Abstract

We present an application of simple closed-form expressions based on a variational approach for field parameters and effective indices for a closed-form analysis of diffused planar single-mode optical waveguides to obtain the characteristics of diffused channel waveguides by a combination of the variational and effective-index methods.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. K. Sharma, S. Sharma, J. P. Meunier, “Design of refractive ion exchange integrated optical components,” IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996).
    [CrossRef]
  2. W.-H. Tsai, S.-C. Chao, M.-S. Wu, “Variational analysis of single mode inhomogeneous planar optical waveguides,” J. Lightwave Technol. 10, 747–751 (1992).
    [CrossRef]
  3. S.-C. Chao, M.-S. Wu, W.-H. Tsai, “Variational analysis of modal coupling efficiency between graded-index waveguides,” J. Lightwave Technol. 12, 1543–1549 (1994).
    [CrossRef]
  4. A. K. Taneja, S. Srivastava, E. K. Sharma, “Closedform expressions for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305–310 (1997).
    [CrossRef]
  5. S. I. Najafi, ed., Introduction to Glass and Integrated Optics (Artech House, London, 1992).
  6. M. Stern, “Finite-difference analysis of planar optical waveguides,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed., Progress in Electromagnetic Research, Vol. 10 (EMW, Cambridge, Mass., 1995), pp. 123–186.
  7. H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York).

1997 (1)

A. K. Taneja, S. Srivastava, E. K. Sharma, “Closedform expressions for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305–310 (1997).
[CrossRef]

1996 (1)

E. K. Sharma, S. Sharma, J. P. Meunier, “Design of refractive ion exchange integrated optical components,” IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996).
[CrossRef]

1994 (1)

S.-C. Chao, M.-S. Wu, W.-H. Tsai, “Variational analysis of modal coupling efficiency between graded-index waveguides,” J. Lightwave Technol. 12, 1543–1549 (1994).
[CrossRef]

1992 (1)

W.-H. Tsai, S.-C. Chao, M.-S. Wu, “Variational analysis of single mode inhomogeneous planar optical waveguides,” J. Lightwave Technol. 10, 747–751 (1992).
[CrossRef]

Chao, S.-C.

S.-C. Chao, M.-S. Wu, W.-H. Tsai, “Variational analysis of modal coupling efficiency between graded-index waveguides,” J. Lightwave Technol. 12, 1543–1549 (1994).
[CrossRef]

W.-H. Tsai, S.-C. Chao, M.-S. Wu, “Variational analysis of single mode inhomogeneous planar optical waveguides,” J. Lightwave Technol. 10, 747–751 (1992).
[CrossRef]

Haruna, M.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York).

Meunier, J. P.

E. K. Sharma, S. Sharma, J. P. Meunier, “Design of refractive ion exchange integrated optical components,” IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996).
[CrossRef]

Nishihara, H.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York).

Sharma, E. K.

A. K. Taneja, S. Srivastava, E. K. Sharma, “Closedform expressions for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305–310 (1997).
[CrossRef]

E. K. Sharma, S. Sharma, J. P. Meunier, “Design of refractive ion exchange integrated optical components,” IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996).
[CrossRef]

Sharma, S.

E. K. Sharma, S. Sharma, J. P. Meunier, “Design of refractive ion exchange integrated optical components,” IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996).
[CrossRef]

Srivastava, S.

A. K. Taneja, S. Srivastava, E. K. Sharma, “Closedform expressions for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305–310 (1997).
[CrossRef]

Stern, M.

M. Stern, “Finite-difference analysis of planar optical waveguides,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed., Progress in Electromagnetic Research, Vol. 10 (EMW, Cambridge, Mass., 1995), pp. 123–186.

Suhara, T.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York).

Taneja, A. K.

A. K. Taneja, S. Srivastava, E. K. Sharma, “Closedform expressions for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305–310 (1997).
[CrossRef]

Tsai, W.-H.

S.-C. Chao, M.-S. Wu, W.-H. Tsai, “Variational analysis of modal coupling efficiency between graded-index waveguides,” J. Lightwave Technol. 12, 1543–1549 (1994).
[CrossRef]

W.-H. Tsai, S.-C. Chao, M.-S. Wu, “Variational analysis of single mode inhomogeneous planar optical waveguides,” J. Lightwave Technol. 10, 747–751 (1992).
[CrossRef]

Wu, M.-S.

S.-C. Chao, M.-S. Wu, W.-H. Tsai, “Variational analysis of modal coupling efficiency between graded-index waveguides,” J. Lightwave Technol. 12, 1543–1549 (1994).
[CrossRef]

W.-H. Tsai, S.-C. Chao, M.-S. Wu, “Variational analysis of single mode inhomogeneous planar optical waveguides,” J. Lightwave Technol. 10, 747–751 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. K. Sharma, S. Sharma, J. P. Meunier, “Design of refractive ion exchange integrated optical components,” IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996).
[CrossRef]

J. Lightwave Technol. (2)

W.-H. Tsai, S.-C. Chao, M.-S. Wu, “Variational analysis of single mode inhomogeneous planar optical waveguides,” J. Lightwave Technol. 10, 747–751 (1992).
[CrossRef]

S.-C. Chao, M.-S. Wu, W.-H. Tsai, “Variational analysis of modal coupling efficiency between graded-index waveguides,” J. Lightwave Technol. 12, 1543–1549 (1994).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

A. K. Taneja, S. Srivastava, E. K. Sharma, “Closedform expressions for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305–310 (1997).
[CrossRef]

Other (3)

S. I. Najafi, ed., Introduction to Glass and Integrated Optics (Artech House, London, 1992).

M. Stern, “Finite-difference analysis of planar optical waveguides,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed., Progress in Electromagnetic Research, Vol. 10 (EMW, Cambridge, Mass., 1995), pp. 123–186.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Typical refractive-index profile of a diffused planar optical waveguide. (b) Cross section view of the diffused channel optical waveguide showing the coordinates used.

