Abstract

Measurements of weak, embedded index structures are important for material characterization of photopolymers, glass and other optical materials as well as for characterization of fabricated structures such as waveguides. We demonstrate an optical diffraction tomography system capable of measuring deeply-buried, weak, fabricated index structures written in a homogeneous volume. High-fidelity cross sections of these weak index structures are constructed by replicating the structure to be measured to form a diffraction grating. The coherent addition of scattering from each of these objects increases the sensitivity of the imaging system. Measurements are made in the far field, without the use of lenses, eliminating phase aberration errors through thick volumes.

© 2007 Optical Society of America

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2007 (1)

2005 (3)

2003 (1)

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

2002 (2)

M. Will, S. Nolte, B. N. Chichkov, and A. Tünnermann, "Optical properties of waveguides fabricated in fused silica by femtosecond laser pulses," Appl. Opt. 41, 4360-4364 (2002).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002).
[CrossRef]

2000 (1)

1998 (1)

1997 (2)

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

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997).
[CrossRef]

1996 (5)

1995 (1)

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

1994 (1)

G. Zhao and P. Mouroulis, ‘‘Diffusion model of hologram formation in dry photopolymer materials,’’J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

1992 (2)

C. J. Cogswell, and J. W. O’Byrne, "High-resolution confocal transmission microscope, Part I: system design," Proc. SPIE 1660, 503-511 (1992).
[CrossRef]

C. J. Cogswell and C. J. R. Sheppard, "Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

1989 (1)

B. L. Booth, "Low loss channel waveguides in polymers," J. Lightwave Technol. 7, 1445-1453 (1989).
[CrossRef]

1983 (1)

S. X. Pan and A. C. Kak, "A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation," IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262-1275 (1983).
[CrossRef]

1982 (1)

A. J. Devaney, "A Filtered Backpropagation Algorithm for Diffraction Tomography," Ultrasonic Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1971 (1)

1970 (1)

Booth, B. L.

B. L. Booth, "Low loss channel waveguides in polymers," J. Lightwave Technol. 7, 1445-1453 (1989).
[CrossRef]

Booth, M. J.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002).
[CrossRef]

Brenner, K.

Chen, B.

Chichkov, B. N.

Cogswell, C. J.

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

C. J. Cogswell, and J. W. O’Byrne, "High-resolution confocal transmission microscope, Part I: system design," Proc. SPIE 1660, 503-511 (1992).
[CrossRef]

C. J. Cogswell and C. J. R. Sheppard, "Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

Colburn, W. S.

Colvin, V. L.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997).
[CrossRef]

Daiber, A. J.

Davis, K. M.

Devaney, A. J.

A. J. Devaney, "A Filtered Backpropagation Algorithm for Diffraction Tomography," Ultrasonic Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Dhal, P. K.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Dhar, L.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Dobler, B.

Fujimoto, J. G.

Grabowski, M. W.

Haines, K. A.

Hariharan, P.

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

Harris, A. L.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997).
[CrossRef]

Hesselink, L.

Hill, A.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Hirao, K.

Horner, M. G.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Ihas, B.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Ingwall, R. T.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Ippen, E. P.

Juskaitis, R.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002).
[CrossRef]

Kagami, M.

Kak, A. C.

S. X. Pan and A. C. Kak, "A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation," IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262-1275 (1983).
[CrossRef]

Kato, S.

Kawasaki, A.

Kawata, S.

Kawata, Y.

Kewitsch, A. S.

Kolb, E. S.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Kowalevicz, A. M.

Larkin, K. G.

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

Larson, R. G.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997).
[CrossRef]

Lawrence, J. R.

Li, H-Y. S.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

McDonald, M. E.

McLeod, R. R.

Messerschmidt, B.

Michaels, D.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Minns, R. A.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Minoshima, K.

Miura, K.

Mouroulis, P.

G. Zhao and P. Mouroulis, ‘‘Diffusion model of hologram formation in dry photopolymer materials,’’J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002).
[CrossRef]

Nolte, S.

O’Byrne, J. W.

C. J. Cogswell, and J. W. O’Byrne, "High-resolution confocal transmission microscope, Part I: system design," Proc. SPIE 1660, 503-511 (1992).
[CrossRef]

Pan, S. X.

S. X. Pan and A. C. Kak, "A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation," IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262-1275 (1983).
[CrossRef]

Robertson, T. L.

