance using a Twyman–Green interferometer,” Meas. Sci. Technol. 18(6), 1724–1728 (2007).
[Crossref]

Zhang, J.

H. Zhang, J. Zhang, and L. Wu, “Evaluation of phase-only liquid crystal spatial light modulator for phase modulation performance using a Twyman–Green interferometer,” Meas. Sci. Technol. 18(6), 1724–1728 (2007).
[Crossref]

Zhang, Z.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33(9), 3018–3022 (1994).
[Crossref]

Zhao, H.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

Zhao, Z.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

Zhuang, Y.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

Adv. Opt. Photonics (3)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Appl. Opt. (8)

Appl. Sci. (1)

R. Li and L. Cao, “Progress in phase calibration for liquid crystal spatial light modulators,” Appl. Sci. 9(10), 2012 (2019).
[Crossref]

J. Disp. Technol. (1)

O. Mendoza-Yero, G. Mínguez-Vega, L. Martínez-León, M. Carbonell-Leal, M. Fernández-Alonso, C. Doñate-Buendía, J. Pérez-Vizcaíno, and J. Lancis, “Diffraction-based phase calibration of spatial light modulators with binary phase fresnel lenses,” J. Disp. Technol. 12(10), 1027–1032 (2016).
[Crossref]

Meas. Sci. Technol. (1)

H. Zhang, J. Zhang, and L. Wu, “Evaluation of phase-only liquid crystal spatial light modulator for phase modulation performance using a Twyman–Green interferometer,” Meas. Sci. Technol. 18(6), 1724–1728 (2007).
[Crossref]

Opt. Commun. (1)

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Opt. Eng. (2)

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. Hazra, “Polarization phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52(3), 035602 (2013).
[Crossref]

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33(9), 3018–3022 (1994).
[Crossref]

Opt. Express (4)

Rev. Sci. Instrum. (1)

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref]

Other (1)

https://holoeye.com/gaea-4k-phase-only-spatial-light-modulator/

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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic of the SeRMI system. OBJ is a microscope objective, PH is a pinhole, A is an aperture, L is a collimating lens, M is a mirror, P is a polarizer and BS is a beam splitter. (b) Flow chart of the calibration procedure.
Fig. 2.
Fig. 2. A comparison between (a) the traditional two-beam interference and (b) the proposed SeRMI.
Fig. 3.
Fig. 3. The fringe patterns for phase shift of 0, π/2, π and 3π/2. Red dotted lines are added to visualize the shift of the fringe. The left correlated term ((e)–(h)) and the right correlated term ((i)–(l)) are derived from the overall intensity distribution on the hologram plane ((a)–(d)).
Fig. 4.
Fig. 4. (a)–(c) The analysis of the SeRMI pattern. (a) The intensity distribution of the interferometric pattern in the image domain. (b) The corresponding frequency domain. (c) The correlated frequency vectors (solid lines) and the complex conjugate terms (dashed lines). (d) – (f) are the same analysis on the traditional two-beam interference for comparison.
Fig. 5.
Fig. 5. (a) The intensity-grayscale response of Canon EOS 550D. The Fourier spectrum of the interference pattern (b) before and (c) after modification. The red impulses belong to the expected intensity distribution while the others are formed by the effect of nonlinearity.
Fig. 6.
Fig. 6. The look-up table derived by the traditional two-beam interference method and the proposed SeRMI method. The captured interferometric patterns and the retrieved phase shifts for grayscale 64, 128 and 192 are shown on the left.
Fig. 7.
Fig. 7. The standard deviation of the traditional two-beam interference method and the proposed SeRMI method.

Tables (1)

Tables Icon

Table 1. Measured phase values for selected grayscale levels of GAEA-2 LC-SLM

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

I ( x , y ) = | U ( x , y ) | 2 = | i = 1 N U i ( x , y ) | 2 = | i = 1 N A i exp [ j φ i ( x , y ) ]   | 2 = i = 1 N A i 2 + 2 1 i < k N A i A k cos [ φ i ( x , y ) φ k ( x , y ) ] = i = 1 N A i 2 + 2 1 i < k N A i A k cos [ 2 π ( f i f k ) r + ( φ i 0 φ k 0 ) ] = i = 1 N A i 2 + 2 1 i < k N A i A k cos ( 2 π f i k r + φ i k )
{ U t e s t ( x , y ) = A t e s t exp [ j ( Δ φ g + φ t e s t 0 ) ] U d i f f ( x , y ) = A d i f f exp [ j ( 2 π f d i f f r + φ d i f f 0 ) ]
I ( x , y , g ) = A t e s t 2 + i = 1 N 1 A d i f f i 2 + 2 i = 1 N 1 A t e s t A d i f f i cos [ 2 π ( f t e s t f d i f f i ) r + Δ φ g ] + 2 1 i < k N 1 A d i f f i A d i f f k cos [ 2 π ( f d i f f i f d i f f k ) r ] = A t e s t 2 + i = 1 N 1 A d i f f i 2 + 2 i = 1 N 1 A t e s t A d i f f i cos φ c o r r i + 2 1 i < k N 1 A d i f f i A d i f f k cos φ u n c o r r i , k
f d i f f L = 1 / P f d i f f R = 1 / P + tan α / P
{ φ c o r r L = 2 π x / P Δ φ g + φ L 0 φ t e s t 0 φ c o r r R = 2 π x / P + 2 π y tan α / P Δ φ g + φ R 0 φ t e s t 0
I ( x , y ) = | U t e s t ( x , y ) + U L ( x , y ) + U R ( x , y ) | 2   = A t e s t 2 + A L 2 + A R 2 + 2 A L A t e s t cos φ c o r r L + 2 A R A t e s t cos φ c o r r R + 2 A R A L cos φ u n c o r r
Δ φ g ( i ) = φ ( g ) φ ( 0 ) = arg F { I ( x , y , g ) } | f = f i arg F { I ( x , y , 0 ) } | f = f i
Δ φ g = 1 2 ( Δ φ g ( 1 ) + Δ φ g ( 2 ) )