Abstract

A quantitative phase measuring technique is presented that estimates the object phase from a series of phase shifted interferograms that are obtained in a common-path configuration with unknown phase shifts. The derived random phase shifting algorithm for common-path interferometers is based on the Generalized Phase Contrast theory [pl. Opt. 40(2), 268 (2001) [CrossRef]  ], which accounts for the particular image formation and includes effects that are not present in two-beam interferometry. It is shown experimentally that this technique can be used within common-path configurations employing nonlinear liquid crystal materials as self-induced phase filters for quantitative phase imaging without the need of phase shift calibrations. The advantages of such liquid crystal elements compared to spatial light modulator based solutions are given by the cost-effectiveness, self-alignment, and the generation of diminutive dimensions of the phase filter size, giving unique performance advantages.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2014 (4)

2013 (2)

2012 (2)

2011 (1)

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

2010 (1)

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

2009 (1)

2008 (1)

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

2004 (3)

2003 (1)

2002 (1)

2001 (2)

1998 (1)

I. Jánossy and L. Szabados, “Optical reorientation of nematic liquid crystals in the presence of photoisomerization,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), 4598–4604 (1998).
[Crossref]

1997 (1)

L. Marrucci and D. Paparo, “Photoinduced molecular reorientation of absorbing liquid crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(2), 1765–1772 (1997).
[Crossref]

1996 (2)

1995 (1)

1987 (1)

1978 (1)

1971 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects - Part I,” Physica 9(7), 686–698 (1942).
[Crossref]

Anderson, C. S.

Arroyo Carrasco, M. L.

R. Porras Aguilar, J. C. Ramirez-San-Juan, O. Baldovino-Pantaleon, D. May-Arrioja, M. L. Arroyo Carrasco, M. D. Iturbe-Castillo, D. Sánchez-de-la-Llave, and R. Ramos-Garcia, “Polarization-controlled contrasted images using dye-doped nematic liquid crystals,” Opt. Express 17(5), 3417–3423 (2009).
[Crossref] [PubMed]

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Ayubi, G. A.

Badizadegan, K.

Baldovino-Pantaleon, O.

Baldovino-Pantaleón, O.

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Bergmann, R. B.

Castillo, M. D. I.

Claus, R. A.

Creath, K.

Dasari, R. R.

Dauwels, J.

Deflores, L. P.

Di Martino, J. M.

Falaggis, K.

Falldorf, C.

Feld, M. S.

Feng, L.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Ferrari, J. A.

Fienup, J. R.

Flores, J. L.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Gao, W.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Glückstad, J.

Guizar-Sicairos, M.

Gureyev, T. E.

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase Odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

Han, B.

Hayslett, C. R.

Hoang, T.

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

Hock Soon, S.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Iturbe-Castillo, M. D.

Iwai, H.

Jánossy, I.

I. Jánossy and L. Szabados, “Optical reorientation of nematic liquid crystals in the presence of photoisomerization,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), 4598–4604 (1998).
[Crossref]

Jingshan, Z.

Kemao, Q.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Koliopoulos, C. L.

Kozacki, T.

Kujawinska, M.

Kwon, O.

Luu, L.

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

Ma, J.

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

Malacara, D.

Marroquin, J. L.

Marrucci, L.

L. Marrucci and D. Paparo, “Photoinduced molecular reorientation of absorbing liquid crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(2), 1765–1772 (1997).
[Crossref]

Martinez-Carranza, J.

May-Arrioja, D.

Mercer, C. R.

Mogensen, P. C.

Nugent, K. A.

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase Odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

Olivos-Pérez, L. I.

Ortega-Martínez, R.

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Paganin, D.

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase Odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

Pan, B.

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

Paparo, D.

L. Marrucci and D. Paparo, “Photoinduced molecular reorientation of absorbing liquid crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(2), 1765–1772 (1997).
[Crossref]

Perciante, C. D.

Popescu, G.

Porras Aguilar, R.

Ramirez-San-Juan, J. C.

Ramos García, R.

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Ramos-Garcia, R.

Ramos-García, R.

Reynoso Lara, E.

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Rodríguez-Rosales, A. A.

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Sánchez-de-la-Llave, D.

Servin, M.

Shagam, R.

Sheppard, C. J. R.

Szabados, L.

I. Jánossy and L. Szabados, “Optical reorientation of nematic liquid crystals in the presence of photoisomerization,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), 4598–4604 (1998).
[Crossref]

Tian, L.

