Abstract

This paper describes research in developing a dynamic quantitative phase imaging microscope providing instantaneous measurements of dynamic motions within and among live cells without labels or contrast agents. It utilizes a pixelated phase mask enabling simultaneous measurement of multiple interference patterns derived using the polarization properties of light to track dynamic motions and morphological changes. Optical path difference (OPD) and optical thickness (OT) data are obtained from phase images. Two different processing routines are presented to remove background surface shape to enable quantification of changes in cell position and volume over time. Data from a number of different moving biological organisms and cell cultures are presented.

©2012 Optical Society of America

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2012 (2)

K. Creath and G. Goldstein, “Dynamic phase imaging and processing of moving biological organisms,” Proc. SPIE 8227, 82270M, 82270M-10 (2012).
[Crossref]

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

2011 (3)

W.-J. Hwang, S. C. Cheng, and C.-J. Cheng, “Efficient phase unwrapping architecture for digital holographic microscopy,” Sensors (Basel) 11(10), 9160–9181 (2011).
[Crossref] [PubMed]

K. Creath, “Dynamic phase imaging utilizing a 4-dimensional microscope system,” Proc. SPIE 7904, 79040O, 79040O-11 (2011).
[Crossref]

J. Dong, R. Lu, Y. Li, and K. Wu, “Automated determination of best focus and minimization of optical path difference in Linnik white light interferometry,” Appl. Opt. 50(30), 5861–5871 (2011).
[Crossref] [PubMed]

2010 (2)

K. Creath, “Dynamic quantitative phase images of pond life, insect wings, and in vitro cell cultures,” Proc. SPIE 7782, 77820B, 77820B-13 (2010).
[Crossref]

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

2008 (3)

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
[Crossref] [PubMed]

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

2006 (3)

B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
[Crossref] [PubMed]

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and intererence microscopy,” J. Opt. A, Pure Appl. Opt. 8(11), 952–958 (2006).
[Crossref]

S. Han, “Interferometric testing through transmissive media,” Proc. SPIE 6293, 629305, 629305-5 (2006).
[Crossref]

2005 (3)

2003 (1)

2001 (2)

1999 (1)

1996 (1)

1993 (1)

P. G. Walker, G. B. Cranney, M. B. Scheidegger, G. Waseleski, G. M. Pohost, and A. P. Yoganathan, “Semiautomated method for noise reduction and background phase error correction in MR phase velocity data,” J. Magn. Reson. Imaging 3(3), 521–530 (1993).
[Crossref] [PubMed]

1992 (1)

K. B. Farr and N. George, “Beamsplitter cube for white light interferometry,” Opt. Eng. 31(10), 2191–2196 (1992).
[Crossref]

1990 (1)

1989 (1)

S. D. Yanowitz and A. M. Bruckstein, “A new method for image segmentation,” Comput. Vis. Graph. Image Process. 46(1), 82–95 (1989).
[Crossref]

1985 (1)

1978 (1)

J. S. Weszka, “A survey of threshold selection techniques,” Comput. Graphics Image Processing 7(2), 259–265 (1978).
[Crossref]

Abdulhalim, I.

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and intererence microscopy,” J. Opt. A, Pure Appl. Opt. 8(11), 952–958 (2006).
[Crossref]

Badizadegan, K.

Berns, M. W.

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
[Crossref] [PubMed]

Brock, N.

Bruckstein, A. M.

S. D. Yanowitz and A. M. Bruckstein, “A new method for image segmentation,” Comput. Vis. Graph. Image Process. 46(1), 82–95 (1989).
[Crossref]

Chen, Z.

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

Chen, Z. P.

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
[Crossref] [PubMed]

Cheng, C.-J.

W.-J. Hwang, S. C. Cheng, and C.-J. Cheng, “Efficient phase unwrapping architecture for digital holographic microscopy,” Sensors (Basel) 11(10), 9160–9181 (2011).
[Crossref] [PubMed]

Cheng, S. C.

W.-J. Hwang, S. C. Cheng, and C.-J. Cheng, “Efficient phase unwrapping architecture for digital holographic microscopy,” Sensors (Basel) 11(10), 9160–9181 (2011).
[Crossref] [PubMed]

Corwin, A. D.

Cranney, G. B.

