Abstract

On the basis of the transfer-matrix technique, the concept of the equivalent attenuated vector is proposed and used to analyze the characteristics of the propagating modes in planar optical waveguides with graded refractive-index profile; an exact analytic equation is obtained for the dispersion equation of arbitrary graded refractive-index planar optical waveguides.

© 1998 Optical Society of America

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References

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  1. E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
    [CrossRef]
  2. A. K. Ghatak, L. A. Kraus, L. A. Kraus, “Propagation of waves in a medium varying transverse to the direction of propagation,” IEEE J. Quantum Electron. QE-10, 456–457 (1974).
  3. M. J. Adams, “The cladded parabolic index profile waveguides analysis and application to strip-geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
    [CrossRef]
  4. C. Zhuangqi, “Dispersion equation of inhomogeneous planar optical waveguide,” Act. Opt. Sin. 14, 12 (1994).
  5. I. C. Joyal, R. L. Gallaya, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Commun. 16, 30–32 (1991).
  6. A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded index waveguides,” Opt. Commun. 12, 329–332 (1974).
    [CrossRef]
  7. K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. LT-4, 201–204 (1986).
  8. S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
    [CrossRef]

1994 (1)

C. Zhuangqi, “Dispersion equation of inhomogeneous planar optical waveguide,” Act. Opt. Sin. 14, 12 (1994).

1991 (1)

I. C. Joyal, R. L. Gallaya, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Commun. 16, 30–32 (1991).

1986 (1)

K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. LT-4, 201–204 (1986).

1982 (1)

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

1978 (1)

M. J. Adams, “The cladded parabolic index profile waveguides analysis and application to strip-geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
[CrossRef]

1974 (2)

A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded index waveguides,” Opt. Commun. 12, 329–332 (1974).
[CrossRef]

A. K. Ghatak, L. A. Kraus, L. A. Kraus, “Propagation of waves in a medium varying transverse to the direction of propagation,” IEEE J. Quantum Electron. QE-10, 456–457 (1974).

1973 (1)

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[CrossRef]

Adams, M. J.

M. J. Adams, “The cladded parabolic index profile waveguides analysis and application to strip-geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
[CrossRef]

Alferness, R. C.

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

Buhl, L. L.

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

Conwell, E. M.

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[CrossRef]

Divino, M. D.

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

Gallaya, R. L.

I. C. Joyal, R. L. Gallaya, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Commun. 16, 30–32 (1991).

Gedeon, A.

A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded index waveguides,” Opt. Commun. 12, 329–332 (1974).
[CrossRef]

Ghatak, A. K.

I. C. Joyal, R. L. Gallaya, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Commun. 16, 30–32 (1991).

A. K. Ghatak, L. A. Kraus, L. A. Kraus, “Propagation of waves in a medium varying transverse to the direction of propagation,” IEEE J. Quantum Electron. QE-10, 456–457 (1974).

Joyal, I. C.

I. C. Joyal, R. L. Gallaya, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Commun. 16, 30–32 (1991).

Korotky, S. K.

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

Kraus, L. A.

A. K. Ghatak, L. A. Kraus, L. A. Kraus, “Propagation of waves in a medium varying transverse to the direction of propagation,” IEEE J. Quantum Electron. QE-10, 456–457 (1974).

A. K. Ghatak, L. A. Kraus, L. A. Kraus, “Propagation of waves in a medium varying transverse to the direction of propagation,” IEEE J. Quantum Electron. QE-10, 456–457 (1974).

Minford, W. J.

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

Mishra, K.

K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. LT-4, 201–204 (1986).

Sharma, A.

K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. LT-4, 201–204 (1986).

Zhuangqi, C.

C. Zhuangqi, “Dispersion equation of inhomogeneous planar optical waveguide,” Act. Opt. Sin. 14, 12 (1994).

Act. Opt. Sin. (1)

C. Zhuangqi, “Dispersion equation of inhomogeneous planar optical waveguide,” Act. Opt. Sin. 14, 12 (1994).

Appl. Phys. Lett. (1)

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. K. Ghatak, L. A. Kraus, L. A. Kraus, “Propagation of waves in a medium varying transverse to the direction of propagation,” IEEE J. Quantum Electron. QE-10, 456–457 (1974).

S. K. Korotky, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
[CrossRef]

J. Lightwave Technol. (1)

K. Mishra, A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. LT-4, 201–204 (1986).

Opt. Commun. (2)

I. C. Joyal, R. L. Gallaya, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Commun. 16, 30–32 (1991).

A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded index waveguides,” Opt. Commun. 12, 329–332 (1974).
[CrossRef]

Opt. Quantum Electron. (1)

M. J. Adams, “The cladded parabolic index profile waveguides analysis and application to strip-geometry lasers,” Opt. Quantum Electron. 10, 17–29 (1978).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Plot of planar waveguides with arbitrary refractive-index profile.

Tables (3)

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Table 1 Exponential Profiles

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Table 2 Gaussian Profiles

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Table 3 Complementary Error Profiles

Equations (37)

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Mi=cos(kih)-1/ki sin(kih)ki sin(kih)cos(kih)
(i=1, 2, , l),
ki=(k0ni2-β2)1/2,
Ey(0)Ey(0)=i=1lMiEy(xt)Ey(xt),
Ey(x)=A0 exp(P0x)(x<0),
P0=(β2-k0n02)1/2,
Ey(xi)=Ai exp[-Pi(x-xi)],
P(0)=-P0,
P(xt)=Pt,
(-P01)M1M2Ml1-Pt=0.
k1h=mπ+tan-1P0k0+tan-1P2k1,
Pi=ki tantan-1Pi+1ki-kih
(i=2, 3, , l),
Pi=P(xi).
tan-1P2k1=tan-1k2k1tantan-1P3k2-k2h=tan-11+k2-k1k1×tantan-1P3k2-k2h=tan-1P3k2-k2h+P2k22-P22k2k1(k2-k1),
tan-1Piki-1=tan-1Pi+1ki-kih+Piki2-Pi2kiki-1(ki-ki-1).
tan-1P2k1=-i=21kih+i=21 Piki2-Pi2(ki-ki-1)+tan-1Piki-1.
0xtk(x)dx 0xtP(x)P2(x)+k2(x) dk(x)dxdx
=mπ+tan-1P0k1+tan-1Ptkt,
k1=[k02n2(0)-β2]1/2,
kt=[k02n2(xt)-β2]1/2.
dP(x)dx=k2(x)+P2(x),
P(0)=-P0,
P(xt)=Pt.
0xtP(x)P2(x)+k2(x)dk(x)dxdx
kh=mπ+tan-1P0k+tan-1P2k,
kk1k2=(k02n12-β2)1/2,
P0=(β2-k02n02)1/2,
P2Pt=(β2-k02n22)1/2;
tan-1Ptkt=tan-1 limxxt+ [β2-k02n2(x)]1/2limxxt- [k02n2(x)-β2]1/2=π/4,
0xtk(x)dx=mπ+π4+tan-1P0k1.
xtxt+1/Ptβ2-k02n2(x)dx=1.
n2(x)=n22+(n12-n22)exp(-x/d)(x>0)n02(x<0),
n2(x)=n22+(n12-n22)exp(-x2/d2)(x>0)n02(x<0),
n2(x)=n22+(n12-n22)erfc(x/d)(x>0)n02(x<0),
V=k0d(n12-n22)1/2
b=n02-n22n12-n22,n0=β/k0.

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