Fig. 2
Fig. 2

neff(x) profile obtained for a Gauss error function channel waveguide with ns=1.50771, w=4.1 μm, h=3.65 μm, Δn=0.028, at γ0=1.214 μm. The thicker curve corresponds to the Gaussian fitted profile.

Fig. 3
Fig. 3

(a) Comparison of normalized field forms obtained by the VEIM and the FDM in the y direction at different values of x for a Gauss error function channel waveguide with parameters as in Fig. 2. The thicker curves correspond to the VEIM results. (b) Comparison of normalized field forms obtained by the VEIM and the FDM in the x direction at different values of y for a Gauss error function channel waveguide with parameters as in Fig. 2. The thicker curves correspond to the VEIM results.  

Fig. 4
Fig. 4

Two-dimensional field obtained for a Gauss error function channel waveguide with parameters as in Fig. 2.

Tables (2)

Tables Icon

Table 1 Empirical Constants for Various Profiles

Tables Icon

Table 2 Comparison of ne for a Channel Waveguide, n2(x, y)=ns2+2nsΔng(ξ) ×exp(-x2/w2), with ns=1.50771, w=4.1 μm, V0=k0h2nsΔn and p0=(ns2-nc2)/2nsΔn

Equations (28)

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

n2(y)=ns2+2nsΔng(ξ),ξ>0nc2,ξ<0,
g(ξ)=exp(-ξ2),Gaussianerfc(ξ),complementaryerrorfunction.exp(-2ξ),exponentialfunction
n2(y)=ns2+2nsΔn exp(-ξ2).
ψ(y)=A(1+κa)exp(-α2a2)exp[-γ(ξ-a)],ξ>aA(1+κξ)exp(-α2ξ2), 0<ξ<a,A exp(κξ),ξ<0
γ=2α2a-κa1+κa.
κ=a0+a1p1/2+a2V+a3Vp1/2;
α=b0+b1p-1/2+b2V+b3p-1/2V
α=b0+b1p-1/2+b2V3/2+b3p-1/2V3/2
α=b0+b1p-1/2+b2V-1/2+b3p-1/2V-1/2
ψ(y)=B exp(α2a2)exp(-2α2a|ξ|),|ξ|>a,
=B exp(-α2ξ2),-a<ξ<a,
α=a0+a1V1/2,
a=b0+b1V.
n2(x, y)=ns2+2nsΔng(ξ)exp(-x2/w2),ξ>0nc2,ξ<0,
V=k0h[2nsΔn exp(-x2/w2)]1/2,
p=(ns2-nc2)2nsΔn exp(x2/w2),
n2(x)=ns2+2nsδn exp(-x2/d2),
δn=[neff2(0)-ns2]/2ns,
b=(r-s)/t,
s=4pακ+1V2 4κα+π2 erf(2αa)(4α2+3κ2)-α exp(-2α2a2)(16α2a-6κ2a+16κα2a2(1+κa)-4κ,
t=4(α2+κ2)ακ+π2 erf(2αa)α2 (4α2+κ2)-2κα exp(-2α2a2)(κa-2)+4α(1+κa)222α2a-κ1+κa exp(-2α2a2),
r=2απ erf[(1+2α2)1/2a][2+κ2(1+2α2)](1+2α2)3/2+4ακ2a1+2α2 exp[-(1+2α2a2)]+8πα(1+κa)2×exp2a(1+2α2)α2a-κ1+κa+κ2(1+κa)2erfc2α2a+a-κ1+κa,
r=4α0a(1+κξ)2 exp(-2α2ξ2)erfc(ξ)dξf+2α(1+κa)2 exp2α2a2-κa1+κa2α2a-κ1+κa×0.27exp22α2a-κ1+κa-exp2α2a-κ1+κa2erfc2α2a-κ1+κa+a,
r=exp(1/2α2)π2 erf2αa+12α-erf12α 41-κ2α22+κ2α2-exp2αa+12α282α2a+42+4κα 1-κ2α2+42+4κα 1-κ2α2+4α(1+κa)21+2α2a-κ1+κa exp[-2a(1+α2a)],
r=4κα+π2 erf(2αa)α2 (4α2+κ2)-2κα exp(-2α2a2)(κa-2).
A=8α/th.
b=π2α2+1 erf(2α2+1a)+π exp[2α2a2(2α2+1)]erfc[a(2α2+1)]-αV2 π2 erf(2αa)1α π2 erf(2αa)+12α2a exp(-2α2a),
 B=1h1α π2 erf(2αa)+12α2a exp(-2α2a)1/2.

Metrics