P. G. Walker, G. B. Cranney, M. B. Scheidegger, G. Waseleski, G. M. Pohost, and A. P. Yoganathan, “Semiautomated method for noise reduction and background phase error correction in MR phase velocity data,” J. Magn. Reson. Imaging 3(3), 521–530 (1993).
[Crossref] [PubMed]

Creath, K.

K. Creath and G. Goldstein, “Dynamic phase imaging and processing of moving biological organisms,” Proc. SPIE 8227, 82270M, 82270M-10 (2012).
[Crossref]

K. Creath, “Dynamic phase imaging utilizing a 4-dimensional microscope system,” Proc. SPIE 7904, 79040O, 79040O-11 (2011).
[Crossref]

K. Creath, “Dynamic quantitative phase images of pond life, insect wings, and in vitro cell cultures,” Proc. SPIE 7782, 77820B, 77820B-13 (2010).
[Crossref]

K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. 29(26), 3823–3827 (1990).
[Crossref] [PubMed]

Dainty, J. C.

Dasari, R. R.

de Boer, M. P.

Deguzman, P. C.

Delisle, C.

Dong, J.

Farr, K. B.

K. B. Farr and N. George, “Beamsplitter cube for white light interferometry,” Opt. Eng. 31(10), 2191–2196 (1992).
[Crossref]

Feld, M. S.

Filippova, N. A.

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

Frank, M.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

Gale, D. M.

Genc, S.

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
[Crossref] [PubMed]

George, N.

K. B. Farr and N. George, “Beamsplitter cube for white light interferometry,” Opt. Eng. 31(10), 2191–2196 (1992).
[Crossref]

Gimzewski, J. K.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

Goldstein, G.

K. Creath and G. Goldstein, “Dynamic phase imaging and processing of moving biological organisms,” Proc. SPIE 8227, 82270M, 82270M-10 (2012).
[Crossref]

Hahn, M. S.

Han, S.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

S. Han, “Interferometric testing through transmissive media,” Proc. SPIE 6293, 629305, 629305-5 (2006).
[Crossref]

Hayes, J.

Holl, M. M. B.

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

Hwang, W.-J.

W.-J. Hwang, S. C. Cheng, and C.-J. Cheng, “Efficient phase unwrapping architecture for digital holographic microscopy,” Sensors (Basel) 11(10), 9160–9181 (2011).
[Crossref] [PubMed]

Jasensky, J.

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

Jones, M. W.

Kakaiya, M.

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

Kekre, H.

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

Khmaladze, A.

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

Kimbrough, B. T.

Klemyashov, I. V.

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

Kothiyal, M. P.

Kretushev, A. V.

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

Li, Y.

Lieber, C. A.

Liu, G. J.

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
[Crossref] [PubMed]

Lobachev, M.

Lu, R.

Lyakin, D.

Mahadevan-Jansen, A.

Matz, R. L.

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

Meier, J. T.

Mikherkee, P.

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

Millerd, J.

Mohanty, S.

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
[Crossref] [PubMed]

Nordin, G. P.

North-Morris, M.

Novak, M.

Pether, M. I.

Pförtner, A.

Pohost, G. M.

P. G. Walker, G. B. Cranney, M. B. Scheidegger, G. Waseleski, G. M. Pohost, and A. P. Yoganathan, “Semiautomated method for noise reduction and background phase error correction in MR phase velocity data,” J. Magn. Reson. Imaging 3(3), 521–530 (1993).
[Crossref] [PubMed]

Raikhlin, N. T.

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

Reed, J.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

Ryabukho, V.

Scheidegger, M. B.

P. G. Walker, G. B. Cranney, M. B. Scheidegger, G. Waseleski, G. M. Pohost, and A. P. Yoganathan, “Semiautomated method for noise reduction and background phase error correction in MR phase velocity data,” J. Magn. Reson. Imaging 3(3), 521–530 (1993).
[Crossref] [PubMed]

Schmit, J.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

Schwider, J.

Seeley, E.

A. Khmaladze, R. L. Matz, J. Jasensky, E. Seeley, M. M. B. Holl, and Z. Chen, “Dual-wavelength digital holographic imaging with phase background subtraction,” Opt. Eng. 51(5), 055801–055808 (2012).
[Crossref]

Shtil, A. A.

