Abstract

Nondestructive noncontact high-resolution optical technique for profiling soft or fluidic boundary of an opaque object is presented. Our technique utilizes the fact that the angle width, the angular separation between two adjacent intensity minima in the forward shadow diffraction, is inversely proportional to the projected width of the object in the same direction. An analytic formula for reconstructing the boundary shape is obtained for an object with two-fold symmetry in terms of the angle widths measured for various rotational angles of the object. The typical error in determining the object shape parameter is less than 0.2%, which corresponds to 20 nm of radial accuracy when applied to an object with a mean radius of 10 microns.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
  2. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
    [CrossRef] [PubMed]
  3. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
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    [CrossRef]
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  12. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy," Appl. Opt. 40, 6177-6186 (2001).
    [CrossRef]
  13. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
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    [CrossRef] [PubMed]
  23. S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
    [CrossRef]
  24. J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
    [CrossRef] [PubMed]
  25. J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
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2008

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

2007

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, "Classical Phase Space Revealed by Coherent Light," Phys. Rev. Lett. 98, 033902 (2007).
[CrossRef] [PubMed]

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

2006

Z. Chen, "Study of a dynamic-shape-curve function for a fused tapering optical fiber," Appl. Opt. 45, 6914-6918 (2006).
[CrossRef] [PubMed]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

2005

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
[CrossRef] [PubMed]

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

2004

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

S. V. Boriskina, P. Sewell, and T. M. Benson, "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

2003

M. Yokota, A. Asaka, and T. Yosino, "Stabilization Improvements of Laser-Diode Closed-Loop Heterodyne Phase-Shifting Interferometer for Surface Profile Measurement," Appl. Opt. 42, 1805-1808 (2003).
[CrossRef] [PubMed]

S. Sonozaki, K. Iwata, and Y. Iwahashi, "Measurement of Profiles along a Circle on Two Flat Surfaces by Use of a Fizeau Interferometer with No Standard," Appl. Opt. 42, 6853-6858 (2003).
[CrossRef] [PubMed]

2002

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

2001

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy," Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

1999

P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, "Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements," Appl. Opt. 38, 5468-5479 (1999).
[CrossRef]

1998

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

1997

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, "Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory," Phys. Rev. E 55, 4458-4474 (1997).
[CrossRef]

J. U. Nöckel and A. D. Stone, "Ray and wave chaos in asymmetric resonant optical cavities," Nature 385, 45-47 (1997).
[CrossRef]

1995

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

1992

T. Noda and S. Kawata, "Separation of phase and absorption images in phase-contrast microscopy," J. Opt. Soc. Am. A 9, 924-931 (1992).
[CrossRef]

M. R. Atkinson, A. E. Dixon, and S. Damaskinos, "Surface-profile reconstruction using reflection differential phase-contrast microscopy," Appl. Opt. 31, 6765-6771 (1992).
[CrossRef] [PubMed]

1985

B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, "Measurement of surface topography of magnetic tapes by Mirau interferometry," Appl. Opt. 24, 1489-1497 (1985).
[CrossRef] [PubMed]

Akkin, T.

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
[CrossRef] [PubMed]

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

An, K.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Asaka, A.

M. Yokota, A. Asaka, and T. Yosino, "Stabilization Improvements of Laser-Diode Closed-Loop Heterodyne Phase-Shifting Interferometer for Surface Profile Measurement," Appl. Opt. 42, 1805-1808 (2003).
[CrossRef] [PubMed]

Assoufid, L.

P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, "Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements," Appl. Opt. 38, 5468-5479 (1999).
[CrossRef]

Atkinson, M. R.

M. R. Atkinson, A. E. Dixon, and S. Damaskinos, "Surface-profile reconstruction using reflection differential phase-contrast microscopy," Appl. Opt. 31, 6765-6771 (1992).
[CrossRef] [PubMed]

Badizadegan, K.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Ben-Messaoud, T.

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

Benson, T. M.

S. V. Boriskina, P. Sewell, and T. M. Benson, "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Bhushan, B.

B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, "Measurement of surface topography of magnetic tapes by Mirau interferometry," Appl. Opt. 24, 1489-1497 (1985).
[CrossRef] [PubMed]

Boriskina, S. V.

S. V. Boriskina, P. Sewell, and T. M. Benson, "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Bresloff, C. J.

P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, "Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements," Appl. Opt. 38, 5468-5479 (1999).
[CrossRef]

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Cense, B.

