Abstract

We consider quasi-phase-matched (QPM) second-harmonic generation in a leaky-waveguide structure in which the fundamental is a guided mode and the second harmonic is radiated into the substrate. Under a no-pump-depletion approximation and using the coupled-mode theory, we show through numerical calculations that by an appropriate choice of the thickness of the index depression region of the leaky waveguide one can achieve conversion efficiencies and tolerance values that are intermediate between QPM and QPM Čerenkov configurations. The proposed configuration thus possesses the advantages of both QPM guided-to-guided and Čerenkov configurations and hence should be of interest in the realization of short-wavelength light sources.

© 1998 Optical Society of America

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  1. G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  2. M. De Micheli, J. Botineau, S. Neves, P. Sibillot, D. B. Ostrowsky, and M. Papuchon, “Extension of second harmonic phase-matching range in lithium niobate guides,” Opt. Lett. 8, 116–118 (1983).
    [CrossRef] [PubMed]
  3. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  4. P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Čerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
    [CrossRef]
  5. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  6. H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Čerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27, 502–508 (1991).
    [CrossRef]
  7. K. Thyagarajan, V. Mahalakshmi, and M. R. Shenoy, “Performance comparison of different configurations for second harmonic generation in planar waveguides,” Int. J. Optoelectron. 8, 319–332 (1993).
  8. Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
    [CrossRef]
  9. A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
    [CrossRef]
  10. K. Thyagarajan, M. Vaya, and A. Kumar, “Coupled mode analysis to study cascading in the QPM Čerenkov regime in waveguides,” Opt. Commun. 140, 316–322 (1997).
    [CrossRef]
  11. J. Olivares, M. A. Diaz-Gracia, and J. M. Cabrera, “Direct measurement of ordinary refractive index of proton exchanged LiNbO3 waveguides,” Opt. Commun. 92, 40–44 (1992).
    [CrossRef]
  12. M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
    [CrossRef]
  13. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
    [CrossRef]

1997

K. Thyagarajan, M. Vaya, and A. Kumar, “Coupled mode analysis to study cascading in the QPM Čerenkov regime in waveguides,” Opt. Commun. 140, 316–322 (1997).
[CrossRef]

1993

K. Thyagarajan, V. Mahalakshmi, and M. R. Shenoy, “Performance comparison of different configurations for second harmonic generation in planar waveguides,” Int. J. Optoelectron. 8, 319–332 (1993).

1992

J. Olivares, M. A. Diaz-Gracia, and J. M. Cabrera, “Direct measurement of ordinary refractive index of proton exchanged LiNbO3 waveguides,” Opt. Commun. 92, 40–44 (1992).
[CrossRef]

1991

H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Čerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27, 502–508 (1991).
[CrossRef]

1990

M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
[CrossRef]

1989

1987

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

1985

A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[CrossRef]

1983

1974

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

1973

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

1970

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Čerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Botineau, J.

Cabrera, J. M.

J. Olivares, M. A. Diaz-Gracia, and J. M. Cabrera, “Direct measurement of ordinary refractive index of proton exchanged LiNbO3 waveguides,” Opt. Commun. 92, 40–44 (1992).
[CrossRef]

De Micheli, M.

De. Micheli, M.

M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
[CrossRef]

Diaz-Gracia, M. A.

J. Olivares, M. A. Diaz-Gracia, and J. M. Cabrera, “Direct measurement of ordinary refractive index of proton exchanged LiNbO3 waveguides,” Opt. Commun. 92, 40–44 (1992).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Furuya, K.

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[CrossRef]

He, Q.

M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
[CrossRef]

Ibukuro, S.

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

Kumar, A.

K. Thyagarajan, M. Vaya, and A. Kumar, “Coupled mode analysis to study cascading in the QPM Čerenkov regime in waveguides,” Opt. Commun. 140, 316–322 (1997).
[CrossRef]

Li, M. J.

M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
[CrossRef]

Mahalakshmi, V.

K. Thyagarajan, V. Mahalakshmi, and M. R. Shenoy, “Performance comparison of different configurations for second harmonic generation in planar waveguides,” Int. J. Optoelectron. 8, 319–332 (1993).

Martin, R. J.

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Čerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[CrossRef]

Neves, S.

Olivares, J.

J. Olivares, M. A. Diaz-Gracia, and J. M. Cabrera, “Direct measurement of ordinary refractive index of proton exchanged LiNbO3 waveguides,” Opt. Commun. 92, 40–44 (1992).
[CrossRef]

Ostrowsky, D. B.

M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
[CrossRef]

M. De Micheli, J. Botineau, S. Neves, P. Sibillot, D. B. Ostrowsky, and M. Papuchon, “Extension of second harmonic phase-matching range in lithium niobate guides,” Opt. Lett. 8, 116–118 (1983).
[CrossRef] [PubMed]

Papuchon, M.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Sasaki, Y.

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

Shenoy, M. R.

K. Thyagarajan, V. Mahalakshmi, and M. R. Shenoy, “Performance comparison of different configurations for second harmonic generation in planar waveguides,” Int. J. Optoelectron. 8, 319–332 (1993).

