Abstract

We demonstrate the use of shallow diffraction gratings for quadrature phase interferometry. A single shallow diffraction grating-based Michelson interferometer yields only trivial (0° or 180°) phase shift between different output ports. In comparison, a combination of two parallel shallow diffraction gratings can be useful to achieve desired phase shifts (e.g., 90° for quadrature phase interferometry). We show that the phase at different output ports of a grating-pair based interferometer can be adjusted by shearing the two gratings with respect to each other. Two harmonically-related diffraction gratings are used to demonstrate phase shift control at the output ports of a modified Michelson interferometer. Our experimental data is in good agreement with theory.

© 2006 Optical Society of America

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  1. J. C. Shaw, "Metrology using differential phase-contrast microscopy," Microelectron. Eng. 13, 527-530 (1991).
    [CrossRef]
  2. P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
    [CrossRef]
  3. F. Zernike, "Phase contrast, a new method for the microsopic observation of transparent objects," Physica 9, 686-698 (1942).
    [CrossRef]
  4. F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
    [CrossRef]
  5. G. Nomarski, and A. R. Weill, "Application à la métallographie des méthodes interférentielles à deux ondes polarisées," Rev. Metall. 2, 121-128 (1955).
  6. W. Shimada, T. Sato, and T. Yatagai, "Optical surface microtopography using phase-shifting Nomarski microscope," Proc. SPIE 1332, 525-529 (1990).
    [CrossRef]
  7. P. Hariharan, and M. Roy, "Achromatic phase-shifting for two-wavelength phase-stepping interferometry," Opt. Comm. 126, 220-222 (1996).
    [CrossRef]
  8. C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
    [CrossRef]
  9. M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
    [CrossRef]
  10. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
    [CrossRef]
  11. H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three-dimensional measurement by differential interference contrast microscope," Opt. Comm. 260, 117-126 (2006).
    [CrossRef]
  12. U. Schnars, and W. Juptner, "Direct Recording of Holograms by a Ccd Target and Numerical Reconstruction," Appl. Opt. 33, 179-181 (1994).
    [CrossRef] [PubMed]
  13. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  14. G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
    [CrossRef]
  15. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
    [CrossRef] [PubMed]
  16. D. S. Marx, and D. Psaltis, "Polarization quadrature measurement of subwavelength diffracting structures," Appl. Opt. 36, 6434-6440 (1997).
    [CrossRef]
  17. D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, "Three-dimensional images generated by quadrature interferometry," Opt. Lett. 23, 783-785 (1998).
    [CrossRef]
  18. M. A. Choma, "Instantaneous quadrature low-coherence interferometry with 3 x 3 fiber-optic couplers," Opt. Lett. 28, 2162 - 2164 (2003).
    [CrossRef] [PubMed]
  19. Z. Yaqoob, J. Fingler, X. Heng, and C. H. Yang, "Homodyne en face optical coherence tomography," Opt. Lett. 31, 1815-1817 (2006).
    [CrossRef] [PubMed]
  20. B. L. Danielson, and C. Y. Boisrobert, "Absolute optical ranging using low coherence interferometry," Appl. Opt. 30, 2975-2979 (1991).
    [CrossRef] [PubMed]
  21. V. V. Tuchin, "Coherence-domain methods in tissue and cell optics," Laser Phys. 8, 807-849 (1998).
  22. S. R. Thurber, A. M. Brodsky, and L. W. Burgess, "Characterization of random media by low-coherence interferometry," Appl. Spectrosc. 54, 1506-1514 (2000).
    [CrossRef]
  23. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
    [CrossRef] [PubMed]
  24. C. M. B. Cordeiro, L. Cescato, A. A. Freschi, and L. F. Li, "Measurement of phase differences between the diffracted orders of deep relief gratings," Opt. Lett. 28, 683-685 (2003).
    [CrossRef] [PubMed]

2006

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three-dimensional measurement by differential interference contrast microscope," Opt. Comm. 260, 117-126 (2006).
[CrossRef]

