Abstract

An approximate closed-form expression for the loss in a planar phase grating is derived by using coupled-mode theory. It is shown that this loss expression can be used to determine the spectral and angular width of a resonant-grating filter. A resonant-grating filter is a free-space optic that takes advantage of grating resonances to create narrow-band reflection peaks. Design characteristics, such as bandwidth, have previously been determined by profiling the resonance in reflectivity with the use of numerically intensive vector-diffraction methods such as rigorous coupled-wave analysis. The coupled-mode approach described here, however, gives the resonant-filter width directly, without the need to profile the resonance. Therefore computation time and hence design time are reduced. In addition, it is shown that the coupled-mode approach provides physical insights into the factors contributing to filter bandwidth.

© 1997 Optical Society of America

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References

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  1. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
    [CrossRef]
  2. T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  3. D. Marcuse, “Mode conversion caused by diameter changes of a round dielectric waveguide,” Bell Syst. Tech. J. 48, 3217–3232 (1969).
    [CrossRef]
  4. D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187–3215 (1969).
    [CrossRef]
  5. D. G. Hall, “Comparison of two approaches to the waveguide scattering problem,” Appl. Opt. 19, 1732–1734 (1980).
    [CrossRef] [PubMed]
  6. H. Nishihara, M. Haruna, T. Suhara, “Theory of gratings in waveguide structures,” in Optical Integrated Circuits, R. E. Fischer, W. J. Smith, eds., McGraw-Hill Optical and Electro-optical Engineering Series (McGraw-Hill, New York, 1989), Chap. 4, pp. 62–95.
  7. W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
    [CrossRef]
  8. H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
    [CrossRef]
  9. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  10. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
    [CrossRef]
  11. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  12. T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 38, 271–299 (1973).
  13. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  14. J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. (Bellingham) 34, 2560–2566 (1995).
    [CrossRef]
  15. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode filters,” Appl. Opt. 23, 2606–2613 (1993).
    [CrossRef]
  16. M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
    [CrossRef]
  17. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  18. R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  19. M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  20. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), Vol. 7, pp. 13–81.
  21. M. Born, E. Wolf, eds., Principles of Optics (Pergamon, Oxford, 1980), p. 453.
  22. R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium on Optical and Acoustical Micro-Electronics (Polytechnic, Brooklyn, 1975), Vol. 23, pp. 359–376.
  23. S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
    [CrossRef] [PubMed]
  24. P. Vincent, M. Neviere, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
    [CrossRef]

1996 (1)

1995 (1)

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. (Bellingham) 34, 2560–2566 (1995).
[CrossRef]

1994 (1)

1993 (2)

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode filters,” Appl. Opt. 23, 2606–2613 (1993).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

1992 (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1989 (1)

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1981 (1)

1980 (1)

1979 (1)

P. Vincent, M. Neviere, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

1977 (1)

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

1976 (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

1973 (2)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 38, 271–299 (1973).

1969 (2)

D. Marcuse, “Mode conversion caused by diameter changes of a round dielectric waveguide,” Bell Syst. Tech. J. 48, 3217–3232 (1969).
[CrossRef]

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187–3215 (1969).
[CrossRef]

1965 (1)

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Bertoni, H. L.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

Cadilhac, M.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Cheo, L. S.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Chu, R. S.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

Erdogan, T.

Gale, M. T.

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Gaylord, T. K.

Hall, D. G.

Haruna, M.

H. Nishihara, M. Haruna, T. Suhara, “Theory of gratings in waveguide structures,” in Optical Integrated Circuits, R. E. Fischer, W. J. Smith, eds., McGraw-Hill Optical and Electro-optical Engineering Series (McGraw-Hill, New York, 1989), Chap. 4, pp. 62–95.

Hessel, A.

Knop, K.

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), Vol. 7, pp. 13–81.

Kong, J. A.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

Magnusson, R.

S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
[CrossRef] [PubMed]

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode filters,” Appl. Opt. 23, 2606–2613 (1993).
[CrossRef]

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Marcuse, D.