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
[CrossRef] [PubMed]

Chang, J.-S.

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Chang, R. K.

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

Chen, Z.

Z. Chen, "Study of a dynamic-shape-curve function for a fused tapering optical fiber," Appl. Opt. 45, 6914-6918 (2006).
[CrossRef] [PubMed]

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Choi, W.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Church, E. L.

P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, "Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements," Appl. Opt. 38, 5468-5479 (1999).
[CrossRef]

Cid, M. G.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Constantinides, G. N.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Cucu, R. G.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Damaskinos, S.

M. R. Atkinson, A. E. Dixon, and S. Damaskinos, "Surface-profile reconstruction using reflection differential phase-contrast microscopy," Appl. Opt. 31, 6765-6771 (1992).
[CrossRef] [PubMed]

Dasari, R. R.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Davé, D. P.

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

de Boer, J. F.

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
[CrossRef] [PubMed]

Diller, K. R.

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

Dixon, A. E.

M. R. Atkinson, A. E. Dixon, and S. Damaskinos, "Surface-profile reconstruction using reflection differential phase-contrast microscopy," Appl. Opt. 31, 6765-6771 (1992).
[CrossRef] [PubMed]

Döbereiner, H.-G.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, "Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory," Phys. Rev. E 55, 4458-4474 (1997).
[CrossRef]

Dobre, G. M.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Evans, E.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, "Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory," Phys. Rev. E 55, 4458-4474 (1997).
[CrossRef]

Faist, G. J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Feld, M. S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

Fukushima, T.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, "Classical Phase Space Revealed by Coherent Light," Phys. Rev. Lett. 98, 033902 (2007).
[CrossRef] [PubMed]

Gintides, D.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Harald, S. G. L.

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

Harayama, T.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, "Classical Phase Space Revealed by Coherent Light," Phys. Rev. Lett. 98, 033902 (2007).
[CrossRef] [PubMed]

Hentschel, M.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, "Classical Phase Space Revealed by Coherent Light," Phys. Rev. Lett. 98, 033902 (2007).
[CrossRef] [PubMed]

Ikeda, T.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

Iwahashi, Y.

S. Sonozaki, K. Iwata, and Y. Iwahashi, "Measurement of Profiles along a Circle on Two Flat Surfaces by Use of a Fizeau Interferometer with No Standard," Appl. Opt. 42, 6853-6858 (2003).
[CrossRef] [PubMed]

Iwata, K.

S. Sonozaki, K. Iwata, and Y. Iwahashi, "Measurement of Profiles along a Circle on Two Flat Surfaces by Use of a Fizeau Interferometer with No Standard," Appl. Opt. 42, 6853-6858 (2003).
[CrossRef] [PubMed]

Joo, C.

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
[CrossRef] [PubMed]

Kato, J.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy," Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

Kattis, S. E.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Kawata, S.

T. Noda and S. Kawata, "Separation of phase and absorption images in phase-contrast microscopy," J. Opt. Soc. Am. A 9, 924-931 (1992).
[CrossRef]

Kim, S. W.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Kim, S.-W.

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

Kiriaki, K.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Koliopoulos, C. L.

B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, "Measurement of surface topography of magnetic tapes by Mirau interferometry," Appl. Opt. 24, 1489-1497 (1985).
[CrossRef] [PubMed]

Kraus, M.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, "Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory," Phys. Rev. E 55, 4458-4474 (1997).
[CrossRef]

Lee, H.-W.

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

Lee, J.-H.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Lee, S. B.

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Lee, S.-B.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

Lee, S.-Y.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

Liang, H.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Lue, N.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Martin, Y.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

Milner, E.

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

Mizuno, J.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy," Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

Moon, H.-J.

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers," Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Moon, S.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. U. Nöckel and A. D. Stone, "Ray and wave chaos in asymmetric resonant optical cavities," Nature 385, 45-47 (1997).
[CrossRef]

Noda, T.

T. Noda and S. Kawata, "Separation of phase and absorption images in phase-contrast microscopy," J. Opt. Soc. Am. A 9, 924-931 (1992).
[CrossRef]

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Ohta, S.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy," Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

Paraskeva, C. A.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Park, B. H.

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett. 30, 2131-2133 (2005).
[CrossRef] [PubMed]

Payatakes, A. C.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Pedro, J.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Podoleanu, A. G.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Polyzos, D.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Popescu, G.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Quantitative phase imaging of live cells using fast Fourier phase microscopy," Appl. Opt. 46,1836-1842 (2007).
[CrossRef] [PubMed]

Rex, N. B.

