Abstract

What we believe to be a new type of resonant coupling of an incident bulk wave into guided modes of a slab with a thick holographic grating is shown to occur in the presence of strong frequency detunings of the Bragg condition. This happens through the reflection of the strongly noneigen +1 diffracted order with the slab–grating boundaries, the resultant reflected waves forming a guided slab mode. Rigorous coupled-wave analysis is used for the numerical analysis of the predicted resonant effects. Possible applications include enhanced options for the design of multiplexing and demultiplexing systems, optical signal-processing devices, optical sensors, and measurement techniques.

© 2006 Optical Society of America

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References

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  1. V. M. Aranovich and D. L. Mills, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).
  2. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
  3. M. C. Hutley, Diffraction Gratings (Academic, 1982).
  4. T. K. Gaylord and M. G. Moharam, "Analysis and application of optical diffraction by gratings," Proc. IEEE 73, 894-938 (1985).
    [CrossRef]
  5. D. K. Gramotnev, "Extremely asymmetrical scattering of slab modes in periodic Bragg arrays," Opt. Lett. 22, 1053-1055 (1997).
    [CrossRef] [PubMed]
  6. D. K. Gramotnev, "Grazing angle scattering of electromagnetic waves in periodic Bragg arrays," Opt. Quantum Electron. 33, 253-288 (2001).
    [CrossRef]
  7. D. K. Gramotnev and T. A. Nieminen, "Rigorous analysis of grazing-angle scattering of electromagnetic waves in periodic gratings," Opt. Commun. 219, 33-48 (2003).
    [CrossRef]
  8. D. F. P. Pile and D. K. Gramotnev, "Second-order grazing-angle scattering in uniform wide holographic gratings," Appl. Phys. B 76, 65-73 (2003).
    [CrossRef]
  9. D. K. Gramotnev and D. F. P. Pile, "Frequency response of second-order extremely asymmetrical scattering in wide uniform holographic gratings," Appl. Phys. B 77, 663-671 (2003).
    [CrossRef]
  10. D. K. Gramotnev, S. J. Goodman, and T. A. Nieminen, "Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes," J. Mod. Opt. 51, 379-397 (2004).
    [CrossRef]
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  12. D. K. Gramotnev, "Frequency response of extremely asymmetrical scattering of electromagnetic waves in periodic gratings," in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), pp. 165-167.
  13. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmittance-matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995).
    [CrossRef]
  14. K. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell. Syst. Tech. J. 48, 2909-2947 (1969).
  15. I. Avrutsky, Department of Electrical and Computer Engineering, Wayne State University Detroit, Mich. (personal communication, 2004).
  16. D. K. Gramotnev, T. A. Nieminen, and T. A. Hopper, "Extremely asymmetrical scattering in gratings with varying mean structural parameters," J. Mod. Opt. 49, 1567-1585 (2002).
    [CrossRef]

2004 (1)

D. K. Gramotnev, S. J. Goodman, and T. A. Nieminen, "Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes," J. Mod. Opt. 51, 379-397 (2004).
[CrossRef]

2003 (3)

D. K. Gramotnev and T. A. Nieminen, "Rigorous analysis of grazing-angle scattering of electromagnetic waves in periodic gratings," Opt. Commun. 219, 33-48 (2003).
[CrossRef]

D. F. P. Pile and D. K. Gramotnev, "Second-order grazing-angle scattering in uniform wide holographic gratings," Appl. Phys. B 76, 65-73 (2003).
[CrossRef]

D. K. Gramotnev and D. F. P. Pile, "Frequency response of second-order extremely asymmetrical scattering in wide uniform holographic gratings," Appl. Phys. B 77, 663-671 (2003).
[CrossRef]

2002 (1)

D. K. Gramotnev, T. A. Nieminen, and T. A. Hopper, "Extremely asymmetrical scattering in gratings with varying mean structural parameters," J. Mod. Opt. 49, 1567-1585 (2002).
[CrossRef]

2001 (1)

D. K. Gramotnev, "Grazing angle scattering of electromagnetic waves in periodic Bragg arrays," Opt. Quantum Electron. 33, 253-288 (2001).
[CrossRef]

1997 (1)

1995 (1)

1985 (1)

T. K. Gaylord and M. G. Moharam, "Analysis and application of optical diffraction by gratings," Proc. IEEE 73, 894-938 (1985).
[CrossRef]

1969 (1)

K. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell. Syst. Tech. J. 48, 2909-2947 (1969).