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

Shibata, K.

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

Sibillot, P.

Stegeman, G. I.

Stolen, R. H.

Suematsu, Y.

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

Tamada, H.

H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Čerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27, 502–508 (1991).
[CrossRef]

Thyagarajan, K.

K. Thyagarajan, M. Vaya, and A. Kumar, “Coupled mode analysis to study cascading in the QPM Čerenkov regime in waveguides,” Opt. Commun. 140, 316–322 (1997).
[CrossRef]

K. Thyagarajan, V. Mahalakshmi, and M. R. Shenoy, “Performance comparison of different configurations for second harmonic generation in planar waveguides,” Int. J. Optoelectron. 8, 319–332 (1993).

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

Tien, P. K.

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Čerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Čerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[CrossRef]

Vaya, M.

K. Thyagarajan, M. Vaya, and A. Kumar, “Coupled mode analysis to study cascading in the QPM Čerenkov regime in waveguides,” Opt. Commun. 140, 316–322 (1997).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Appl. Phys. Lett.

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Čerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[CrossRef]

IEEE J. Quantum Electron.

H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Čerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27, 502–508 (1991).
[CrossRef]

Y. Suematsu, Y. Sasaki, K. Furuya, K. Shibata, and S. Ibukuro, “Optical second-harmonic generation due to guided-wave structure consisting of quartz and glass film,” IEEE J. Quantum Electron. QE-10, 222–229 (1974).
[CrossRef]

M. J. Li, M. De. Micheli, Q. He, and D. B. Ostrowsky, “Čerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides,” IEEE J. Quantum Electron. 26, 1384–1393 (1990).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Int. J. Optoelectron.

K. Thyagarajan, V. Mahalakshmi, and M. R. Shenoy, “Performance comparison of different configurations for second harmonic generation in planar waveguides,” Int. J. Optoelectron. 8, 319–332 (1993).

J. Lightwave Technol.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. LT-5, 660–667 (1987).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

K. Thyagarajan, M. Vaya, and A. Kumar, “Coupled mode analysis to study cascading in the QPM Čerenkov regime in waveguides,” Opt. Commun. 140, 316–322 (1997).
[CrossRef]

J. Olivares, M. A. Diaz-Gracia, and J. M. Cabrera, “Direct measurement of ordinary refractive index of proton exchanged LiNbO3 waveguides,” Opt. Commun. 92, 40–44 (1992).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

A. K. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Waveguide configuration for a QPM leaky waveguide. XYZ represent the principal axes of the crystal, and xyz is the waveguide coordinate system. θc is the phase-matched Čerenkov angle, and Λ is the grating period. (b) Refractive-index profile of the leaky-waveguide structure, where ne1f and ne1s, ne2 f and ne2s, and ne3f, and ne3s are the extraordinary refractive indices of the fundamental and the SH, respectively, of regions 1, 2, and 3, respectively. nc is the cover’s refractive index, and d and t are thicknesses of regions 1 and 3, respectively.

Fig. 2
Fig. 2

SHG efficiency as a function of grating period Λ for a leaky waveguide with parameters listed in the text. In curves (1)–(3) d33(3)=d33(2) and t=1.35 μm.

Fig. 3
Fig. 3

SHG efficiency as a function of grating period Λ. In curves (1)–(3) t=0 μm.

Fig. 4
Fig. 4

SHG efficiency as a function of grating period Λ. Curves (1), (2), (3), and (4) are for d=0.395, 0.4, 0.405, and 0.41 μm, respectively. In all these curves t=1.35 μm.

Fig. 5
Fig. 5

SHG efficiency as a function of grating period for different thicknesses of the index depression layer. Curves (1), (2), (3), and (4) are for t=1.35 μm, 1.6 μm, 2.1 μm, and ∞, respectively. The other parameters are mentioned in the text. Curve (4), corresponding to t=, is the same as in the QPM guided–guided interaction case.

Fig. 6
Fig. 6

SHG efficiency as a function of grating period for t=0.04, 0.01, 0 μm. The other parameters are listed in the text. Curve (3) is the same as in the QPMC case.

Fig. 7
Fig. 7

Variation of the effective index of leaky mode neff(l) with thickness t of region 3. The solid curve represents neff(l) at those grating periods for which the SHG efficiency peaks, and the crosses on the curve indicate the neff(l) obtained by the matrix method.12

Fig. 8
Fig. 8

Variation of peak conversion efficiency ηp and tolerance (FWHM) with the thickness of the index depression region (region 3).

Fig. 9
Fig. 9

Solid curve, variation of reflectivity of a three-layered structure with the thickness of the index depression region (region 3). Inset, a three-layered structure with the incident, reflected, and transmitted waves; numbers represent the different regions and θ is the angle of incidence. The dashed curve shows the variation of peak conversion efficiency ηp with the thickness of region 3 for a structure shown in Fig. 1(a).    

