Abstract

Giving a new physical interpretation to the principle of longitudinal coherence control, we propose an improved method for synthesizing a spatial coherence function along the longitudinal axis of light propagation. By controlling the irradiance of an extended quasi-monochromatic spatially incoherent source with a spatial light modulator, we generated a special optical field that exhibits high coherence selectively for a specific pair of points at specified locations along the axis of beam propagation. This function of longitudinal coherence control provides new possibilities for dispersion-free measurements in optical tomography and profilometry. A quantitative experimental proof of principle is presented.

© 2002 Optical Society of America

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References

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2000 (1)

1999 (1)

1996 (1)

1995 (2)

J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

J. Rosen, B. Salik, A. Yariv, “Pseudo-nondiffracting beams generated by radial harmonic functions,” J. Opt. Soc. Am. A 12, 2446–2457 (1995).
[CrossRef]

1994 (2)

K. Hotate, T. Okugawa, “Optical information-processing by synthesis of the coherence function,” J. Lightwave Technol. 12, 1247–1255 (1994).
[CrossRef]

J. E. Biegen, “Determination of the phase change on reflection from two-beam interference,” Opt. Lett. 19, 1690–1692 (1994).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (1)

1987 (1)

1982 (1)

1974 (1)

1972 (1)

1966 (1)

Biegen, J. E.

Brangaccio, D. J.

Bruning, J. H.

Carr, S.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, & Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, & Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Davies, D. E. N.

de Boer, J. F.

Dresel, T.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Flournoy, P. 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gallagher, J. E.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hausler, G.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Herriott, D. J.

Hotate, K.

K. Hotate, T. Okugawa, “Optical information-processing by synthesis of the coherence function,” J. Lightwave Technol. 12, 1247–1255 (1994).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Ina, H.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1981), Chap. 14, p. 292.

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, & Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Kobayashi, S.

Lee, B. S.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, 1995), Chap. 4, p. 149.

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, & Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

McClure, R. W.

McCutchen, C. W.

Milner, T. E.

Nelson, J. S.

Okugawa, T.

K. Hotate, T. Okugawa, “Optical information-processing by synthesis of the coherence function,” J. Lightwave Technol. 12, 1247–1255 (1994).
[CrossRef]

Pashley, D. H.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Rosen, J.

Rosenfeld, D. P.

Salik, B.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Strand, T. C.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Takeda, M.

J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[CrossRef]

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

M. Takeda, “The philosophy of fringes—analogies and dualities in optical metrology,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1997), pp. 17–26.

Venzke, H.

Wang, X. J.

White, A. D.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1981), Chap. 14, p. 292.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, 1995), Chap. 4, p. 149.

Wyntjes, G.

Yariv, A.

Youngquist, R. C.

Zhang, Y.

Appl. Opt. (6)

J. Lightwave Technol. (1)

K. Hotate, T. Okugawa, “Optical information-processing by synthesis of the coherence function,” J. Lightwave Technol. 12, 1247–1255 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

Opt. Lett. (2)

Science (1)

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other (4)

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, & Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, 1995), Chap. 4, p. 149.

M. Takeda, “The philosophy of fringes—analogies and dualities in optical metrology,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1997), pp. 17–26.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1981), Chap. 14, p. 292.

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Figures (10)

Fig. 1
Fig. 1

Optical system for measuring the longitudinal complex degree of coherence. Abbreviations are defined in text.

Fig. 2
Fig. 2

Generation of Heiginger fringes of equal inclination and the inverse process of beam propagation.

Fig. 3
Fig. 3

Schematic illustration of the experimental system: C1, C2, collimator lenses; GG1, GG2, ground glass; transducer; BS1, BS2, beam splitters; other abbreviations defined in text.

Fig. 4
Fig. 4

(a) Haidinger fringes recorded for Δz = 8 mm. (b) Source irradiance distribution designed to have the same shape as the Haidinger fringes. (c) Longitudinal degree of coherence.