Z. Yaqoob, J. Fingler, X. Heng, and C. H. Yang, "Homodyne en face optical coherence tomography," Opt. Lett. 31, 1815-1817 (2006).
[CrossRef] [PubMed]

2005

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

2003

2002

P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
[CrossRef]

2000

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

S. R. Thurber, A. M. Brodsky, and L. W. Burgess, "Characterization of random media by low-coherence interferometry," Appl. Spectrosc. 54, 1506-1514 (2000).
[CrossRef]

1999

1998

1997

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

D. S. Marx, and D. Psaltis, "Polarization quadrature measurement of subwavelength diffracting structures," Appl. Opt. 36, 6434-6440 (1997).
[CrossRef]

1996

P. Hariharan, and M. Roy, "Achromatic phase-shifting for two-wavelength phase-stepping interferometry," Opt. Comm. 126, 220-222 (1996).
[CrossRef]

1994

1991

J. C. Shaw, "Metrology using differential phase-contrast microscopy," Microelectron. Eng. 13, 527-530 (1991).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

B. L. Danielson, and C. Y. Boisrobert, "Absolute optical ranging using low coherence interferometry," Appl. Opt. 30, 2975-2979 (1991).
[CrossRef] [PubMed]

1990

W. Shimada, T. Sato, and T. Yatagai, "Optical surface microtopography using phase-shifting Nomarski microscope," Proc. SPIE 1332, 525-529 (1990).
[CrossRef]

1955

G. Nomarski, and A. R. Weill, "Application à la métallographie des méthodes interférentielles à deux ondes polarisées," Rev. Metall. 2, 121-128 (1955).

1942

F. Zernike, "Phase contrast, a new method for the microsopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
[CrossRef]

Allman, B. E.

P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
[CrossRef]

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

Badizadegan, K.

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

Barone-Nugent, E. D.

P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
[CrossRef]

Best, C. A.

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

Bevilacqua, F.

Boisrobert, C. Y.

Brodsky, A. M.

Burgess, L. W.

Cescato, L.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Choma, M. A.

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Cordeiro, C. M. B.

Cuche, E.

Danielson, B. L.

Dasari, R. R.

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

Depeursinge, C.

Devaney, A. J.

DiMarzio, C. A.

Fekete, P. W.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

Feld, M. S.

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

Fingler, J.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Freschi, A. A.

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Gaudette, T. J.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hariharan, P.

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

P. Hariharan, and M. Roy, "Achromatic phase-shifting for two-wavelength phase-stepping interferometry," Opt. Comm. 126, 220-222 (1996).
[CrossRef]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Heng, X.

Hogenboom, D. O.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Ikeda, T.

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

Ishiwata, H.

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three-dimensional measurement by differential interference contrast microscope," Opt. Comm. 260, 117-126 (2006).
[CrossRef]

Itoh, M.

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three-dimensional measurement by differential interference contrast microscope," Opt. Comm. 260, 117-126 (2006).
[CrossRef]

Juptner, W.

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Li, L. F.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Lindberg, S. C.

Marx, D. S.

McMahon, P. J.

P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
[CrossRef]

Nomarski, G.

G. Nomarski, and A. R. Weill, "Application à la métallographie des méthodes interférentielles à deux ondes polarisées," Rev. Metall. 2, 121-128 (1955).

Nugent, K. A.

P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
[CrossRef]

Popescu, G.

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

Psaltis, D.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Roy, M.

P. Hariharan, and M. Roy, "Achromatic phase-shifting for two-wavelength phase-stepping interferometry," Opt. Comm. 126, 220-222 (1996).
[CrossRef]

Sato, T.

W. Shimada, T. Sato, and T. Yatagai, "Optical surface microtopography using phase-shifting Nomarski microscope," Proc. SPIE 1332, 525-529 (1990).
[CrossRef]

Schnars, U.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Shaw, J. C.

J. C. Shaw, "Metrology using differential phase-contrast microscopy," Microelectron. Eng. 13, 527-530 (1991).
[CrossRef]

Sheppard, C. J. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

Shimada, W.