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

Rylander, C. G.

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

Saunders, D.

H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, "En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation," Opt. Express 13, 6133-6144 (2005).
[CrossRef] [PubMed]

Seifert, U.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, "Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory," Phys. Rev. E 55, 4458-4474 (1997).
[CrossRef]

Sewell, P.

S. V. Boriskina, P. Sewell, and T. M. Benson, "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Shim, J.-B.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Sonozaki, S.

S. Sonozaki, K. Iwata, and Y. Iwahashi, "Measurement of Profiles along a Circle on Two Flat Surfaces by Use of a Fizeau Interferometer with No Standard," Appl. Opt. 42, 6853-6858 (2003).
[CrossRef] [PubMed]

Stone, A. D.

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, "High-Power Directional Emission from Microlasers with Chaotic Resonators," Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. U. Nöckel and A. D. Stone, "Ray and wave chaos in asymmetric resonant optical cavities," Nature 385, 45-47 (1997).
[CrossRef]

Takacs, P. Z.

P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, "Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements," Appl. Opt. 38, 5468-5479 (1999).
[CrossRef]

Tanaka, T.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, "Classical Phase Space Revealed by Coherent Light," Phys. Rev. Lett. 98, 033902 (2007).
[CrossRef] [PubMed]

Tsinopoulos, S. V.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Tureci, H. E.

S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Am. B 21, 923-934 (2004).
[CrossRef]

Welch, A. J.

C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

Wickramasinghe, H. K.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

Wortis, M.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, "Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory," Phys. Rev. E 55, 4458-4474 (1997).
[CrossRef]

Wyant, J. C.

B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, "Measurement of surface topography of magnetic tapes by Mirau interferometry," Appl. Opt. 24, 1489-1497 (1985).
[CrossRef] [PubMed]

Yamaguchi, I.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy," Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

Yang, H.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, "Uncertainty-limited turnstile transport in deformed microcavities," Phys. Rev. Lett. 100, 174102 (2008).
[CrossRef] [PubMed]

Yang, J.

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).
[CrossRef]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, "Development of a deformation-tunable quadrupolar microcavity," Rev. Sci. Instrum. 77, 083103 (2006).
[CrossRef]

Yannopoulos, S. N.

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, "Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results," Appl. Opt. 37, 7310-7319 (1998).
[CrossRef]

Yokota, M.

M. Yokota, A. Asaka, and T. Yosino, "Stabilization Improvements of Laser-Diode Closed-Loop Heterodyne Phase-Shifting Interferometer for Surface Profile Measurement," Appl. Opt. 42, 1805-1808 (2003).
[CrossRef] [PubMed]

Yosino, T.

M. Yokota, A. Asaka, and T. Yosino, "Stabilization Improvements of Laser-Diode Closed-Loop Heterodyne Phase-Shifting Interferometer for Surface Profile Measurement," Appl. Opt. 42, 1805-1808 (2003).
[CrossRef] [PubMed]

Zenhausern, F.

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Supplementary Material (2)

» Media 1: MOV (2469 KB)     
» Media 2: MOV (1312 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Diffraction patterns of an absorbing cylindrical object with the size parameter x=180. The real part of its index of refraction is 1.361 while its imaginary part is 0.0022 (T=21%, Qabs =310), 0.0096 (T=0.1%, Qabs =71) or 0.019 (T=10-4%, Qabs =36). Also in animation. [Media 1]

Fig. 2.
Fig. 2.

Measurement geometry for boundary profiling. Angle β is the rotation angle of the major axis of the object with respect to the forward direction and the polar angle α corresponds to the border between the shadow and illuminated regions on the object surface. The border angle ϕb measured from the forward direction is related to α by ϕb = αβ. Projected width w is also denoted.

Fig. 3.
Fig. 3.

(a) Reconstructed boundary profile (in black) obtained from calculated diffraction patterns for an opaque cylindrical object whose boundary (shown in red) is described by Eq. (13) with η 1=16.00% and η 2=1.02%. (b) Projected width w as a function of the object rotation angle β. The difference between the reconstructed width (in black) and the actual width (in red) is also shown (in blue). By fitting the reconstructed boundary of (a) with Eq. (13), we obtain η 1=15.90(±0.02)% and η 2=0.98(±0.05)%.

Fig. 4.
Fig. 4.