Schild, H. G.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Schilling, M. L.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997).
[CrossRef]

Schnoes, M.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Schomberger, G.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Schreiber, H.

Setthachayanon, S.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Sharma, V.

Sheppard, C. J. R.

C. J. Cogswell and C. J. R. Sheppard, "Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

Sheridan, J. T.

Singer, W.

Slagle, T.

Smith, N. I.

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

Sochava, S. L.

Stamnes, J. J.

B. Chen and J. J. Stamnes, "Validity of diffraction tomography based on the first Born and the first Rytov approximations," Appl. Opt. 37, 2996-3006 (1998).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

Sugimoto, N.

Sullivan, A. C.

Tünnermann, A.

Waldman, R. A.

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Wedberg, T. C.

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

Will, M.

Wilson, T.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002).
[CrossRef]

T. Wilson, Y. Kawata, and S. Kawata, "Readout of three-dimensional optical memories," Opt. Lett. 21, 1003-1005 (1996).
[CrossRef] [PubMed]

Wilson, W. L.

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

Wolf, E.

Yariv, A.

Yonemura, M.

Zhao, G.

G. Zhao and P. Mouroulis, ‘‘Diffusion model of hologram formation in dry photopolymer materials,’’J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Acoust. Speech Signal Process. (1)

S. X. Pan and A. C. Kak, "A computation study of reconstruction algorithms for Diffraction Tomography: Interpolation versus Filtered Backpropagation," IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262-1275 (1983).
[CrossRef]

J. Appl. Phys. (1)

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, "Quantitative model of volume hologram formation in photopolymers," J. Appl. Phys. 81, 5913-5923 (1997).
[CrossRef]

J. Lightwave Technol. (1)

B. L. Booth, "Low loss channel waveguides in polymers," J. Lightwave Technol. 7, 1445-1453 (1989).
[CrossRef]

J. Microsc. (1)

C. J. Cogswell and C. J. R. Sheppard, "Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

J. Mod. Opt. (1)

G. Zhao and P. Mouroulis, ‘‘Diffusion model of hologram formation in dry photopolymer materials,’’J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (5)

Proc. SPIE (4)

C. J. Cogswell, and J. W. O’Byrne, "High-resolution confocal transmission microscope, Part I: system design," Proc. SPIE 1660, 503-511 (1992).
[CrossRef]

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

M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, W. L. Wilson, "Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003).
[CrossRef]

R. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H-Y. S. Li, R. A. Minns, and H. G. Schild, "Cationic ring-opening photopolymerization methods for holography," Proc. SPIE 2689, 127-141 (1996).
[CrossRef]

Proceedings of the National Academy of Sciences (1)

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, "Adaptive aberration correction in a confocal microscope," Proceedings of the National Academy of Sciences 99, 5788-5792 (2002).
[CrossRef]

Pure Appl. Opt. (1)

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

Ultrasonic Imaging (1)

A. J. Devaney, "A Filtered Backpropagation Algorithm for Diffraction Tomography," Ultrasonic Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (8)

R. T. Weverka, K. Wagner, R. R. Mcleod, K. Wu, and C. Garvin, "Low-Loss Acousto-Optic Photonic Switch," in Acousto-Optic Signal Processing Theory and Implementation, N. J. Berg and J. M. Pellegrino, eds. (Marcel Dekker, 1996), pp. 479 - 573.

M. M. Woolfson, An introduction to X-ray crystallography, ed. 2 (Cambridge University Press, 1997).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, ed. 7 (Cambridge University Press, 1999).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

InPhase Technologies, Tapestry Media, www.inphase-technologies.com.

M. Pluta, Advanced Light Microscopy Vol. 2. Specialized Methods, (Elsevier, NY 1989), pp. 146-197.

S. V. King, Department of Electrical and Computer Engineering, University of Colorado, Campus Box 425, Boulder, CO 80309, USA, A. Libertun, C. Preza, R. Piestun, and C. J. Cogswell are preparing a manuscript to be called "Quantitative phase microscopy through differential interference imaging."

C. J. R. Sheppard and D. M. Shotton, Confocal laser scanning microscopy, (BIOS Scientific, 1997).

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Figures (8)

Fig. 1.
Fig. 1.

Arcs in Fourier space representing the allowed propagating modes in the material at various incident electric field angles. The Fourier transform of the scattered field can only be measured along these arcs, resulting in an irregularly sampled grid of data.