D. Marcuse, “Mode conversion caused by diameter changes of a round dielectric waveguide,” Bell Syst. Tech. J. 48, 3217–3232 (1969).
[CrossRef]

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187–3215 (1969).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Mizrahi, V.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

Moharam, M. G.

Morf, R. H.

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Neviere, M.

P. Vincent, M. Neviere, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

Nevière, M.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Nishihara, H.

H. Nishihara, M. Haruna, T. Suhara, “Theory of gratings in waveguide structures,” in Optical Integrated Circuits, R. E. Fischer, W. J. Smith, eds., McGraw-Hill Optical and Electro-optical Engineering Series (McGraw-Hill, New York, 1989), Chap. 4, pp. 62–95.

Noponen, E.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. (Bellingham) 34, 2560–2566 (1995).
[CrossRef]

Oliner, A. A.

Petit, R.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Saarinen, J.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. (Bellingham) 34, 2560–2566 (1995).
[CrossRef]

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

Sipe, J. E.

T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

Suhara, T.

H. Nishihara, M. Haruna, T. Suhara, “Theory of gratings in waveguide structures,” in Optical Integrated Circuits, R. E. Fischer, W. J. Smith, eds., McGraw-Hill Optical and Electro-optical Engineering Series (McGraw-Hill, New York, 1989), Chap. 4, pp. 62–95.

Tamir, T.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 38, 271–299 (1973).

Turunen, J.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. (Bellingham) 34, 2560–2566 (1995).
[CrossRef]

Ulrich, R.

R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium on Optical and Acoustical Micro-Electronics (Polytechnic, Brooklyn, 1975), Vol. 23, pp. 359–376.

Vincent, P.

P. Vincent, M. Neviere, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

Wang, S. S.

S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
[CrossRef] [PubMed]

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode filters,” Appl. Opt. 23, 2606–2613 (1993).
[CrossRef]

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

P. Vincent, M. Neviere, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Bell Syst. Tech. J. (2)

D. Marcuse, “Mode conversion caused by diameter changes of a round dielectric waveguide,” Bell Syst. Tech. J. 48, 3217–3232 (1969).
[CrossRef]

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187–3215 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

J. Lightwave Technol. (1)

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (1)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Opt. Eng. (Bellingham) (1)

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. (Bellingham) 34, 2560–2566 (1995).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 38, 271–299 (1973).

Philos. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Other (5)

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

H. Nishihara, M. Haruna, T. Suhara, “Theory of gratings in waveguide structures,” in Optical Integrated Circuits, R. E. Fischer, W. J. Smith, eds., McGraw-Hill Optical and Electro-optical Engineering Series (McGraw-Hill, New York, 1989), Chap. 4, pp. 62–95.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), Vol. 7, pp. 13–81.

M. Born, E. Wolf, eds., Principles of Optics (Pergamon, Oxford, 1980), p. 453.

R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium on Optical and Acoustical Micro-Electronics (Polytechnic, Brooklyn, 1975), Vol. 23, pp. 359–376.

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Figures (11)

Fig. 1
Fig. 1

Example of a transverse-electric (TE) resonance reflection peak. Resonances are characterized by a narrow (in angle or wavelength) reflection peak, which, when designed to support only zero-order diffraction, can theoretically reach 100%. For this example the following structural parameters were used: nc=1.0, ns=1.45, nav=1.6, δnf=0.08, λ=0.6 µm, Λ =0.38 µm, and h=0.17 µm (see Fig. 2).

Fig. 2
Fig. 2

Structural model used for coupled-mode analysis of coupling loss. A fundamental mode with amplitude A0 is incident on a periodic refractive index region nf(z) from a uniform region with refractive index nav. As the mode enters the periodic region, it is scattered into the radiation modes designated by Bp(ρ).