(a) Scheme for measuring angle width Δθ for various rotational angle β of a liquid-jet microcavity (oval shape). A laser beam is indicated by a thick arrow. (b) Angle width versus object rotation angle measured for a liquid-jet microcavity of a mean radius of 14 μm with an Ar-ion laser (λ =514 nm). The angle widths were measured around at θ =5° in order to avoid the strong forward intensity peak. (c) Reconstructed boundary profile (black) and a fit (red). By fitting the reconstructed boundary with Eq. (13), we obtain (quadrupole : octapole)=93.2 : 6.8 with a total deformation of 17.7(±0.3)%. (a) and (b) also in animation. [Media 2]

Fig. 5.
Fig. 5.

A plane electromagnetic wave with a wave vector k 0 is incident on an opaque object on the xy-plane. The boundary of the object is described by r(ϕ ) with ϕ the polar angle and the unit vector normal to the boundary. The polar angle is measured with respect to the direction of k 0. The major axis of the object is rotated by β from the k 0 direction. The region of the object specified by the condition ϕ b2 < ϕ < ϕ b1 is shadowed. The diffracted wave with a wave vector k in the direction of θ is considered.

Fig. 6.
Fig. 6.

Comparison of the simulated angle width Δθ (black dots) and the inverse w −1 of the projected width (red dots) as a function of object rotation angle β for a totally absorbing quadrupole-deformed object whose boundary is given by Eq. (12). The deformation parameter η is varied from 4% to 22% while the size parameter and the absorption Q are respectively fixed at 200 and 100. The angle widths are measured in the direction of θ 0=5° and the projected widths are measured in the direction of θ 0/2. We observe an excellent agreement between them.

Tables (1)

Tables Icon

Table 1. Reconstructed shape parameters of various pure quadrupolar objects (2 ≤ η ≤ 19) with a common size parameter of 200. Reconstructed boundary profiles are intentionally fitted to the quadru-octapole shape of Eq. (13) in order to test the robustness and sensitivity of the present reconstruction technique. The resulting fit parameters are listed.

Equations (51)