Fig. 5
Fig. 5

(a) Haidinger fringes recorded for Δz = 5 mm. (b) Source irradiance distribution designed to have the same shape as the Haidinger fringes. (c) Longitudinal degree of coherence. Optical path difference, zero; Δz = 0 at position z = 10 mm on the horizontal axis.

Fig. 6
Fig. 6

(a) Haidinger fringes recorded for Δz = 2 mm. (b) Source irradiance distribution designed to have the same shape as the Haidinger fringes. (c) Longitudinal degree of coherence. Optical path difference, zero; Δz = 0 at position z = 10 mm on the horizontal axis.

Fig. 7
Fig. 7

Relationship between the optical path difference for the first coherence peak and effective spatial frequency.

Fig. 8
Fig. 8

Longitudinal degree of coherence when a small tilt angle was introduced. The contrast peak at z = 10 mm corresponds the position of Δz = 0.

Fig. 9
Fig. 9

Relationship between degree of coherence and tilt angle.

Fig. 10
Fig. 10

Relationship between degree of coherence and decentering.

Equations (18)

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ux, y, z= usxs, ysjλfexpj 2πz+2fλ-j 2πλf× xsx+ysy-j πzλf2xs2+ys2,
Ix, y, L=usxs, ysjλfexpj 2πL+2fλ-j 2πλfxsx+ysy-j πLλf2xs2+ys2+ usxs, ysjλfexpj 2πL+2Δz+2fλ-j 2πλfxsx+ysy-j πL+2Δzλf2× xs2+ys22dxsdys,
I x, y, L=B1+ μ2Δzcos-4πΔzλ +ϕ2Δz,
μ2Δz=  Isxs, ysexpj 2πΔzλf2xs2+ys2dxsdys Isxs, ysdxsdys.
ux, y, -Δz0=expjk2f-Δz0jλf usxs, ys×exp- j2πλfxsx+ysy+jπΔz0λf2xs2+ys2dxsdys.
u0, 0, -Δz0=expjk2f-Δz0jλf usxs, ys×expjπΔz0λf2xs2+ys2dxsdys.
Isrs= 1/21+cos2πγrs2+β,
μΔz=0R2 Isρsexpj 2πΔzρsλf2dρs0R2 Isρsdρ,
Isρs=1+cos2πγρs+β/2,
μΔz  expjπR2Δzλf2sincπR2Δzλf2 * 2δ Δz+expjβδΔz+γλf2+exp-jβδΔz-γλf2,
ACH¯=AH¯=2Δz cos θ2Δz1-θ222Δz 1-r22f2,
Ir=121+cos2πΔzλf2 r2- 4πΔzλ.
Ixs, ys  1+cos2πγxs2+ys2+A0xs2-ys2+B0xsys+β,
Ix, y, L=usxs, ysjλfexpj 2πL+2fλ-j 2πλfxsx+ysy-j πLλf2xs2+ys2+ usxs, ysjλfexpj 2πL+2Δz+2fλ-j 2πλfxs-αxx+ys-αyy-j πL+2Δzλf2xs-αx2+ys-αy22dxsdys.
Ix, y, L=A 1+ μ2Δzcos-4πΔzλ+ϕ 2Δz-2πλfαxx+αyy+πL+2Δzλf2αx2+αy2,
μαx, αy, 2Δz=  Isxs, ysexpj2πΔzλf2xs2+ys2- j2πL+2Δzλf2xsαx+ysαydxsdys Isxs, ysdxsdys.
μΔx, Δy, 2Δz=  Isxs, ysexpj2πΔzλf2xs2+ys2- j2πλfxsΔx+ysΔydxsdys Isxs, ysdxsdys.
μ0= 02π0R Isrsexp-j 2πLλf2 αrs cosξ-ηrsdrsdη02π0R Isrsrsdrsdη= 2J12πLRα/λf22πLRα/λf2 = 2J14πLRθ/λf4πLRθ/λf,

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