Fig. 3
Fig. 3

Illustration of grating coupling from the fundamental mode with z-propagation coefficient β0 to a radiation mode with propagation coefficient between -(2π/λ)ns and +(2π/λ)ns. Note that |(2π/λ)ns|<β0<|(2π/λ)nf|, where ns>nc.

Fig. 4
Fig. 4

Typical configuration of a resonant-grating filter being excited at resonance by an incident plane wave with tangential wave number given by kz. Through grating coupling the incident field excites a guided mode that rescatters above and below the guide, constructively interfering with the direct field reflection and destructively interfering with the direct field transmission.

Fig. 5
Fig. 5

Illustration showing how the grating creates an infinite set of complex poles. If the period is chosen properly [such as to match the coupling condition, Eq. (43)], a spatial order of the fundamental mode can fall within a region excitable by a plane wave with tangential wave number, kz.

Fig. 6
Fig. 6

The resonance reflection over angle (in wave-number units) can be characterized by a Lorentzian. With this model the full width at half-maximum (FWHM) in kz units is given by 2γ.

Fig. 7
Fig. 7

(a) Plot comparing resonant width in degrees from coupled-mode (CM) theory and the Lorentzian model with the actual FWHM derived from RCWA. The plot shows excellent agreement over a broad depth range. Some deviation at h =0.2 µm is found to be the result of an asymmetrical resonance peak as shown in (b). Other depth points, such as h=0.3 µm, that show an excellent fit are characterized by reflection peaks with significant Lorentzian character as shown in (c). Note that nc=1.0, ns=1.45, nav=1.6, λ=0.6 µm, ΛG=0.38 µm, and δnf=0.08.

Fig. 8
Fig. 8

Plot comparing resonant spectral width in micrometers versus film layer depth h, using both RCWA and the CM approach. It is possible to approximate the spectral width from the coupling-loss value γ by modification of the original Lorentzian expression, Eq. (47), to Eq. (53) and by use of wavelengths, λPEAK, that satisfy the coupling relation, Eq. (52). Note that nc=1.0, ns=1.45, nav=1.6, θ=10°, ΛG=0.38 µm, and δnf=0.08.

Fig. 9
Fig. 9

Coupling loss is given by the overlap of the bound mode and the radiation mode within the film layer. As the depth h increases from zero, the bound mode eventually emerges from cutoff and, initially, is not well confined to the film layer, thus giving a low coupling loss. As the depth continues to increase, nulls of the radiation field begin to overlap with the bound mode, creating nulls in the coupling loss. Finally, at higher depths, the orthogonality of the two modes becomes more pronounced, thereby decreasing the overall magnitude of the coupling loss.

Fig. 10
Fig. 10

Illustration of coupling at normal incidence. Two modes are interacting by both being coupled through the grating to the normally propagating radiation mode.

Fig. 11
Fig. 11

Plot of angular width versus depth showing a discrepancy in the coupling equations versus the FWHM data from RCWA at a film layer depth of h=0.3 µm. This depth corresponds to coupling at normal incidence to the guide (x direction). Near normal incidence, two bound modes begin to interact. The interaction was not taken into account in the coupling-loss equations derived in Section 2, and therefore the equations should be used with caution when coupling near normal incidence.