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I ( θ ) [ sin ( πd sin θ / λ ) πd sin θ / λ ] 2 ,
Δ θ = λ d cos θ ,
Q abc = m r 2 m i ,
x = 2 πr λ
Δ θ = λ 2 w ,
w ( β ) = r ( α ) sin ( α β ) .
tan β = d [ r ( α ) sin α ] d [ r ( α ) cos α ] .
w ( β ) = w β + w α α β = r ( α ) cos ( α β ) + { d [ r ( α ) sin α ] cos β d [ r ( α ) cos α ] sin β } .
w ( β ) = r ( α ) cos ( α β ) .
α = β arctan w ( β ) w ( β )
r ( β arctan w ( β ) w ( β ) ) = w ( β ) sin [ arctan w ( β ) w ( β ) ] .
r ( ψ ) = a ( 1 + η cos 2 ψ ) ,
r ( ψ ) = a ( 1 + η 1 cos 2 ψ + η 2 cos 4 ψ ) .
E i 4 π [ ω z ̂ ( n ̂ × B s ) + z ̂ ( k × ( n ̂ × E s ) ] e i k r ( ϕ ) dl ,
E sh = E 0 4 πi z ̂ sh ( n ̂ × ( k 0 × z ̂ ) + k × ( n ̂ × z ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi z ̂ sh ( k 0 ( n ̂ z ̂ ) z ̂ ( n ̂ k 0 ) + n ̂ ( k ̂ z ̂ ) z ̂ ( k ̂ n ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi sh ( k + k 0 ) n ̂ e i ( k 0 k ) r ( ϕ ) dl ,
E ill = E 0 4 πi z ̂ ill R ( n ̂ × ( k 0 × z ̂ ) k × ( n ̂ × z ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi z ̂ ill R ( k 0 ( n ̂ z ̂ ) z ̂ ( n ̂ k 0 ) n ̂ ( k z ̂ ) + z ̂ ( k n ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi ill R ( k 0 k ) n ̂ e i ( k 0 k ) r ( ϕ ) dl
k = x ̂ k cos θ + y ̂ k sin θ ,
r ( ϕ ) = x ̂ r ( ϕ ) cos ϕ + y ̂ r ( ϕ ) sin ϕ ,
dl = [ d ( r cos ϕ ) ] 2 + [ d ( r sin ϕ ) ] 2 ,
n ̂ = [ x ̂ d ( r sin ϕ ) y ̂ d ( r cos ϕ ) ] dl .
[ ( 1 cos θ ) cos ϕ sin θ sin ϕ ] = sin θ [ sin ( ϕ θ ) + sin ϕ ]
E sh = E 0 k 4 πi ϕ b2 ϕ b1 e ikr ( ϕ ) [ ( 1 cos θ ) cos ϕ sin θ sin ϕ ] d { r ( ϕ ) [ sinϕ ( 1 + cos θ ) cos ϕ sin θ ] }
= E 0 k 4 πi ϕ b2 ϕ b1 e −ikr ( ϕ ) sin θ 1 + cos θ [ sin ( ϕ θ ) + sin ϕ ] d { r ( ϕ ) [ sin ( ϕ θ ) + sin ϕ ] } ,
E ill = E 0 k 4 πi ϕ b1 ϕ b 2 + 2 π R ( ϕ ) e ikr ( ϕ ) ( ( 1 cos θ ) cos ϕ sin θ sin ϕ ) d { r ( ϕ ) [ sin ϕ ( 1 cos θ ) + cos ϕ sin θ ] }
= E 0 k 4 πi ϕ b1 ϕ b 2 + 2 π R ( ϕ ) e ikr ( ϕ ) sin θ 1 + cos θ [ sin ( ϕ θ ) + sin ϕ ] d { r ( ϕ ) [ sin ( ϕ θ ) sin ϕ ] } .
E sh E 0 k 2 πi ϕ b 2 ϕ b 1 e ikθr ( ϕ ) sin ( ϕ θ 2 ) d { r ( ϕ ) sin ( ϕ θ 2 ) } E 0 k 2 πi X ,
E ill θ E 0 k 4 πi ϕ b 1 ϕ b 2 + 2 π e ikθr ( ϕ ) sin ( ϕ θ 2 ) d { r ( ϕ ) cos ϕ } θ E 0 k 4 πi Y .
I = E 2 X 2 + θ 2 Y 2 X 2
E ( θ ) E 0 k 2 πi e ikr ( ϕ b 2 ) θ sin ( ϕ b 2 θ 2 ) e ikr ( ϕ b 1 ) θ sin ( ϕ b 1 θ 2 ) ikθ .
E ( θ ) E 0 πi sin [ kr ( ϕ b ) θ sin ( ϕ b θ 2 ) ] θ .
I ( θ ) sin 2 [ kr ( ϕ b ) θ sin ( ϕ b θ 2 ) ] θ 2 .
e ikθy 2 Δx 2 + e ikθy 1 Δx 1
( e ikθy 1 e ikθy 1 ) Δ x 1 i 2 sin y 1
r ( ϕ b + θ 0 2 ) sin ( ϕ b ) r ( ϕ b ) sin ( ϕ b θ 0 2 ) θ 0 2 2 .
Δθ = π kw
d d ϕ r ( ϕ ) | ϕ b = r ´ ( ϕ b ) sin ϕ b + r ( ϕ b ) cos ϕ b = 0
D [ r ( ϕ b ) + r ´ ( ϕ b ) θ ] sin ϕ b r ( ϕ b ) [ sin ( ϕ b ) cos θ cos ϕ b sin θ ]
= r ( ϕ b ) sin ϕ b ( 1 cos θ ) + r ( ϕ b ) sin ϕ b θ + r ( ϕ b ) cos ϕ b sin θ
r ( ϕ b ) sin ϕ b ( θ 2 2 ) + [ r ( ϕ b ) sin ϕ b + r ( ϕ b ) cos ϕ b ] θ r ( ϕ b ) cos ϕ b ( θ 3 6 )
= r ( ϕ b ) sin ϕ b ( θ 2 2 ) r ( ϕ b ) cos ϕ b ( θ 3 6 )
2 ,
w ( β ) = r ( α ) sin ( α β θ 0 2 ) r ( α + θ 0 2 ) sin ( α β ) ,
w ( θ 0 2 ) = r ( α + θ 0 2 ) sin ( α + θ 0 2 ) = r ( α ) sin α ,
w ( β ) = r ( α ) sin ( α β ) .
w ( β ) = r ( α ) sin ( α β ) .
E ( θ ) E 0 2 πθ [ e ikr ( ϕ b 2 ) θ sin ( ϕ b 2 θ 2 ) e ikr ( ϕ b 1 ) θ sin ( ϕ b 1 θ 2 ) ] ,
I ( θ ) sin 2 [ ( w 1 + w 2 ) 2 ] θ 2

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