Abstract

We present a novel matrix method that allows the straightforward determination of exact propagation constants as well as the field configuration in the region that is beyond the turning point at the substrate side for arbitrarily graded-index planar waveguides.

© 1999 Optical Society of America

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References

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  1. A. Gedeon, “Comparison between rigorous theory and WKB analysis of modes in graded index waveguides,” Opt. Commun. 12, 329–332 (1974).
    [CrossRef]
  2. A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1987).
    [CrossRef]
  3. R. Srivastava, C. K. Kao, R. V. Rmaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. 5, 1605–1609 (1987).
    [CrossRef]
  4. I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Lett. 16, 30–32 (1991).
    [CrossRef] [PubMed]
  5. I. C. Goyal, R. Jindal, A. K. Ghatak, “Planar optical waveguides with arbitrary index profile: an accurate method of analysis,” J. Lightwave Technol. 15, 2179–2182 (1997).
    [CrossRef]
  6. L. Zhan, Z. Cao, “Exact dispersion equation of a graded refractive-index optical waveguide based on the equivalent attenuated vector,” J. Opt. Soc. Am. A 15, 713–716 (1998).
    [CrossRef]

1998

1997

I. C. Goyal, R. Jindal, A. K. Ghatak, “Planar optical waveguides with arbitrary index profile: an accurate method of analysis,” J. Lightwave Technol. 15, 2179–2182 (1997).
[CrossRef]

1991

1987

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

R. Srivastava, C. K. Kao, R. V. Rmaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. 5, 1605–1609 (1987).
[CrossRef]

1974

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

Cao, Z.

Gallawa, R. L.

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. Goyal, R. Jindal, A. K. Ghatak, “Planar optical waveguides with arbitrary index profile: an accurate method of analysis,” J. Lightwave Technol. 15, 2179–2182 (1997).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Lett. 16, 30–32 (1991).
[CrossRef] [PubMed]

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

Goyal, I. C.

I. C. Goyal, R. Jindal, A. K. Ghatak, “Planar optical waveguides with arbitrary index profile: an accurate method of analysis,” J. Lightwave Technol. 15, 2179–2182 (1997).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Methods of analyzing planar optical waveguides,” Opt. Lett. 16, 30–32 (1991).
[CrossRef] [PubMed]

Jindal, R.

I. C. Goyal, R. Jindal, A. K. Ghatak, “Planar optical waveguides with arbitrary index profile: an accurate method of analysis,” J. Lightwave Technol. 15, 2179–2182 (1997).
[CrossRef]

Kao, C. K.

R. Srivastava, C. K. Kao, R. V. Rmaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. 5, 1605–1609 (1987).
[CrossRef]

Rmaswamy, R. V.

R. Srivastava, C. K. Kao, R. V. Rmaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. 5, 1605–1609 (1987).
[CrossRef]

Shenoy, M. R.

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

Srivastava, R.

R. Srivastava, C. K. Kao, R. V. Rmaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. 5, 1605–1609 (1987).
[CrossRef]

Thyagarajan, K.

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

Zhan, L.

J. Lightwave Technol.

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

R. Srivastava, C. K. Kao, R. V. Rmaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. 5, 1605–1609 (1987).
[CrossRef]

I. C. Goyal, R. Jindal, A. K. Ghatak, “Planar optical waveguides with arbitrary index profile: an accurate method of analysis,” J. Lightwave Technol. 15, 2179–2182 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Lett.

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Figures (1)

Fig. 1
Fig. 1

Plot of planar waveguide with arbitrary index profile.

Tables (6)

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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

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Table 4 Comparison at V=4.0

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Table 5 Exponentially Attenuated Coefficient and Equivalent Index beyond the Turning Point

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Table 6 Calculating Results

Equations (38)

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n2(x)=n22+(n12-n22)f(x/d)(x>0)n02(x<0),
Mi=cos(kih)-1/ki sin(kih)ki sin(kih)cos(kih),
(i=1, 2, , l),
Mj=cosh(αjh)-1/αj sinh(αjh)-αj sinh(αjh)cosh(αjh),
(j=l+1, l+2, , l+m)
ki=[k02n2(xi)-β2]1/2,
αj=[β2-k02n2(xj)]1/2.
Ey(0)Ey(0)=i=1lMij=l+1l+mMjEy(xs)Ey(xs),
Ey(x)=A0 exp(P0x)(x<0)As exp[-Ps(x-xs)](x>xs),
P0=(β2-k02n02)1/2,
Ps=(β2-k02ns2)1/2.
(-P01)i=1lMij=l+1l+mMj1-Ps=0.
(-P01)i=1lMi1-Pt=0,
Pt=Pl+1,
Pj=αj sinh(αjh)+Pj+1αj cosh(αjh)cosh(αjh)+Pj+1αj sinh(αjh),
(j=l+1, l+2, , l+m),
Pl+m+1=Ps.
Ey(x)=At exp[-Pt(x-xt)](x>xt),
Pt=(β2-k02neq2)1/2.
P0+P1=0,
Pi=ki tantan-1Pi+1ki-kih,(i=1, 2, 3, , l),
Pl+1=Pt.
Φi=tan-1Piki,
Φi=miπ+tan-1Pi+1ki-kih,=miπ+tan-1ki+1ki tan Φi+1-kih
(mi=0, 1, 2, ;i=1, 2, , l-1).
kih+Φi+1-tan-1ki+1ki tan Φi+1
=miπ+(Φi+1-Φi).
klh=mlπ+tan-1Ptkl-Φl.
i=1lkih+i=1l-1Φi+1-tan-1ki+1ki tan Φi+1
=mπ+tan-1Ptkl-Φ1.
Φ1=-tan-1P0k1.
0xtk(x)dx+Φ(r)=mπ+tan-1P0k1+tan-1Ptkl
(m=0, 1, 2, ),
Φ(r)=i=1l-1Φi+1-tan-1ki+1ki tan Φi+1.
V=k0d(n12-n22)1/2,
b=ne2-n22n12-n22,ne=β/k0.
f(x/d)=exp(-x/d)exponentialexp(-x2/d2)Gaussianerfc(-x/d)complimentaryerror,
n2(x)=n02(x<0)n12-(n12-n22)(x/d)2(0<x<d)n22(x>d).

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