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

nf(x, z)=nav+δnf cos2πΛGzz0, 0xh,
Δε(x, z)=ε0[n2(x, z)actual-n2(x, z)ideal]2ε0n(x, z)ideal[n(x, z)actual-n(x, z)ideal],
Δε(x, z)2ε0navδnf cos2πΛGz0xh0,x<0, x>h.
Et=ν[Aν exp(iβνz)+Bν exp(-iβνz)]Eν(x)exp(-iωt)+pdρ[Ap(ρ)exp(iβpρz)+Bp(ρ)exp(-iβpρz)]Epρ(x)exp(-iωt),
Ht=ν[Aν exp(iβνz)-Bν exp(-iβνz)]Hν(x)exp(-iωt)+pdρ[Ap(ρ)exp(iβpρz)-Bp(ρ)exp(-iβpρz)]Hpρ(x)exp(-iωt).
ρ=ksρ=ns2k02-βpρ2,
dAμdz=iν{Aν(z)(Kνμt+Kνμz)exp[i(βν-βμ)z]+Bν(z)(Kνμt-Kνμz)exp[-(βν+βμ)z]}+ipdρ{Ap(ρ: z)(Kpρ.μt+Kpρ:μz)×exp[i(βpρ-βμ)z]+Bp(ρ: z)×(Kpρ.μt-Kpρ:μz)exp[-i(βpρ+βμ)z]},
dBμdz=-iν{Aν(z)(Kνμt-Kνμz)exp[i(βν+βμ)z]+Bν(z)(Kνμt+Kνμz)exp[-i(βν-βμ)z]}-ipdρ{Ap(ρ: z)(Kpρ:μt-Kpρ:μz)×exp[i(βpρ+βμ)z]+Bp(ρ: z)×(Kpρ:μt+Kpρ:μz)exp[-i(βpρ-βμ)z]},
dAq(ρ)dz=iν{Aν(z)(Kν:qρt+Kν:qρz)×exp[i(βν-βqρ)z]+Bν(z)×(Kν:qρt-Kν:qρz)exp[-i(βν+βqρ)z]}+ipdρ{Ap(ρ: z)(Kpρ:qρt+Kpρ:qρz)×exp[i(βpρ-βqρ)z]+Bp(ρ: z)×(Kpρ:qρt-Kpρ:qρz)exp[-i(βpρ+βqρ)z]},
dBq(ρ)dz=-iν{Aν(z)(Kν:qρt-Kν:qρz)×exp[i(βν+βqρ)z]+Bν(z)×(Kν:qρt+Kν:qρz)exp[-i(βν-βqρ)z]}-ipdρ{Ap(ρ: z)(Kpρ:qρt-Kpρ:qρz)×exp[i(βpρ+βqρ)z]+Bp(ρ: z)×(Kpρ:qρt+Kpρ:qρz)exp[-i(βpρ-βqρ)z]},
Kνμt=ω4-Δε(x, z)Etν·Etμ* dx,
Kν:qρt=ω4-Δε(x, z)Etν·Et,qρ* dx,
Kνμz=ω4- εΔεε+ΔεEzνEzμ* dx,
Kν:qρz=ω4- εΔεε+ΔεEzνEz,qρ* dx.
Kν:qρt=ω4- Δε(x, z)EyνEy,qρ* dx,
Eyν(c)=Eν(c) exp[-γc(x-h)]x>h,
Eyν(f)=Eν(f) cos(κf x-ϕs)0<x<h,
Eyν(s)=Eν(s) exp(γcx)x<0,
  Odd modes (p odd) Even modes (p even)
Ey,pρ(c)=Epρ(c) sin[κc(x-h)+ϕc]Ey,pρ(c)=Epρ(c) cos[κc(x-h)+ϕc](x>h),
Ey,pρ(f)=Epρ(f) sin(κf x-ϕ)Ey,pρ(f)=Epρ(f) cos(κf x-ϕ)(x<0<h),
Ey,pρ(s)=Epρ(s) sin(κsx-ϕs)Ey,pρ(s)=Epρ(s) cos(κsx-ϕs)(x<0), 
κi=iγi=ni2k02-β2,ϕi=tan-1γiκf
i=c, f, or s,
ϕ=12tan-1sin(2κfh)cos(2κfh)+κs(1-κf2/κs2)κc(1-κf2/κc2.
Kν:qρt=κν:qρexpi 2πΛGz+exp-i 2πΛGz,