Aranovich, V. M.

V. M. Aranovich and D. L. Mills, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).

Avrutsky, I.

I. Avrutsky, Department of Electrical and Computer Engineering, Wayne State University Detroit, Mich. (personal communication, 2004).

Gaylord, T. K.

Goodman, S. J.

D. K. Gramotnev, S. J. Goodman, and T. A. Nieminen, "Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes," J. Mod. Opt. 51, 379-397 (2004).
[CrossRef]

Gramotnev, D. K.

D. K. Gramotnev, S. J. Goodman, and T. A. Nieminen, "Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes," J. Mod. Opt. 51, 379-397 (2004).
[CrossRef]

D. F. P. Pile and D. K. Gramotnev, "Second-order grazing-angle scattering in uniform wide holographic gratings," Appl. Phys. B 76, 65-73 (2003).
[CrossRef]

D. K. Gramotnev and D. F. P. Pile, "Frequency response of second-order extremely asymmetrical scattering in wide uniform holographic gratings," Appl. Phys. B 77, 663-671 (2003).
[CrossRef]

D. K. Gramotnev and T. A. Nieminen, "Rigorous analysis of grazing-angle scattering of electromagnetic waves in periodic gratings," Opt. Commun. 219, 33-48 (2003).
[CrossRef]

D. K. Gramotnev, T. A. Nieminen, and T. A. Hopper, "Extremely asymmetrical scattering in gratings with varying mean structural parameters," J. Mod. Opt. 49, 1567-1585 (2002).
[CrossRef]

D. K. Gramotnev, "Grazing angle scattering of electromagnetic waves in periodic Bragg arrays," Opt. Quantum Electron. 33, 253-288 (2001).
[CrossRef]

D. K. Gramotnev, "Extremely asymmetrical scattering of slab modes in periodic Bragg arrays," Opt. Lett. 22, 1053-1055 (1997).
[CrossRef] [PubMed]

D. K. Gramotnev, "Frequency response of extremely asymmetrical scattering of electromagnetic waves in periodic gratings," in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), pp. 165-167.

Grann, E. B.

Hopper, T. A.

D. K. Gramotnev, T. A. Nieminen, and T. A. Hopper, "Extremely asymmetrical scattering in gratings with varying mean structural parameters," J. Mod. Opt. 49, 1567-1585 (2002).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, 1982).

Kogelnik, K.

K. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell. Syst. Tech. J. 48, 2909-2947 (1969).

Mills, D. L.

V. M. Aranovich and D. L. Mills, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).

Moharam, M. G.

Nieminen, T. A.

D. K. Gramotnev, S. J. Goodman, and T. A. Nieminen, "Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes," J. Mod. Opt. 51, 379-397 (2004).
[CrossRef]

D. K. Gramotnev and T. A. Nieminen, "Rigorous analysis of grazing-angle scattering of electromagnetic waves in periodic gratings," Opt. Commun. 219, 33-48 (2003).
[CrossRef]

D. K. Gramotnev, T. A. Nieminen, and T. A. Hopper, "Extremely asymmetrical scattering in gratings with varying mean structural parameters," J. Mod. Opt. 49, 1567-1585 (2002).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

Pile, D. F. P.

