Abstract

We develop new generalized four-wave-model-based waveguide mode equations for both isotropic and anisotropic systems by taking into account the influence of the incident light. These new mode equations eliminate the inherent deficiency in the conventional waveguide model, in which the action of incident light was neglected. Further, a peak-value-search (PVS) numerical method is developed to solve the four-wave-model-based mode equations. The PVS method has significant advantages in that accurate refractive index and thickness can be obtained without prior knowledge of the thickness of the air gap.

© 2004 Optical Society of America

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References

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  1. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
    [CrossRef] [PubMed]
  2. R. Ulrich, R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12, 2901–2908 (1973).
    [CrossRef] [PubMed]
  3. R. Th. Kersten, “Prism–film coupler as a precision instrument,” Opt. Acta 22, 503–513 (1975).
  4. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, Mass., 1991).
  5. D. G. Dalgoutte, “A high efficiency thin grating coupler for integrated optics,” Opt. Commun. 8, 124–127 (1973).
    [CrossRef]
  6. R. Th. Kersten, “A new method for measuring refractive index and thickness of liquid and deposited solid thin films,” Opt. Commun. 13, 327–329 (1975).
    [CrossRef]
  7. P. K. Tien, R. Ulrich, “Theory of prism–film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  8. P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
    [CrossRef]
  9. Manufacturer’s Bulletin, Metricon PC-2010 Prism Coupler (Metricon Corporation, Pennington, N.J., 1992).
  10. R. Th. Kersten, “Numerical solution of the mode-equation of planar dielectric waveguides to determine their refractive index and thickness by means of a prism–film coupler,” Opt. Commun. 9, 427–431 (1973).
    [CrossRef]
  11. Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
    [CrossRef]
  12. T. Liu, R. J. Samuels, “Improved refractive index from a planar leaky wave-guide coupler,” in Thin Films for Optical Devices and Materials for Optical Limiting, Vol. 597 of MRS Symposium Proceedings, K. Nashimoto, R. Pachter, B. W. Wessels, J. Shmulovich, A. K.-Y. Jen, K. Lewis, R. Sutherland, J. W. Perry, eds. (Materials Research Society, Boston, Mass., 2000), pp. 57–62.
  13. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  14. A. Varsicek, Optics of Thin Films, translated by H. Watney-Kaczer (Interscience, New York, 1960).
  15. O. S. Heavens, Optical Properties of Thin Solid Films (Academic, New York, 1955).
  16. L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (Institute of Physics, Philadelphia, Pa., 1994).
  17. T. Liu, “Novel methods for determining the optical constants of anisotropic polymer films—new application of prism wave-guide coupling,” Ph.D. thesis (Georgia Institute of Technology, Atlanta, Ga., 2002).
  18. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).
  19. H. Wang, “Determination of refractive indices and thickness of absorbing crystalline thin films by using prism coupler,” Fiber Integr. Opt. 13, 293–308 (1994).
    [CrossRef]
  20. P. K. Tien, G. Smolinsky, R. J. Martin, “Thin organosilicon films for integrated optics,” Appl. Opt. 11, 637–642 (1972).
    [CrossRef] [PubMed]

1997 (1)

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

1994 (1)

H. Wang, “Determination of refractive indices and thickness of absorbing crystalline thin films by using prism coupler,” Fiber Integr. Opt. 13, 293–308 (1994).
[CrossRef]

1975 (2)

R. Th. Kersten, “A new method for measuring refractive index and thickness of liquid and deposited solid thin films,” Opt. Commun. 13, 327–329 (1975).
[CrossRef]

R. Th. Kersten, “Prism–film coupler as a precision instrument,” Opt. Acta 22, 503–513 (1975).

1973 (3)

D. G. Dalgoutte, “A high efficiency thin grating coupler for integrated optics,” Opt. Commun. 8, 124–127 (1973).
[CrossRef]

R. Ulrich, R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12, 2901–2908 (1973).
[CrossRef] [PubMed]

R. Th. Kersten, “Numerical solution of the mode-equation of planar dielectric waveguides to determine their refractive index and thickness by means of a prism–film coupler,” Opt. Commun. 9, 427–431 (1973).
[CrossRef]

1972 (1)

1971 (1)

1970 (1)

1969 (1)

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Dalgoutte, D. G.

D. G. Dalgoutte, “A high efficiency thin grating coupler for integrated optics,” Opt. Commun. 8, 124–127 (1973).
[CrossRef]

Guo, S. Y.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Academic, New York, 1955).