Equations (37)

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βs=2βf+K,
βs=2k0ne2s cos θc
Ex=Exf+Exs,
Exf=1/2[A(z)Exf(x)exp[i(ωt-βfz)]+c.c.],
Exs=1/2B(Δ;z)Exs(Δ; x)×exp[i(2ωt-βsz)]dΔ+c.c.,
B(Δ; z)=-iA2X(βs)exp(-iΔβz/2) sin(Δβz/2)Δβ/2,
X(βs)=μ0βf20βs  d¯33nef4 HyfHyfHys*dx.
P2ω=0|B(Δ;z)|2dΔ=0A2X(βs) sin(Δβz/2)Δβ/22dΔ.
η=P2ωPω=Pω[X(βs)]2z2 sin2(Δβz/2)(Δβz/2)2 no2sne2s2 βsΔ dβs,
Hyf=Bf exp(-δfx) ;x0Bf[cos κfx-δfκf no1f2 sin κfx] ;-dx0Bf{B1 cosh[(x+d)γf]+B2 sinh[(x+d)γf]} ;-hx-dBfA1 exp[σf(h+x)] ;x-h,
h=t+d,
δf=(βf2-k02)1/2,
κf=no1fne1f (ne1f2k02-βf2)1/2,
γf=no3fne3f (βf2-ne3f2k02)1/2,
σf=no2 fne2 f (βf2-ne2 f2k02)1/2,
B1=cos κfd+δfκf no1f2 sin κfd,
B2=κfno3f2γfno1f2 sin κfd-δfγf no3f2 cos κfd,
A1={B1 cosh[(d-h)γf]+B2 sinh[(d-h)γf]},
A2=δfκf no1f2,
Bf=4ω0βf (B12+B22) sinh[(h-d)2γf]2γfne3f2-B1B2γfne3f2 cosh[(h-d)2γf]+sin(2κfd)2κfne1f2×(1-A22)-A2κfne1f2 cos(2κfd)+A12σfne2 f2+A2κfne1f2+(h-d)ne3f2 (B12-B22)+B1B2γfne3f2+dne1f2 (1+A22)+1δf-11/2.
Hys=Bs exp(-γx)x0Bs(cos κx-γκ no1s2 sin κx)-dx0Bs{b1cosh[(x+d)δ]+b2sinh[(x+d)δ]}-hx-dBscos[Δ(h+x)]{b1 cosh[(d-h)δ]+b2 sinh[(d-h)δ]}+sin[Δ(h+x)]  ×b2 δno2s2Δno3s2cosh[(d-h)δ]+b1 δno2s2Δno3s2sinh[(d-h)δ]x-h,
γ=(βs2-ks2)1/2,
κ=no1sne1s (ne1s2ks2-βs2)1/2,
δ=no3sne3s (βs2-ne3s2ks2)1/2,
Δ=no2sne2s (ne2s2ks2-βs2)1/2,
ks=2k0,
b1=cos κd+γκ no1s2 sin κd,
b2=κn03s2δno1s2 sin κd-γδ no3s2 cos κd,
Bs=4ne2s2(2ω)0πβs {b1 cosh[δ(d-h)]+b2×sinh[δ(d-h)]}2+b1 δno2s2Δno3s2 sinh[(d-h)δ]+b2 δno2s2Δno3s2 cosh[(d-h)δ]2-11/2.
X(βs)=μ0βf20βs -+ d¯33nef4 HyfHyfHys*dx=μ0βf20βs --h d33(2)nef4 HyfHyfHys*dx+-h-d d33(3)nef4 HyfHyfHys*dx+-d0 d33(1)nef4 HyfHyfHys*dx=μ0βf20βs BsBf2d33(2)I2ne2 f4+d33(3)I3ne3f4+d33(1)I1ne1f4,
I1=1+b422κ {sin κd+b3(1-cos κd)}+sin(2κf+κ)d4(2κf+κ) (1-b42-2b4b3)+sin(2κf-κ)d4(2κf-κ) (1-b42+2b4b3)+1-cos(2κf+κ)d4(2κf+κ) (2b4+b3-b42b3)+1-cos(2κf-κ)d4(2κf-κ) (2b4-b3+b42b3),
I2=A122σfa1-a2Δ4σf2+Δ2,
I3=-sinh[δ(d-h)]2δ (B12b1-B22b1)-cosh[δ(d-h)]2δ (B12b2-B22b2)-cosh[(2γf-δ)(d-h)]4(2γf-δ) (-B12b2-B22b2+2B1B2b1)-cosh[(2γf+δ)(d-h)]4(2γf+δ)×(B12b2+B22b2+2B1B2b1)-sinh[(2γf+δ)(d-h)]4(2γf+δ) (B12b1+B22b1+2B1B2b2)-sinh[(2γf-δ)(d-h)]4(2γf-δ) ×(B12b1+B22b1-2B1B2b2)+4B1B2b1γf-δb2(B12+B22)2(4γf2-δ2)+(B12-B22)b12δ,
b3=no1s2 γκ,
b4=no1f2 δfkf,
a1=b1 cosh[(d-h)δ]+b2 sinh[(d-h)δ],
a2=b1 δno2s2Δno3s2 sinh[(d-h)δ]+b2 δno2s2Δno3s2 ×cosh[(d-h)δ].

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