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

Sinclair, M. B.

Singh, S.

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

Teitell, M. A.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

Thepade, S.

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

Troke, J. J.

J. Reed, M. Frank, J. J. Troke, J. Schmit, S. Han, M. A. Teitell, and J. K. Gimzewski, “High throughput cell nanomechanics with mechanical imaging interferometry,” Nanotechnology 19(23), 235101 (2008).
[Crossref] [PubMed]

Tychinsky, V. P.

V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

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V. P. Tychinsky, A. V. Kretushev, I. V. Klemyashov, T. V. Vyshenskaya, N. A. Filippova, N. T. Raikhlin, and A. A. Shtil, “Quantitative real-time analysis of nucleolar stress by coherent phase microscopy,” J. Biomed. Opt. 13(6), 064032 (2008).
[Crossref] [PubMed]

Wadhwa, S.

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

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P. G. Walker, G. B. Cranney, M. B. Scheidegger, G. Waseleski, G. M. Pohost, and A. P. Yoganathan, “Semiautomated method for noise reduction and background phase error correction in MR phase velocity data,” J. Magn. Reson. Imaging 3(3), 521–530 (1993).
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[Crossref]

Yoganathan, A. P.

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Appl. Opt. (8)

Appl. Spectrosc. (1)

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J. S. Weszka, “A survey of threshold selection techniques,” Comput. Graphics Image Processing 7(2), 259–265 (1978).
[Crossref]

Comput. Vis. Graph. Image Process. (1)

S. D. Yanowitz and A. M. Bruckstein, “A new method for image segmentation,” Comput. Vis. Graph. Image Process. 46(1), 82–95 (1989).
[Crossref]

Int. J. Comput. Appl. (1)

H. Kekre, S. Thepade, P. Mikherkee, M. Kakaiya, S. Wadhwa, and S. Singh, “Image retrieval with shape features extracted using gradient operators and slope magnitude technique with BTC,” Int. J. Comput. Appl. 6, 28–33 (2010).

J. Biomed. Opt. (2)

L. F. Yu, S. Mohanty, G. J. Liu, S. Genc, Z. P. Chen, and M. W. Berns, “Quantitative phase evaluation of dynamic changes on cell membrane during laser microsurgery,” J. Biomed. Opt. 13(5), 050508 (2008).
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Supplementary Material (10)

Media 1: MOV (734 KB)     
Media 2: MOV (2470 KB)     
Media 3: MOV (1897 KB)     
Media 4: MOV (879 KB)     
Media 5: MOV (2390 KB)     
Media 6: MOV (3176 KB)     
Media 7: MOV (1590 KB)     
Media 8: MOV (1559 KB)     
Media 9: MOV (415 KB)     
Media 10: MOV (3767 KB)     

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Figures (13)
Fig. 1
Fig. 1 Optical schematic and photograph of dynamic interference microscope.

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Fig. 2
Fig. 2 (left) Scanning electron micrograph (SEM) image of a wire grid polarizer (Courtesy of Moxtek). (right) Illustration of the pixelated phase mask that is bonded to the detector array having 4 different polarizations comprising a unit cell of 2 × 2 different relative phase values.

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Fig. 3
Fig. 3 Optical path length through test sample includes the coverslip, liquid and objects. Graph (bottom) shows an optical thickness profile for the section within the highlighted area.

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Fig. 4
Fig. 4 Frames from a time series of a drop of cold water dropped into cuvette of warm water. (a) Interference fringes. (b) Wrapped phase modulo 2π. (c) Phase images with a range (high-low) of 1.75 waves. (d) Cross-section profile of horizontal line through center of 2nd phase image in (c).

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Fig. 5
Fig. 5 Improved imaging by reducing spatial coherence using a rotating diffuser. Irradiance of a VLSI step height standard phase object (a) without diffuser, and (b) with diffuser. Measured phase of object (c) without diffuser and (d) with diffuser. (e) Comparison of a phase line profile through dashed lines in (c) without diffuser and (d) with diffuser showing improvement by reducing spatial coherence.