Kersten, R. Th.

R. Th. Kersten, “A new method for measuring refractive index and thickness of liquid and deposited solid thin films,” Opt. Commun. 13, 327–329 (1975).
[CrossRef]

R. Th. Kersten, “Prism–film coupler as a precision instrument,” Opt. Acta 22, 503–513 (1975).

R. Th. Kersten, “Numerical solution of the mode-equation of planar dielectric waveguides to determine their refractive index and thickness by means of a prism–film coupler,” Opt. Commun. 9, 427–431 (1973).
[CrossRef]

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Liu, T.

T. Liu, R. J. Samuels, “Improved refractive index from a planar leaky wave-guide coupler,” in Thin Films for Optical Devices and Materials for Optical Limiting, Vol. 597 of MRS Symposium Proceedings, K. Nashimoto, R. Pachter, B. W. Wessels, J. Shmulovich, A. K.-Y. Jen, K. Lewis, R. Sutherland, J. W. Perry, eds. (Materials Research Society, Boston, Mass., 2000), pp. 57–62.

T. Liu, “Novel methods for determining the optical constants of anisotropic polymer films—new application of prism wave-guide coupling,” Ph.D. thesis (Georgia Institute of Technology, Atlanta, Ga., 2002).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, Mass., 1991).

Martin, R. J.

P. K. Tien, G. Smolinsky, R. J. Martin, “Thin organosilicon films for integrated optics,” Appl. Opt. 11, 637–642 (1972).
[CrossRef] [PubMed]

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

Mu, X. D.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Ren, Q.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Samuels, R. J.

T. Liu, R. J. Samuels, “Improved refractive index from a planar leaky wave-guide coupler,” in Thin Films for Optical Devices and Materials for Optical Limiting, Vol. 597 of MRS Symposium Proceedings, K. Nashimoto, R. Pachter, B. W. Wessels, J. Shmulovich, A. K.-Y. Jen, K. Lewis, R. Sutherland, J. W. Perry, eds. (Materials Research Society, Boston, Mass., 2000), pp. 57–62.

Smolinsky, G.

Tien, P. K.

Torge, R.

Trivedi, S. B.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Ulrich, R.

Varsicek, A.

A. Varsicek, Optics of Thin Films, translated by H. Watney-Kaczer (Interscience, New York, 1960).

Wang, H.

H. Wang, “Determination of refractive indices and thickness of absorbing crystalline thin films by using prism coupler,” Fiber Integr. Opt. 13, 293–308 (1994).
[CrossRef]

Wang, Z. G.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Ward, L.

L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (Institute of Physics, Philadelphia, Pa., 1994).

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Xu, D.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Zhang, G. H.

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

P. K. Tien, R. Ulrich, R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14, 291–294 (1969).
[CrossRef]

Fiber Integr. Opt. (1)

H. Wang, “Determination of refractive indices and thickness of absorbing crystalline thin films by using prism coupler,” Fiber Integr. Opt. 13, 293–308 (1994).
[CrossRef]

J. Mater. Sci. Lett. (1)

Q. Ren, Z. G. Wang, S. Y. Guo, X. D. Mu, G. H. Zhang, D. Xu, S. B. Trivedi, “Measurement of PT-PEK-c-polymer film parameters using the quasi-waveguide m-line method,” J. Mater. Sci. Lett. 16, 1389–1391 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. Th. Kersten, “Prism–film coupler as a precision instrument,” Opt. Acta 22, 503–513 (1975).

Opt. Commun. (3)

D. G. Dalgoutte, “A high efficiency thin grating coupler for integrated optics,” Opt. Commun. 8, 124–127 (1973).
[CrossRef]

R. Th. Kersten, “A new method for measuring refractive index and thickness of liquid and deposited solid thin films,” Opt. Commun. 13, 327–329 (1975).
[CrossRef]

R. Th. Kersten, “Numerical solution of the mode-equation of planar dielectric waveguides to determine their refractive index and thickness by means of a prism–film coupler,” Opt. Commun. 9, 427–431 (1973).
[CrossRef]

Other (9)

Manufacturer’s Bulletin, Metricon PC-2010 Prism Coupler (Metricon Corporation, Pennington, N.J., 1992).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, Mass., 1991).