κν:qρε0ωnav4δnf 0h EyνEy,qρ* dx.
dA0dz=i p  κpρ:0 exp(-i2δz)Bp(ρ: z)dρ,
dBp(ρ)dz=-iκ0:pρ exp(i2δz)A0(z)p=even, odd,
2δβ0+βpρ-2π/ΛG
dA0dz-i2A0(z) p  |κ0:pρ|2δdρ,
nsk00 dρ |κ0:pρ|2δ=2 0nsk0 βpρns2k02-βpρ2×|κ0:pρ|2βpρ+β0-2π/ΛGdβpρ.
nsk00 dρ |κ0:pρ|2δ
=2P0nsk0 βpρns2k02-βpρ2×|κ0:pρ|2βpρ+β0-2π/ΛGdβpρ-2πiβpρns2k02-βpρ2κ0:pρ|2βpρ=2π/ΛG-β0,
dA0/dz=iΔβA0-γA0,
Δβ-Pp 0nsk0 βpρns2k02-βpρ2×|κ0:pρ|2βpρ+β0-2π/ΛGdβpρ,
γπ pβpρns2k02-βpρ2|κ0,pρ|2βpρ=2π/ΛG-β0 .
A0(z)=A0(0)exp(-γz)exp(iΔβz).
κ0:pρ=k02navδnfπheff 0β0βρ1kf02-kfρ2A/CB/Dp evenp odd,
A=[kf0 sin(kf0h)cos(kfρh)-kfρ cos(kf0h)sin(kfρh)]×cos ϕ0 cos ϕρ+[kfρ sin(kf0h)cos(kfρh)-kf0 cos(kf0h)sin(kfρh)]×sin ϕ0 sin ϕρ+[kfρ cos(kf0h)cos(kfρh)+kf0 sin(kf0h)sin(kfρh)-kfρ]cos ϕ0 sin ϕρ-[kf0 cos(kf0h)cos(kfρh)+kfρ sin(kf0h)×sin(kfρh)-kf0]sin ϕ0 cos ϕρ,
B=[kfρ cos(kf0h)cos(kfρh)+kf0 sin(kf0h)sin(kfρh)-kfρ]cos ϕ0 cos ϕρ+[kf0 cos(kf0h)cos(kfρh)+kfρ sin(kf0h)sin(kfρh)-kf0]×sin ϕ0 sin ϕρ-[kf0 sin(kf0h)cos(kfρh)+kfρ cos(kf0h)sin(kfρh)]×cos ϕ0 sin ϕρ+[kfρ sin(kf0h)cos(kfρh)-kf0 cos(kf0h)sin(kfρh)]sin ϕ0 cos ϕρ,
C=cos2 ϕρ+kcρksρcos2(kfρh-ϕρ)+kfρ2ksρ2sin2 ϕρ+kcρksρkfρ2kcρ2sin2(kfρh-ϕρ)1/2,
D=sin2 ϕρ+kcρksρsin2(kfρh-ϕρ)+kfρ2ksρ2cos2 ϕρ+kcρksρkfρ2kcρ2cos2(kfρh-ϕρ)1/2,
ki0=ni2k02-β02,kiρ=ni2k02-βρ2
i=c, f, or s.
γ=πβρksρ[κ0:(even)ρ2+κ0:(odd)ρ2]|βρ=2π/Λ-β0.
γ=k04nav2δnf2heffβ0ksρ(kf02-kfρ2)2A2C2+B2D2.
βpρ=2πΛG-β0=2πΛG-2πλNr,
kz=β0+n 2πΛG,n integer,
β˜=βpρ+iγ,
r=|c|kz-β˜,
r=|c|kz-(2π/ΛG-β0)-iγ.
R=|r|2=|c|2[kz-(2π/ΛG-β0)]2+γ2.
Δθ=180π λ2π cos θpeakΔkz=180π λγπ cos θpeak,
θpeak=sin-1(Nr-λ/ΛG).
nav=nH+nL2,
δnf=2π(nH-nL).
ε=|sin θ-[Nr(λ)-λ/ΛG]|.
R=|c|2(ΛGλpeak/2π)2[λ-ΛG(sin θ+Nr)]2+ΛG2Ni2,
Δλ=λpeakΛGπγλpeak.
limα -αα x Etν·Et,qρ*0.
β=1/2(2π/ΛG)
β=(2π/ΛG),

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