W. Shimada, T. Sato, and T. Yatagai, "Optical surface microtopography using phase-shifting Nomarski microscope," Proc. SPIE 1332, 525-529 (1990).
[CrossRef]

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Thurber, S. R.

Tuchin, V. V.

V. V. Tuchin, "Coherence-domain methods in tissue and cell optics," Laser Phys. 8, 807-849 (1998).

Weill, A. R.

G. Nomarski, and A. R. Weill, "Application à la métallographie des méthodes interférentielles à deux ondes polarisées," Rev. Metall. 2, 121-128 (1955).

Yang, C. H.

Yaqoob, Z.

Yatagai, T.

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three-dimensional measurement by differential interference contrast microscope," Opt. Comm. 260, 117-126 (2006).
[CrossRef]

W. Shimada, T. Sato, and T. Yatagai, "Optical surface microtopography using phase-shifting Nomarski microscope," Proc. SPIE 1332, 525-529 (1990).
[CrossRef]

Zernike, F.

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
[CrossRef]

F. Zernike, "Phase contrast, a new method for the microsopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

Appl. Opt.

Appl. Spectrosc.

J. Biomed. Opt.

G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Erythrocyte structure and dynamics quantified by Hilbert phase microscopy," J. Biomed. Opt. 10, (2005).
[CrossRef]

Laser Phys.

V. V. Tuchin, "Coherence-domain methods in tissue and cell optics," Laser Phys. 8, 807-849 (1998).

Microelectron. Eng.

J. C. Shaw, "Metrology using differential phase-contrast microscopy," Microelectron. Eng. 13, 527-530 (1991).
[CrossRef]

Opt. Comm.

P. Hariharan, and M. Roy, "Achromatic phase-shifting for two-wavelength phase-stepping interferometry," Opt. Comm. 126, 220-222 (1996).
[CrossRef]

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three-dimensional measurement by differential interference contrast microscope," Opt. Comm. 260, 117-126 (2006).
[CrossRef]

Opt. Lett.

Oxford

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. of Microsc.-Oxford 199, 79-84 (2000).
[CrossRef]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. of Microsc.-Oxford 214, 7-12 (2004).
[CrossRef]

P. J. McMahon, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, "Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM," J. Microsc.-Oxford 206, 204-208 (2002).
[CrossRef]

Physica

F. Zernike, "Phase contrast, a new method for the microsopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
[CrossRef]

Proc. SPIE

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, "Quantitative DIC microscopy using a geometric phase shifter," Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

W. Shimada, T. Sato, and T. Yatagai, "Optical surface microtopography using phase-shifting Nomarski microscope," Proc. SPIE 1332, 525-529 (1990).
[CrossRef]

Rev. Metall.

G. Nomarski, and A. R. Weill, "Application à la métallographie des méthodes interférentielles à deux ondes polarisées," Rev. Metall. 2, 121-128 (1955).

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1.
Fig. 1.

Spatial phase modulation in a sinusoidal phase grating.

Fig. 2.
Fig. 2.

(a, b) Schematic of a Michelson interferometer using a single shallow diffraction grating, G1. (a) The transmitted sample beam and the diffracted reference beam are shown in black and red solid lines, respectively. (b) shows dashed black and red lines representative of coincident sample and reference beam at output ports I, II, and III of the interferometer. (c) Phase shift of the diffracted beam with respect to the undiffracted beam during the first diffraction. (d,e) Phase shifts of the diffracted sample and reference beams, respectively, during the second diffraction. x 1 is the actuation of grating G1 along the x-direction for the experiment, whereas d 1 and d 2 represent path lengths of sample and reference arms, respectively. M i : ith Mirror; BS: Beam splitter.

Fig. 3.
Fig. 3.

Measured phase shifts between different ports of a single grating-based Michelson interferometer versus grating displacement (x 1) along the x-direction.

Fig. 4.
Fig. 4.