T. Liu, R. J. Samuels, “Improved refractive index from a planar leaky wave-guide coupler,” in Thin Films for Optical Devices and Materials for Optical Limiting, Vol. 597 of MRS Symposium Proceedings, K. Nashimoto, R. Pachter, B. W. Wessels, J. Shmulovich, A. K.-Y. Jen, K. Lewis, R. Sutherland, J. W. Perry, eds. (Materials Research Society, Boston, Mass., 2000), pp. 57–62.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

A. Varsicek, Optics of Thin Films, translated by H. Watney-Kaczer (Interscience, New York, 1960).

O. S. Heavens, Optical Properties of Thin Solid Films (Academic, New York, 1955).

L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (Institute of Physics, Philadelphia, Pa., 1994).

T. Liu, “Novel methods for determining the optical constants of anisotropic polymer films—new application of prism wave-guide coupling,” Ph.D. thesis (Georgia Institute of Technology, Atlanta, Ga., 2002).

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of (a) a prism waveguide coupler and (b) a reflectance pattern for determining the mode angles.

Fig. 2
Fig. 2

Influence of the thin-film extinction coefficient on the Nm values (TE modes) of a model prism waveguide coupler (d3=0 μm) predicted by thin-film optics. In this figure, one can see the inherent discrepancies between the Nm values predicted by thin-film optics (crosses) and those calculated by the conventional prism waveguide coupler mode equations (filled circles). Similar discrepancies also exist for the Nm values of the TM modes.

Fig. 3
Fig. 3

Schematic wave diagrams on the prism/air gap/film interface for a prism waveguide coupler: (a) three-wave model, (b) four-wave model.

Fig. 4
Fig. 4

Configuration of an anisotropic prism waveguide coupler system in reference coordinates XYZ. (a) TE-polarized incident light with the electric field E (⊗) perpendicular to the XY plane. Only the z-direction refractive index can be measured as represented by the shaded circles. (b) TM-polarized incident light with electric field E (→) parallel to the XY plane. Only the x- and y-direction refractive indices can be measured as represented by the shaded arrows.

Fig. 5
Fig. 5

Boundaries of the zero region of the SSD for a model prism waveguide coupler: (a) d3=0-μm TE mode, (b) d3=0-μm TM mode, (c) d3=0.04-μm TE mode, (d) d3=0.04-μm TM mode.

Fig. 6
Fig. 6

 E and H relationships on the film–air-gap–prism interface for a prism waveguide coupler. (a) TE incident light, (b) TM incident light.

Tables (5)

Tables Icon

Table 1 Parameters for the Model Prism Waveguide Coupler and the Nm Values Determined with Thin-Film Optics

Tables Icon

Table 2 Mode Order Calculated with the Conventional and Improved Prism Waveguide Coupler Mode Equations by Use of Nm Values Determined from Thin-Film Optics

Tables Icon

Table 3 Relationship between the True (n2, d2) and the (n2, d2) at Which 2n2/d22 Shows a Peak for a Model Prism Waveguide Coupler

Tables Icon

Table 4 Comparison of Three-Wave and Four-Wave Mode Equations on Determining the Refractive Index of a Thin Film by Use of the Prism Waveguide Coupler

Tables Icon

Table 5 Determination of Refractive Index of Thin Film n2

Equations (57)