D. F. P. Pile and D. K. Gramotnev, "Second-order grazing-angle scattering in uniform wide holographic gratings," Appl. Phys. B 76, 65-73 (2003).
[CrossRef]

D. K. Gramotnev and D. F. P. Pile, "Frequency response of second-order extremely asymmetrical scattering in wide uniform holographic gratings," Appl. Phys. B 77, 663-671 (2003).
[CrossRef]

Pommet, D. A.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Appl. Phys. B (2)

D. F. P. Pile and D. K. Gramotnev, "Second-order grazing-angle scattering in uniform wide holographic gratings," Appl. Phys. B 76, 65-73 (2003).
[CrossRef]

D. K. Gramotnev and D. F. P. Pile, "Frequency response of second-order extremely asymmetrical scattering in wide uniform holographic gratings," Appl. Phys. B 77, 663-671 (2003).
[CrossRef]

Bell. Syst. Tech. J. (1)

K. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell. Syst. Tech. J. 48, 2909-2947 (1969).

J. Mod. Opt. (2)

D. K. Gramotnev, T. A. Nieminen, and T. A. Hopper, "Extremely asymmetrical scattering in gratings with varying mean structural parameters," J. Mod. Opt. 49, 1567-1585 (2002).
[CrossRef]

D. K. Gramotnev, S. J. Goodman, and T. A. Nieminen, "Grazing-angle scattering of electromagnetic waves in gratings with varying mean parameters: grating eigenmodes," J. Mod. Opt. 51, 379-397 (2004).
[CrossRef]

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

Opt. Commun. (1)

D. K. Gramotnev and T. A. Nieminen, "Rigorous analysis of grazing-angle scattering of electromagnetic waves in periodic gratings," Opt. Commun. 219, 33-48 (2003).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

D. K. Gramotnev, "Grazing angle scattering of electromagnetic waves in periodic Bragg arrays," Opt. Quantum Electron. 33, 253-288 (2001).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, "Analysis and application of optical diffraction by gratings," Proc. IEEE 73, 894-938 (1985).
[CrossRef]

Other (6)

I. Avrutsky, Department of Electrical and Computer Engineering, Wayne State University Detroit, Mich. (personal communication, 2004).

V. M. Aranovich and D. L. Mills, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

M. C. Hutley, Diffraction Gratings (Academic, 1982).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

D. K. Gramotnev, "Frequency response of extremely asymmetrical scattering of electromagnetic waves in periodic gratings," in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), pp. 165-167.

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Figures (6)

Fig. 1
Fig. 1

Thick holographic grating with the period Λ, amplitude ε g , and reciprocal lattice vector q (q = 2π∕Λ) in a guiding dielectric slab of width L. All the materials are assumed to be linear, isotropic, and lossless; the dielectric permittivities in front and behind the guiding slab are ε1 and ε3, respectively. The mean permittivity in the slab is ε2. A bulk TE electromagnetic plane wave with the wave vector k 10 and the amplitude E 10 is incident onto the slab at the angle θ10 with respect to the x axis. The axes of coordinates are also presented.

Fig. 2
Fig. 2

Dependencies of the relative amplitudes of the +1 diffracted order |E 21E 10| on angle of scattering θ21 (i.e., angle between the vector k 21 and the x axis in the grating): (a) at the front and rear grating boundaries (i.e., at x = 0; L) and (b) in the middle of the grating (i.e., at x = L∕2). The slanting angle for grating fringes and the period Λ are adjusted for each angle of scattering θ21 so that the +1 diffracted order satisfies the Bragg condition precisely (i.e., k 21 = ε2 1/2ω∕c). The structural and wave parameters are L = 10 μm, λ0 (vacuum) = 1 μm, ε2 = 5, ε1 = ε3 = 4.8492 (the critical angle in the slab is 80°), ε g = 2 × 10−3, and the angle of incidence θ10 = 45°.