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Fig. 6
Fig. 6 Images of a protozoa determined from pixelated phase data. (a) Brightfield (irradiance/intensity). (b) Phase contrast (interference – a single interferogram). (c) Phase gradient magnitudes (simulated dark field). (d) Simulated DIC (x gradient). (e) Pseudo-colored optical thickness (from phase). (f) 3D optical thickness (from phase).

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Fig. 7
Fig. 7 Sample images of a time series movie of a large paramecium taken with 50× magnification at 660 nm. (a) Phase contrast (interference) (Media 1, movie showing interferogram images). (b) Simulated DIC (x gradient) (Media 2, movie showing simulated DIC). (c) Phase gradient magnitudes (simulated dark field) (Media 3, movie showing simulated dark field images). (d) Optical thickness (Media 4, movie showing optical thickness). Note motion of cilia in media files.

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Fig. 8
Fig. 8 Human breast cancer cells taken with 20× magnification at 660 nm. (a) Interference pattern showing 4 fringes of tilt. (b) Modulo 2π wrapped phase. (c) Unwrapped optical thickness in nm from phase. (d) Best-fit plane removed. (e) Optical thickness with manual background removal using the areas defined by the white rectangles (see text).

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Fig. 9
Fig. 9 Time series of 3D phase images of another breast cancer cell culture taken at 20×. All images have the same optical thickness pseudocolor scale from 0 to 550 nm. (a) Cells in original media. (b) After contact with purified water the cells osmotically swell. (c) After more purified water the cells continue to swell and flatten. (d) After contact with NaOH the cells are beginning to break down. These movies were taken with sampling times of a few seconds over 23 minutes (Media 5).

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Fig. 10
Fig. 10 Time series of phase images of breast cancer cells as they dissolve after contact with pure water, NaOH and Alconox®. Imaging area is 200 × 300 μm. Data are taken as 20× with a 660 nm source and 2 ms exposures. Samples above are every 10 seconds from a 78 second long movie (Media 6).

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Fig. 11
Fig. 11 Three consecutive phase images of a swimming paramecium from a time series movie. (a)–(c) Raw unwrapped optical thickness from phase (Media 7). (d)–(f) After simple automated background leveling using histogram filtering (Media 8).

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Fig. 12
Fig. 12 Rotifer tail imaged with 50× at 660 nm. (a) Optical thickness calculated from raw phase (movie in Media 9). (b) Final binary mask corresponding to background area for fitting Zernike surface. (c) Best fit plane (synthetic 3-term Zernike surface) corresponding to background surface to remove. (d) Optical thickness after automatic background removal (movie in Media 10). (e) Gradient magnitudes from (a). (f)–(i) Iterations 1 to 4 to determine area to mask (see text).

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Fig. 13
Fig. 13 Automatic background removal from cropped portion of data set from Fig. 8. (a) Raw unwrapped optical thickness data. (b) Gradient magnitudes. (c) Binary mask after 4 iterations. (d) Zernike surface generated after fitting tilt, curvature and cylinder to unmasked points. (e) Optical thickness with background removed.

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Equations (10)

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

I ( x , y ) = 1 2 { I O + I R + 2 I O I R cos [ 2 Δ ϕ ( x , y ) + α i ] } ,
A ( x , y ) = 1 2 { I O + I R + 2 I O I R cos [ 2 Δ ϕ ( x , y ) ] } ,
B ( x , y ) = 1 2 { I O + I R + 2 I O I R cos [ 2 Δ ϕ ( x , y ) + π 2 ] } ,
C ( x , y ) = 1 2 { I O + I R + 2 I O I R cos [ 2 Δ ϕ ( x , y ) + π ] } ,
and  D ( x , y ) = 1 2 { I O + I R + 2 I O I R cos [ 2 Δ ϕ ( x , y ) + 3 π 2 ] } .
2 Δ ϕ ( x , y ) = A T A N 2 [ D ( x , y ) − B ( x , y ) A ( x , y ) − C ( x , y ) ] ,
O P D ( x , y ) = Δ ϕ ( x , y ) 4 π .
O T ( x , y ) = λ ⋅ O P D ( x , y ) ,
and  O T ( x , y ) = O P L O ( x , y ) − O P L R ( x , y ) ,
P G M = ( ∂ ϕ ∂ x ) 2 + ( ∂ ϕ ∂ y ) 2 .

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