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Γ=4πλ d2n22-Nm2,
Φ21(TE)=2 tan-1Nm2-n12n22-Nm21/2,
Φ21(TM)=2 tan-1n22n12Nm2-n12n22-Nm21/2,
r23(TE)=|r23(TE)|exp[-Φ23(TE)]=Nm2-n32n42-Nm2-Nm2-n32n22-Nm2-i tanh2πλ d3Nm2-n321+Nm2-n32n42-Nm2Nm2-n32n22-Nm2Nm2-n32n42-Nm2+Nm2-n32n22-Nm2-i tanh2πλ d3Nm2-n321-Nm2-n32n42-Nm2Nm2-n32n22-Nm2,
r23(TM)=|r23(TM)|exp[-Φ23(TM)]=n42Nm2-n32n32n42-Nm2-n22Nm2-n32n32n22-Nm2-i tanh2πλ d3Nm2-n321+n42Nm2-n32n32n42-Nm2n22Nm2-n32n32n22-Nm2n42Nm2-n32n32n42-Nm2-n22Nm2-n32n32n22-Nm2-i tanh2πλ d3Nm2-n321-n42Nm2-n32n32n42-Nm2n22Nm2-n32n32n22-Nm2.
Γ-Φ23-Φ21=2mπ,
Nm=n4sin θm.
r4321=r43+r32exp(2iγ3)+r21exp[2i(γ2+γ3)]+r43r32r21exp(2iγ2)1+r43r32exp(2iγ3)+r32r21exp(2iγ3)+r43r21exp[2i(γ2+γ3)],
γk=2πλ dknk2-N2,
N=n4sin θ.
rij(TE)=ni2-N2-nj2-N2ni2-N2+nj2-N2
rij(TM)=nj2ni2-N2-ni2nj2-N2nj2ni2-N2+ni2nj2-N2
R=Ir/Ii=|r4321|2.
m=Γ-Φ23-Φ212π.
r234=r23+r34exp(2iγ3)1+r23r34exp(2iγ3),
r234w=|r234w|exp(-iΦ234w)=1r21exp(iΓ).
r21(TE)=|r21(TE)|exp[-iΦ21(TE)]=n2z2-Nm(TE)2-n1z2-Nm(TE)2n2z2-Nm(TE)2+n1z2-Nm(TE)2,
Γ(TE)=4πλ d2n2z2-Nm(TE)2,
Nm(TE)=n4zsin θm
r21(TM)
=|r21(TM)|exp[-iΦ21(TM)]
=n1xn1yn2x2-Nm(TM)2-n2xn2yn1x2-Nm(TM)2n1xn1yn2x2-Nm(TM)2+n2xn2yn1x2-Nm(TM)2,
 
Γ(TM)
=4πλ d2n2yn2xn2x2-Nm(TM)2,
Nm(TM)
=n4xn4yn4x2+(n4y2-n4x2)sin2 θmsin θm
Γ-Φ234w-Φ21=2mπ.
SSD(n2, d2)=m=0m[Γ(n2, d2; λ, Nm)-Φ234w(n2, d2; λ, n1, Nm)-Φ21(n2; n1, Nm)-2mπ]2.
Eaz+Ebz=Eez+Efz,
Eez+Efz=Ecz+Edz,
Hay+Hby=Hey+Hfy,
Hey+Hfy=Hcy+Hdy.
Eay+Eby=Eey+Efy,
Eey+Efy=Ecy+Edy,
Haz+Hbz=Hez+Hfz,
Hez+Hfz=Hcz+Hdz.
Hz=-ε0μ0nz2-Nm2Ez,
Hx=ε0μ0 NmEz,
Nm=n4zsin θm
Ex=-μ0ε0Nmnx2 Hz,
Ey=μ0ε0nx2-Nm2nxny Hz,
Nm=n4xn4yn4x2+(n4y2-n4x2)sin2 θmsin θm
r23(TE)4w=|Ec||Ed|=EczEdz=(A*δ+B)1+EbzEaz+n4z2-Nm2n3z2-Nm2 (A*δ-B)1-EbzEaz(B*δ+A)1+EbzEaz+n4z2-Nm2n3z2-Nm2 (B*δ-A)1-EbzEaz,
A=1+n3z2-Nm2n2z2-Nm2,
B=1-n3z2-Nm2n2z2-Nm2,
δ=expi 2πλ 2d3n3z2-Nm2,
 r23(TM)4w=|Ec||Ed|=EcyEdy=(A*δ-B)1-EbyEay+n4xn4yn3xn3yn3x2-Nm2n4x2-Nm2 (A*δ+B)1+EbyEay(B*δ-A)1-EbyEay+n4xn4yn3xn3yn3x2-Nm2n4x2-Nm2 (B*δ+A)1+EbyEay,
A=1+n3xn3yn2x2-Nm2n2xn2yn3x2-Nm2,
B=-1+n3xn3yn2x2-Nm2n2xn2yn3x2-Nm2,
δ=expi 2πλ 2d3n3yn3xn3x2-Nm2.
r4321={r43+r32exp(2iγ3)+r21exp[2i(γ2+γ3)]+r43r32r21exp(2iγ2)}{1+r43r32exp(2iγ3)+r32r21exp(2iγ2)+r43r21exp[2i(γ2+γ3)]},
rij=niz2-Nm2-njz2-Nm2niz2-Nm2+njz2-Nm2,
γi=2πλ diniz2-Nm2,
rij=njxnjynix2-Nm2-nixniynjx2-Nm2njxnjynix2-Nm2+nixniynjx2-Nm2,
γi=2πλ diniynixnix2-Nm2,
r234w=1r21exp(i2γ2),

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