(a, b). Modified Michelson interferometer design based on two harmonically-related shallow transmission gratings, where d 1 is the inter-grating distance and θ is the angle of diffraction. The sample and reference beams are shown as solid black and dashed red-green lines, respectively. (c)-(h) Phase shifts of the diffracted and undiffracted light during the first and second diffractions at gratings G1 and G2. Parameters x 1 and x 2 correspond to the actuations of grating G1 and G2, respectively, whereas d 3 and d 4 represent the path lengths of the sample and reference arms, respectively. M i : ith Mirror; BS: Beam splitter.

Fig. 5.
Fig. 5.

Schematics labeling different ports of gratings G1 and G2.

Fig. 6.
Fig. 6.

Measured phase shifts between (a) ports I & II and (b) ports I & III of a harmonically-related gratings based modified Michelson interferometer versus shearing of grating G2 with respect to G1. The locations where phase difference between port I and III are equal 90° and 270° are labeled.

Tables (1)

Tables Icon

Table 1. Measured efficiencies of the gratings used in the setup shown in Figs. 4(a), and 4(b). These diffraction efficiencies were used to determine the theoretical phase shifts between different output ports of the modified Michelson interferometer.

Equations (41)

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t ( x ) = exp { cos ( 2 π Λ ( x + x o ) ) } ,
t ( x ) = m = J m ( α ) exp { jm ( 2 π Λ x + ξ ( x o ) + π 2 ) } ,
ϕ ( x o ) = { m { ξ ( x o ) + π 2 } , m 1 m { ξ ( x o ) + π 2 } . m 1
E I = E 1 exp { i ( 2 k d 1 ξ 1 ( x 1 ) + π 2 ) } Field comp. from the sample arm
+ E 2 exp { i ( 2 k d 2 + ξ 1 ( x 1 ) + π 2 ) } , Field comp. from the reference arm
i I = 2 A 1 cos { 2 k ( d 2 d 1 ) + 2 ξ 1 ( x 1 ) } ,
i II = 2 A 2 cos { 2 k ( d 2 d 1 ) + 2 ξ 1 ( x 1 ) + π } ,
i III = 2 A 3 cos { 2 k ( d 2 d 1 ) + 2 ξ 1 ( x 1 ) + π } ,
E I = exp { i 2 k ( d 2 + d 4 ) }
× [ E I , 1 exp { i ( ξ 1 ( x 1 ) + π 2 ) } Field comp. from ref. arm via M 1 , M 4 , M 1
+ E I , 2 exp { i ( ξ 1 ( x 1 ) ξ 2 ( x 2 ) + 2 π ) } Field comp. from ref. arm via M 1 , M 4 , M 2
+ E I , 3 exp { i ( ξ 1 ( x 1 ) ξ 2 ( x 2 ) + π ) } Field comp. from ref. arm via M 2 , M 4 , M 1
+ E I , 4 exp { i ( 3 ξ 1 ( x 1 ) 2 ξ 2 ( x 2 ) + π 2 ) } ] Field comp. from ref. arm via M 2 , M 4 , M 2