Treviño Palacios, C. G.

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Treviño-Palacios, C. G.

Vaughan, J. C.

Vo, M.

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

von Kopylow, C.

Waller, L.

Wang, H.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Wang, Z.

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Wyant, J. C.

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects - Part I,” Physica 9(7), 686–698 (1942).
[Crossref]

Appl. Opt. (10)

K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987).
[Crossref] [PubMed]

C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34(32), 7474–7485 (1995).
[Crossref] [PubMed]

C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35(10), 1633–1642 (1996).
[Crossref] [PubMed]

M. Servin, D. Malacara, and J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35(22), 4343–4348 (1996).
[Crossref] [PubMed]

J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40(2), 268–282 (2001).
[Crossref] [PubMed]

D. Sánchez-de-la-Llave and M. D. I. Castillo, “Influence of illuminating beyond the object support on Zernike-type phase contrast filtering,” Appl. Opt. 41(14), 2607–2612 (2002).
[Crossref] [PubMed]

J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10(9), 2113–2118 (1971).
[Crossref] [PubMed]

C. G. Treviño-Palacios, M. D. Iturbe-Castillo, D. Sánchez-de-la-Llave, R. Ramos-García, and L. I. Olivos-Pérez, “Nonlinear common-path interferometer: an image processor,” Appl. Opt. 42(25), 5091–5095 (2003).
[Crossref] [PubMed]

T. Kozacki, K. Falaggis, and M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
[Crossref] [PubMed]

G. A. Ayubi, C. D. Perciante, J. L. Flores, J. M. Di Martino, and J. A. Ferrari, “Generation of phase-shifting algorithms with N arbitrarily spaced phase-steps,” Appl. Opt. 53(30), 7168–7176 (2014).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

T. Hoang, Z. Wang, M. Vo, J. Ma, L. Luu, and B. Pan, “Phase extraction from optical interferograms in presence of intensity nonlinearity and arbitrary phase shifts,” Appl. Phys. Lett. 99(3), 031104 (2011).
[Crossref]

J. Opt. Soc. Am. A (3)

Mol. Cryst. Liq. Cryst. (Phila. Pa.) (1)

A. A. Rodríguez-Rosales, R. Ortega-Martínez, M. L. Arroyo Carrasco, E. Reynoso Lara, C. G. Treviño Palacios, O. Baldovino-Pantaleón, R. Ramos García, and M. D. Iturbe-Castillo, “Neither Kerr Nor Thermal Nonlinear Response of Dye Doped Liquid Crystal Characterized by the Z-Scan Technique,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 489(1), 9 (2008).
[Crossref]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lasers Eng. (1)

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. Hock Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Opt. Lett. (6)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (2)

I. Jánossy and L. Szabados, “Optical reorientation of nematic liquid crystals in the presence of photoisomerization,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), 4598–4604 (1998).
[Crossref]

L. Marrucci and D. Paparo, “Photoinduced molecular reorientation of absorbing liquid crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(2), 1765–1772 (1997).
[Crossref]

Phys. Today (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A Phase Odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects - Part I,” Physica 9(7), 686–698 (1942).
[Crossref]

Other (4)

D. Malacara-Hernandez, Optical Shop Testing (John Wiley & Sons, Inc., 2007).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Pergamon, NewYork, 1988), 26, 350–393.

J. Glückstad and D. Palima, Generalized Phase Contrast, Springer Series in Optical Sciences (Springer Netherlands, 2009), Vol. 146, pp. 7–12.

I.-C. Khoo, Liquid Crystals (John Wiley & Sons, Inc., 2007).

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Figures (7)

Fig. 1
Fig. 1

Schematic of a CP-configuration having a circular phase filter at the Fourier plane.

Fig. 2
Fig. 2

Phase reconstructions in a common path configuration described by Eq. (1) for a Lena phantom with a maximum phase difference of 1.7π for the case of N = 4, and θn = (n-1) π/2, with n = 0, 1, 2, 3 when using (a) interferometry based reconstruction (direct solution to Eq. (6)) and (c) the CP-RPSA presented here. The corresponding error for (a) and (c) is shown in (b) and (d), respectively. The case of (c,d) is shown in (e,f) but having AWGN with σΙ = 0.05. All units are in radians.

