Abstract

Rigorous electromagnetic theory is combined with a phenomenological approach to permit optimization of grating-enhanced second-harmonic generation (SHG) in optical waveguides. Provided that the absorption losses in the optically nonlinear layer are not high, maximum SHG is observed when phase matching occurs between the incident wave at the pump frequency and guided waves at both the pump and the signal frequencies. Different coupling mechanisms are considered, and a procedure for determining the optimal groove depth and period of the grating is discussed. The phenomenological approach permits deeper physical insight into the problem and a considerable saving of computation time. Direct phase matching is shown to result in stronger SHG than indirect phase matching (performed through the grating vector), even if the former includes coupling between waveguide modes of different orders.

© 1995 Optical Society of America

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References

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  1. D. S. Chemla and J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, New York, 1987), Vols. 1 and 2.
  2. R. Ulrich, "Nonlinear optical organics and devices," in Organic Materials for Nonlinear Optics, R. A. Hann and D. Bloor, eds. (Royal Society of Chemistry, Cambridge, 1988), pp. 241–263.
  3. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  4. G. I. Stegeman, "Introduction to nonlinear guided wave optics," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 11–27.
    [CrossRef]
  5. A. D. Boardman, K. Booth, and P. Egan, "Optical guided waves, linear and nonlinear surface plasmons," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 201–230.
    [CrossRef]
  6. T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
    [CrossRef]
  7. J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).
  8. F. Kajzar, "Organic molecules for guided wave quadratic and cubic optics," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 87–111.
    [CrossRef]
  9. E. Popov, M. Nevière, R. Reinisch, J.-L. Coutaz, and J. F. Roux, "Grating enhanced second harmonic generation in polymer waveguides: role of losses," Appl. Opt. 34, 3398–3405 (1995).
    [CrossRef] [PubMed]
  10. M. Kull, J. L. Coutaz, and R. Meyrueix, "Experimental results of second harmonic generation from a polyurethane waveguide on grating coupler," Opt. Lett. 16, 1930–1932 (1991).
    [CrossRef] [PubMed]
  11. E. Popov and M. Nevière, "Surface-enhanced second harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches," J. Opt. Soc. Am. B 11, 1555–1564 (1994).
    [CrossRef]
  12. M. Nevière, E. Popov, and R. Reinisch, "Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and the zeros of the scattering operator," J. Opt. Soc. Am. A 12, 513–523 (1995).
    [CrossRef]
  13. M. Nevière, "The homogeneous problem," in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 5.
    [CrossRef]
  14. R. Reinisch, "Bistabilité optique par génération de second harmonique sur des coupleurs àp prisme ou à réséau," presented at the 14e Journées Nationales d'Optique guideé, October 25–26, 1994, Besançon, France.

1995 (2)

1994 (1)

1992 (1)

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

1991 (1)

1989 (1)

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

1985 (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Barzoukoz, M.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Blanchard-Desce, M.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Boardman, A. D.

A. D. Boardman, K. Booth, and P. Egan, "Optical guided waves, linear and nonlinear surface plasmons," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 201–230.
[CrossRef]

Booth, K.

A. D. Boardman, K. Booth, and P. Egan, "Optical guided waves, linear and nonlinear surface plasmons," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 201–230.
[CrossRef]

Coutaz, J. L.

Coutaz, J.-L.

Egan, P.

A. D. Boardman, K. Booth, and P. Egan, "Optical guided waves, linear and nonlinear surface plasmons," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 201–230.
[CrossRef]

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Hasegawa, M.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Hasegawa, T.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Ishikawa, K.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Kajzar, F.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

F. Kajzar, "Organic molecules for guided wave quadratic and cubic optics," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 87–111.
[CrossRef]

Kanetake, T.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Kobayashi, M.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Koda, T.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Kubodera, K.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Kull, M.

Lehn, J. M.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Messier, J.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Meyrueix, R.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Nevière, M.

Popov, E.

Reinisch, R.

Roux, J. F.

Sentein, C.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Stegeman, G. I.

G. I. Stegeman, "Introduction to nonlinear guided wave optics," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 11–27.
[CrossRef]

Takeda, K.

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

Ulrich, R.

R. Ulrich, "Nonlinear optical organics and devices," in Organic Materials for Nonlinear Optics, R. A. Hann and D. Bloor, eds. (Royal Society of Chemistry, Cambridge, 1988), pp. 241–263.

Zyss, J.

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. Kanetake, K. Ishikawa, T. Hasegawa, T. Koda, K. Takeda, M. Hasegawa, K. Kubodera, and M. Kobayashi, "Nonlinear optical properties of highly oriented polydiacetylene evaporated films," Appl. Phys. Lett. 54, 2287–2295 (1989).
[CrossRef]

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

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

Nonlin. Opt. (1)

J. Messier, F. Kajzar, C. Sentein, M. Barzoukoz, J. Zyss, M. Blanchard-Desce, and J. M. Lehn, "Nonlinear optical susceptibilities of asymmetric push–pull polyenes," Nonlin. Opt. 2, 53–62 (1992).

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (7)

G. I. Stegeman, "Introduction to nonlinear guided wave optics," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 11–27.
[CrossRef]

A. D. Boardman, K. Booth, and P. Egan, "Optical guided waves, linear and nonlinear surface plasmons," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 201–230.
[CrossRef]

D. S. Chemla and J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, New York, 1987), Vols. 1 and 2.

R. Ulrich, "Nonlinear optical organics and devices," in Organic Materials for Nonlinear Optics, R. A. Hann and D. Bloor, eds. (Royal Society of Chemistry, Cambridge, 1988), pp. 241–263.