Fig. 3
Fig. 3

Dependencies of the relative amplitudes of the +1 diffracted order |E 21E 10| on frequency detuning Δω: (a) at the front and rear grating boundaries (i.e., at x = 0; L) and (b) in the middle of the grating (i.e., at x = L∕2). The structural parameters are the same as in Fig. 2. However, this time we fix the grating orientation and period so that the +1 diffracted order satisfies the Bragg condition at θ21 = 70° and ω = ω0 ∼ 1.88 × 1015 rad∕s (λ0 = 1 μm).

Fig. 4
Fig. 4

Scheme of coupling between the bulk waves and slab modes by means of the interaction of the noneigen +1 diffracted order with the grating boundaries: k 200 and k 210 are the wave vectors of the 0th and +1 orders in the slab when the Bragg condition is satisfied (Δω = 0 and ω = ω0); k 20 and k 21 are the wave vectors of the 0th and +1 orders at nonzero frequency detuning Δω < 0; k 21r and k 21 r are the wave vectors of the waves resulting from the reflection of the noneigen +1 diffracted order (with the amplitude E 21s and the wave vector k 21) from the grating boundaries: k 21ry = k 21 ry = k 21y (Snell's law) and k 21r = k 21 r = (ω0 + Δω)(ε2)1∕2c (i.e., the reflected waves are eigen for the slab). When the angle of propagation of these reflected waves θ r corresponds to one of the guided slab modes, resonant generation of this mode occurs with a strong resonant increase of the amplitude of the +1 diffracted order in the grating (Figs. 3, 5, and 6). This is because the reflected waves are formally included in the +1 diffracted order [see Eqs. (2) and (5)].

Fig. 5
Fig. 5

Dependencies of the relative amplitudes of the +1 diffracted order |E 21E 10| on frequency detuning Δω: (a) at the front and rear grating boundaries (i.e., at x = 0; L) and (b) in the middle of the grating (i.e., at x = L∕2). The structural and wave parameters are the same as in Fig. 3, except for the increased grating amplitude: ε g = 0.08.

Fig. 6
Fig. 6

Dependencies of x on the relative amplitudes of the +1 diffracted order |E 21E 10| inside the grating and slab for the three leftmost resonances in Fig. 5(a): (a) ε g = 0.08, (b) ε g = 0.1. Curves 1, 2, and 3 in (a) correspond to the leftmost, second leftmost, and third leftmost resonances in Fig. 5(a). Resonant detunings are ΔωI ≈ −3.71 × 1014 rad∕s, Δω2 ≈ −3.67 × 1014 rad∕s, Δω3 ≈ −3.59 × 1014 rad∕s for curves 1–3, respectively, in both (a) and (b).

Equations (14)

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ε ( x , y )
= { ε 1   for   x < 0 ε 2 + ε g exp ( i q r ) + ε g * exp ( i q r )      for   0 < x < L , ε 3   for   x > L
E ( x , y ) = n = + E 2 n ( x ) exp ( i k 2 n x x + i k 2 n y y i ω t ) ,
k 2 n = k 20 n q ,
k 21 r x = ε 2 ω 2 / c 2 k 21 y 2 k 21 x .
E 21 ( x ) exp ( i k 21 r i ω t ) = [ E 21 s ( x ) + E 21 r exp ( i k 21 r x i k 21 x ) + E 21 r exp ( i k 21 r x i k 21 x ) ] exp ( i k 21 r i ω t ) ,
k 21         2 = k 210             2 + Δ k 20         2 2 k 210 Δ k 20 cos ( 2 α ) ,
Δ k 20 = k 200 k 20 = Δ ω c ε 2 .
2 α = θ 210 θ 200 .
q 2 = k 21         2 + k 20         2 2 k 21 k 20 cos ( β ) .
θ 21 = θ 200  +  β .
k 21 y = k 21 sin θ 21 = k 21 r y ,
k 21 r x = ε 2 ω 2 / c 2 k 21         2 sin 2 θ 21 .
θ r = arctan ( k 21 r y k 21 r x ) .

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