Fig. 3
Fig. 3

Phase reconstructions for the case of Fig. 2, but having various phase shift errors. Figure 3a shows the PSE that is obtained by the CP-RPSA for a given PSE of the measurement system. Figure 3b shows the RMSE of the reconstructed phase when using the CP-RPSA algorithm (blue) or when solving Eq. (3) directly (red). All units are in radians. Each dot corresponds to one out of 100 simulations.

Fig. 4
Fig. 4

Schematic of the nonlinear phase contrast microscope with a polarization controlled phase shift.

Fig. 5
Fig. 5

Measurement of a step-like phase object with a maximum step of 2.8 radians (a,d), 1.5 radians (b,e), and 0.7 radians (c,f). The phases in (a,b,c) and (d,e,f) correspond to the direct solution of Eq. (6) and the GPC based CP-RPSA solution, respectively. A phase cut through the center of the object (g, h, i) and a histogram of the phase values (j, k, l) is shown in the equivalent figure row. A median filter of size 9x9 and an offset correction has been applied to the figures (g) to (l) for better visualization. The measurements have been conducted in the setup of Fig. 4 using a wavelength of 633nm with K ~0.761. All units are in radians.

Fig. 6
Fig. 6

Sinusoidal behaviour of the estimated phase shifts θn for the case of Fig. 5d

Fig. 7
Fig. 7

Quantitative phase imaging for an unlabelled biological object (protozoan) using the measurement system of Fig. 4. Fig (a), (b), and (c) show the recorded intensities at the phase shift angles of −0.490, −0.117, and 0.142 radians, where only the first 120 gray levels are shown for better visualization. The wrapped and unwrapped phase in radians is shown in (d) and (e) respectively. The phase in (d) is used to simulate equivalent DIC view, which is shown in (f).

Tables (1)

Tables Icon

Table 1 Comparison of the real and recovered phase shift as well as RMSE of the recovered phase when using CP-RPSA algorithm with N = 4 (cases 1-4) and N = 5 (cases 5-8) for various levels of noise (σΙ). All units are in radians.

Equations (18)

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I n (A β 0 ) 2 | exp(iφ)+C[ β 0 1 β 1 exp(i θ n )1] | 2 ,
I m,n (A β 0 ) 2 [ q 1 + q 2 cos φ m q 3 cos θ n + q 3 (cos θ n cos φ m +sin θ n sin φ m )],
I m =Qq p CPm ,
Q=[ 1 cos θ 0 sin θ 0 1 cos θ 1 sin θ 1 1 cos θ (N1) sin θ (N1) ],
q=[ q 1 q 3 0 q 2 q 3 0 0 0 q 3 ],
I m =Q [ A 2 A 2 γcos φ m A 2 γsinφ m ] T ,
S f = n=0 N1 ( f (1) +cos θ n f (2) +sin θ n f (3) I m,n ) 2 ,
Aq p CPm = [ n=0 N1 I m,n n=0 N1 I m,n cos θ n n=0 N1 I m,n sin θ n ] T ,
A=( N n=0 N1 cos θ n n=0 N1 sin θ n n=0 N1 cos θ n n=0 N1 cos 2 θ n n=0 N1 sin θ n cos θ n n=0 N1 sin θ n n=0 N1 sin θ n cos θ n n=0 N1 sin 2 θ n ).
p CPm = [Aq] 1 [ n=0 N1 I m,n n=0 N1 I m,n cos θ n n=0 N1 I m,n sin θ n ] T .
I m,n = Q n q p CPm .
I m,n = p CPm T q T Q n T = g (1) +cos φ m g (2) +sin φ m g (3) ,
S g = m=0 M ( g (1) +cos φ m g (2) +sin φ m g (3) I m,n ) 2
B q T h = [ m=0 M1 I m,n m=0 M1 I m,n cos φ m m=0 M1 I m,n sin φ m ] T ,
B=( M m=0 M1 cos φ m m=0 M1 sin φ m m=0 M1 cos φ m m=0 M1 cos 2 φ m m=0 M1 sin φ m cos φ m m=0 M1 sin φ m m=0 M1 sin φ m cos φ m m=0 M1 sin 2 φ m ).
h = [B q T ] 1 [ m=0 M1 I m,n m=0 M1 I m,n cos φ m m=0 M1 I m,n sin φ m ] T .
PSE= n=1 N ( θ n,real θ n,ideal ) 2 .
θ n Φ N cos 2 Ψ n + Φ P sin 2 Ψ n

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