F. Kajzar, "Organic molecules for guided wave quadratic and cubic optics," in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), pp. 87–111.
[CrossRef]

M. Nevière, "The homogeneous problem," in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 5.
[CrossRef]

R. Reinisch, "Bistabilité optique par génération de second harmonique sur des coupleurs àp prisme ou à réséau," presented at the 14e Journées Nationales d'Optique guideé, October 25–26, 1994, Besançon, France.

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Figures (13)

Fig. 1
Fig. 1

Schematic of a layer with nonlinear properties, deposited upon a relief grating, together with the incident wave and the propagating diffracted orders at ω1 and ω2 in the cladding (air).

Fig. 2
Fig. 2

Groove depth dependence of the absolute values of the amplitudes of the diffracted 0th (dashed curve) and +1st (solid curve) orders at ω2 in the cladding. Sinusoidal grating with d = 0.47 μm, n2(ω1) = 1.5785 + i4 × 10−5, n2(ω2) = 1.6705 + i2.7 × 10−5, n3(ω1) = 0.129 + i6.83, n3(ω2) = 0.05 + i2.87, t2 = 0.838 mm, λ = 1 μm, and TM polarization. Direct coupling 1–2 (i.e., between the first waveguide mode at ω1 and the second one at ω2).

Fig. 3
Fig. 3

Real (thick curves) and imaginary (thin curves) parts of the mode propagation constants at ω1 (solid curves) and ω2 (dashed curves) as a function of the groove depth, obtained by a rigorous electromagnetic theory. The parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Flow chart of the optimization process.

Fig. 5
Fig. 5

Maximum values of the amplitude | a + 1 NL| of the +1st diffracted cladding order (solid curve and circles) as a function of the grating period and the corresponding groove depth hmax (dashed curve and asterisks) at which these maxima occur. The parameters are the same as in Fig. 2. The curves are obtained according to the phenomenological approach, and the markers by rigorous theory.

Fig. 6
Fig. 6

(a) Input (ĉ1) and output (ĉ2) coupling coefficients calculated by a code for gratings in linear optics, corresponding to Figs. 24. (b) Comparison between the values of c ^ 1 2 c ^ 2 J ^ 1 - 2 (squares) as obtained by use of Eq. (10) and the rigorous results with the nonlinear code and the product c ^ 1 2 c ^ 2, calculated from the data presented in (a) (solid curve).

Fig. 7
Fig. 7

Imaginary part of p 2 ω 2 as a function of the grating period. The data correspond to those in Fig. 2, except for h = 0.02 μm.

Fig. 8
Fig. 8

Same as Fig. 4 except for the direct mode interaction 1–1. n2(ω2) = 1.56415 + i2.7 × 10−5 and t2 = 1.505 μm.

Fig. 9
Fig. 9

Same as Fig. 4 except for the direct mode interaction 1–3. t2 = 1.513 μm.

Fig. 10
Fig. 10

Same as Fig. 4 except for the direct mode interaction 1–4. t2 = 2.066 μm.

Fig. 11
Fig. 11

Same as Fig. 4 except for the direct mode interaction 2–4. t2 = 1.13 μm.

Fig. 12
Fig. 12

Schematic representation of wave interaction in the indirect 1–1 coupling. The incident wave number (sin θi) is coupled through Kin to the mode at ω1, which is then coupled to the mode at ω2 through the third Fourier harmonic of the profile function Kcoup. The mode at ω2 is radiated through the first Fourier harmonic of the profile Kout.

Fig. 13
Fig. 13

Angular dependence of a 0 NL, corresponding to the coupling of Fig. 11. The refractive indices of the media are the same as in Fig. 2, t2 = 1.88 μm, d = 0.47 μm, and the groove is characterized by two (1 and 3) Fourier harmonics with amplitudes h1 = 0.03183 μm and h3 = h1/3.

Equations (13)

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n 2 ( ω 1 ) = 1.5785 + i 4 × 10 - 5 , n 2 ( ω 2 ) = 1.6705 + i 2.7 × 10 - 5 .
n 1 ( ω j ) sin θ N j ω j = n 1 ( ω 1 ) sin θ 0 + N j λ j d ,
sin θ i Re ( p m 1 ω 1 ) - N 1 λ 1 d ,
a g 1 = c 1 ( sin θ 0 - p m 1 ω 1 + N 1 λ 1 d ) ,
a NL = a flat NL + c 1 2 J 1 - 2 c 2 ( sin θ 0 - p m 1 ω 1 + N 1 λ 1 d ) 2 ( sin θ 0 - p m 2 ω 2 + N 2 λ 2 d ) ,
Re ( p m 1 ω 1 ) = Re ( p m 2 ω 2 ) ,
Re ( p m 1 ω 1 ) = Re ( p m 2 ω 2 ) + N λ 2 d .
p m j ω j r j + i ( δ j + γ j h 2 ) ,
c j = c ^ j h N j ,             j = 1 , 2 ,
J = J ^ h N ,
Max θ i a + 1 NL ( h ) = | a + 1 NL ( 0 ) + i c ^ 1 2 c ^ 2 J ^ h 3 ( δ 1 + γ 1 h 2 ) 2 ( δ 2 + γ 2 h 2 ) | ,
γ 1 γ 2 h 4 - h 2 3 ( δ 1 γ 2 - δ 2 γ 1 ) - δ 1 δ 2 = 0.
a + 1 NL ( h s ) = c ^ 1 2 c ^ 2 J ^ h s 3 [ sin θ 0 - r 1 + λ 1 d - i ( δ 1 + γ 1 h s 2 ) ] 2 [ sin θ 0 - r 2 + 2 λ 2 d - i ( δ 2 + γ 2 h s 2 ) ] .

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