Fig. 2.
Fig. 2.

Three-dimensional direct-write lithography system used for writing deeply-buried index structures in a volume photopolymer. The sample consists of 1 mm of photopolymer between two 1 mm thick pieces of glass. Green light is focused into the photopolymer with a molded asphere operating at 0.3 NA. The sample is moved perpendicular to the direction of propagation of the writing beam using high precision stages.

Fig. 3.
Fig. 3.

Optical diffraction tomography experimental set up. Tomography measurements are taken after the samples are fully cured. A frequency-doubled Nd:YAG laser at 532 nm is used to probe the diffraction gratings. The sample is rotated to probe the grating at a variety of angles. Measurements are taken of each order at each angle with a high-dynamic-range power meter.

Fig. 4.
Fig. 4.

Tomographic reconstruction of the cross section of the index of a single line of a diffraction grating written in photopolymer with the system shown in Fig. 2. The peak index change is 1.6 × 10-3. Cross sections of the reconstructed index are shown in Fig. 5.

Fig. 5.
Fig. 5.

Cross sections of the index change (shown in Fig. 4) written in the photopolymer in (a) x and (b) z at the peak of the index change. The cross section of δn in the x plane is taken at the center of the index in z, where z = 0. Similarly, the cross section of δn in the z plane is taken at x = 0. The small ripples of the index in the x cross section are due to the finite range of spatial frequency data supplied to the tomographic reconstruction algorithm.

Fig. 6.
Fig. 6.

Full-width at half-maximum (FWHM) measurements of cross sections of the reconstructed index as a function of data points used in the reconstruction algorithm. The slopes of both curves are approaching zero, indicating that additional samples will not change the feature size significantly.

Fig. 7.
Fig. 7.

Comparison of experimental diffraction efficiency and efficiency calculated from a fast-Fourier-transform beam-propagation method. There are no fit parameters used and so this calculation confirms both the shape and amplitude of the experimental reconstruction.

Fig. 8.
Fig. 8.

Comparison of (a) phase-shifted Differential Interference Contrast image and (b) d/dx of an optical diffraction tomography reconstruction of a diffraction grating with a 35 μm period.

Equations (16)

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( 2 + k 0 2 n 0 2 ) E ( x , z ) = k 0 2 n o 2 ( 1 n 2 ( x , z ) n 0 2 ) E ( x , z ) ,
1 n 2 ( x , z ) n 0 2 2 δ n ( x , z ) n 0 .
( 2 + k 0 2 n 0 2 ) E sc ( x , z ) = 2 k 0 2 n 0 δ n ( x , z ) E inc ( x , z ) .
E sc ( k x , L ) = i k 0 2 n 0 k z E inc ( k x , z = 0 ) * k x [ δ n obj ( k x , k z k ) ] exp [ i k z L ] ,
δ n tot ( x , z ) = δ n obj ( x , z ) * x comb ( x ) ,
comb ( x ) = n = δ ( x n Λ )
1 Λ comb ( k x ) = 1 Λ m = δ ( k x 2 πm Λ ) .
E inc ( x ) = A inc exp [ ( x w ) 2 ]
E inc ( k x ) = A inc w 2 exp [ ( k x w 2 ) 2 ] ,
E sc ( k x , L ) = i k 0 2 n 0 k z A inc 2 w Λ exp [ ( k x w 2 ) 2 ] * k x [ δ n obj ( k x , k z k ) comb ( k x ) ] exp ( i k z L ) .
E sc ( k x , L ) = i k 0 2 n 0 k z A inc 2 w Λ exp [ i k z L ] ×
m = exp { [ ( k x 2 π m Λ ) w 2 ] 2 } δ n obj ( 2 π m Λ , k z k ) .
P m = ( m 1 2 ) 2 π Λ ( m + 1 2 ) 2 π Λ E sc ( k x , L ) 2 d k x
( k 0 2 n 0 k z A inc 2 w Λ ) 2 δ n obj 2 ( m 2 π Λ , k z k ) exp { 2 [ ( k x 2 π m Λ ) w 2 ] 2 } d k x
P m ( k 0 2 n 0 k z 1 Λ ) 2 A inc 2 w π 2 δ n obj 2 ( m 2 π Λ , k z k ) .
δ n obj ( k x = m 2 π Λ , k z k ) = k z k 0 2 n 0 Λ P m P inc .

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