+ E I , 5 exp { i 2 k ( d 1 + d 3 ) } exp { i ( ξ 1 ( x 1 ) + π 2 ) } Field comp. from the sample arm
i I ( x 1 , x 2 ) = 2 E ( x 1 , x 2 ) E I , 5 cos [ 2 k ( d 4 d 3 + d 2 d 1 ) + ϕ ( x 1 , x 2 ) 2 { ξ 2 ( x 2 ) + ξ 1 ( x 1 ) } ] ,
E ( x 1 , x 2 ) = F 1 2 ( x 1 , x 2 ) + F 2 2 ( x 1 , x 2 ) , ϕ ( x 1 , x 2 ) = tan 1 [ F 2 ( x 1 , x 2 ) F 1 ( x 1 , x 2 ) ]
F 1 ( x 1 , x 2 ) = E I , 1 cos { 4 ξ 1 ( x 1 ) + 2 ξ 2 ( x 2 ) } + ( E I , 2 E I , 3 ) sin { 2 ξ 1 ( x 1 ) + ξ 2 ( x 2 ) } + E I , 4 ,
F 2 ( x 1 , x 2 ) = E I , 1 sin { 4 ξ 1 ( x 1 ) + 2 ξ 2 ( x 2 ) } ( E I , 2 + E I , 3 ) cos { 2 ξ 1 ( x 1 ) + ξ 2 ( x 2 ) } .
E I , 1 = η 1,24 η 2,43 η 2,34 η 1,43 ,
E I , 2 = η 1,24 η 2,43 η 2,36 η 1,63 ,
E I , 3 = η 1,26 η 2,63 η 2,34 η 1,43 ,
E I , 4 = η 1,26 η 2,63 η 2,36 η 1,63 ,
E I , 5 = η 1,25 η 2,52 η 2,25 η 1,53 ,
i II ( x 1 , x 2 ) = 2 E ( x 1 , x 2 ) E II , 5 cos [ 2 k ( d 4 d 3 + d 2 d 1 ) + ϕ ( x 1 , x 2 ) 2 { ξ 2 ( x 2 ) + ξ 1 ( x 1 ) } ] ,
E ( x 1 , x 2 ) = F 3 2 ( x 1 , x 2 ) + F 4 2 ( x 1 , x 2 ) , ϕ ( x 1 , x 2 ) = tan 1 [ F 4 ( x 1 , x 2 ) F 3 ( x 1 , x 2 ) ]
F 3 ( x 1 , x 2 ) = E II , 1 cos { 4 ξ 1 ( x 1 ) + 2 ξ 2 ( x 2 ) } + ( E II , 2 + E II , 3 ) sin { 2 ξ 1 ( x 1 ) + ξ 2 ( x 2 ) } + E II , 4 ,
F 4 ( x 1 , x 2 ) = E II , 1 sin { 4 ξ 1 ( x 1 ) + 2 ξ 2 ( x 2 ) } ( E II , 2 + E II , 3 ) cos { 2 ξ 1 ( x 1 ) + ξ 2 ( x 2 ) } .
E II , 1 = η 1,24 η 2,43 η 2,34 η 1,42 ,
E II , 2 = η 1,24 η 2,43 η 2,36 η 1,62 ,
E II , 3 = η 1,26 η 2,63 η 2,34 η 1,42 ,
E II , 4 = η 1,26 η 2,63 η 2,36 η 1,62 ,
E II , 5 = η 1,25 η 2,52 η 2,25 η 1,52 .
i III ( x 1 , x 2 ) = 2 E ( x 1 , x 2 ) E III , 5 cos [ 2 k ( d 4 d 3 + d 2 d 1 ) + ϕ ( x 1 , x 2 ) 2 { ξ 2 ( x 2 ) + ξ 1 ( x 1 ) } + π ] ,
E ( x 1 , x 2 ) = F 5 2 ( x 1 , x 2 ) + F 6 2 ( x 1 , x 2 ) , ϕ ( x 1 , x 2 ) = tan 1 [ F 6 ( x 1 , x 2 ) F 5 ( x 1 , x 2 ) ]
F 5 ( x 1 , x 2 ) = E III , 1 cos { 4 ξ 1 ( x 1 ) + 2 ξ 2 ( x 2 ) } + ( E III , 2 E III , 3 ) sin { 2 ξ 1 ( x 1 ) + ξ 2 ( x 2 ) } + E III , 4 ,
F 6 ( x 1 , x 2 ) = E III , 1 sin { 4 ξ 1 ( x 1 ) + 2 ξ 2 ( x 2 ) } + ( E III , 3 E III , 2 ) cos { 2 ξ 1 ( x 1 ) + ξ 2 ( x 2 ) } .
E III , 1 = η 1,24 η 2,43 η 2,34 η 1,41 ,
E III , 2 = η 1,24 η 2,43 η 2,36 η 1,61 ,
E III , 3 = η 1,26 η 2,63 η 2,34 η 1,41 ,
E III , 4 = η 1,26 η 2,63 η 2,36 η 1,61 ,
E III , 5 = η 1,25 η 2,52 η 2,25 η 1,51 .

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