Abstract

A number of analysis techniques aimed at determining the characteristics of optical guided waves propagating in lossy structures are examined. The exact theory is used as a guide to assess the validity of several approximate methods based on two basic approaches: (a) geometrical optics and (b) perturbation calculations. The limitations of the conventional perturbation techniques are specified. We present a generalized procedure that permits an accurate description of metal boundaries at optical frequencies. In this case, TM modes differ from their TE counterparts by a field buildup near conducting walls and by the existence of an additional surface plasma mode. The dependence of attenuation coefficients on film thickness and mode order are discussed. The use of low-index dielectric buffers to reduce ohmic losses is considered. It is found that, with increasing buffer thickness, TMN modes undergo a continuous transformation to become TMN+1.

© 1973 Optical Society of America

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References

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  1. S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
    [CrossRef]
  2. D. Marcuse, Bell Syst. Tech. J. 48, 3187, 3233 (1969).
  3. W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
    [CrossRef]
  4. D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
    [CrossRef]
  5. J. J. Burke, Appl. Opt. 9, 2444 (1970).
    [CrossRef] [PubMed]
  6. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  7. R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966), Sec. 3.2.
  8. D. P. Gia Russo, J. H. Harris, Appl. Opt. 10, 2787 (1971).
  9. P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  10. J. Kane, H. Oesterberg, J. Opt. Soc. Am. 54, 347 (1964).
    [CrossRef]
  11. H. K. V. Lotsch, J. Opt. Soc. Am, 58, 551 (1968).
    [CrossRef]
  12. B. R. Horowitz, T. Tamir, J. Opt. Soc. Am. 61, 586, (1971).
    [CrossRef]
  13. W. B. Gandrud, IEEE J. Quantum Electron. QE-7, 580 (1971).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965), Chap. 13.
  15. D. Grey, Ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1963), Sec. 6g.
  16. R. K. Williardson, A. Beer, Eds., Semiconductors and Semimetals (Academic Press, New York, 1967), Vol. 3, Chap. 12.
  17. D. W. Wilmot, E. R. Schineller, J. Opt. Soc. Am. 56, 839 (1966).
    [CrossRef]
  18. A MWVE calculation was recently performed by Chang and Loh (Ref. 24) in the case of Al electrodes @10.6 μ. At that wavelength the coefficient Δ ≃ 1.3, indicating that their results should be in excess by a factor of Δ + 1 ≃ 2.3. (The actual figure depends, on the thickness of the waveguide, and we found it to be always <2.4.)
  19. V. A. Pruzhanovskii, Sov. Phys. Tech. Phys., 13, 784, (1968); see also: E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 763 (1972).
    [CrossRef]
  20. This numbering system is consistent with our convention that, as i2 → π/2, the real part of the phase shift approaches or for either polarization.
  21. A. Otto, W. Sohler, Opt. Commun. 3, 254 (1971).
    [CrossRef]
  22. T. Takano, J. Hamasaki, IEEE J. Quantum Electron. QE-8, 206 (1972).
    [CrossRef]
  23. The author is indebted to Eli Burstein for pointing out that the attenuation of this mode decreases rapidly in the ir.
  24. W. S. C. Chang, K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
    [CrossRef]
  25. G. E. Smith, IEEE J. Quantum Electron, QE-4, 288 (1967).
  26. O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965) Sec. 4–4.
  27. The author is indebted to an anonymous reviewer for suggesting this simple procedure.

1972 (3)

S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
[CrossRef]

T. Takano, J. Hamasaki, IEEE J. Quantum Electron. QE-8, 206 (1972).
[CrossRef]

W. S. C. Chang, K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

1971 (5)

A. Otto, W. Sohler, Opt. Commun. 3, 254 (1971).
[CrossRef]

D. P. Gia Russo, J. H. Harris, Appl. Opt. 10, 2787 (1971).

W. B. Gandrud, IEEE J. Quantum Electron. QE-7, 580 (1971).
[CrossRef]

B. R. Horowitz, T. Tamir, J. Opt. Soc. Am. 61, 586, (1971).
[CrossRef]

P. K. Tien, Appl. Opt. 10, 2395 (1971).
[CrossRef] [PubMed]

1970 (2)

1969 (1)

D. Marcuse, Bell Syst. Tech. J. 48, 3187, 3233 (1969).

1968 (2)

H. K. V. Lotsch, J. Opt. Soc. Am, 58, 551 (1968).
[CrossRef]

V. A. Pruzhanovskii, Sov. Phys. Tech. Phys., 13, 784, (1968); see also: E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 763 (1972).
[CrossRef]

1967 (2)

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

G. E. Smith, IEEE J. Quantum Electron, QE-4, 288 (1967).

1966 (1)

1965 (1)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

1964 (1)

Anderson, W. W.

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965), Chap. 13.

Burke, J. J.

Chang, W. S. C.

W. S. C. Chang, K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966), Sec. 3.2.

Gandrud, W. B.

W. B. Gandrud, IEEE J. Quantum Electron. QE-7, 580 (1971).
[CrossRef]

Gia Russo, D. P.

D. P. Gia Russo, J. H. Harris, Appl. Opt. 10, 2787 (1971).

Hamasaki, J.

T. Takano, J. Hamasaki, IEEE J. Quantum Electron. QE-8, 206 (1972).
[CrossRef]

Harris, J. H.

D. P. Gia Russo, J. H. Harris, Appl. Opt. 10, 2787 (1971).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965) Sec. 4–4.

Horowitz, B. R.

Kane, J.

Loh, K. W.

W. S. C. Chang, K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

Lotsch, H. K. V.

H. K. V. Lotsch, J. Opt. Soc. Am, 58, 551 (1968).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 48, 3187, 3233 (1969).

McKenna, J.

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

Miller, S. E.

S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
[CrossRef]

Nelson, D. F.

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

Oesterberg, H.

Otto, A.

A. Otto, W. Sohler, Opt. Commun. 3, 254 (1971).
[CrossRef]

Pruzhanovskii, V. A.

V. A. Pruzhanovskii, Sov. Phys. Tech. Phys., 13, 784, (1968); see also: E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 763 (1972).
[CrossRef]

Schineller, E. R.

Smith, G. E.

G. E. Smith, IEEE J. Quantum Electron, QE-4, 288 (1967).

Sohler, W.

A. Otto, W. Sohler, Opt. Commun. 3, 254 (1971).
[CrossRef]

Takano, T.

T. Takano, J. Hamasaki, IEEE J. Quantum Electron. QE-8, 206 (1972).
[CrossRef]

Tamir, T.

Tien, P. K.

Ulrich, R.

Wilmot, D. W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965), Chap. 13.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 48, 3187, 3233 (1969).

IEEE J. Quantum Electron (1)

G. E. Smith, IEEE J. Quantum Electron, QE-4, 288 (1967).

IEEE J. Quantum Electron. (5)

W. S. C. Chang, K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
[CrossRef]

T. Takano, J. Hamasaki, IEEE J. Quantum Electron. QE-8, 206 (1972).
[CrossRef]

W. B. Gandrud, IEEE J. Quantum Electron. QE-7, 580 (1971).
[CrossRef]

J. Appl. Phys. (1)

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

J. Opt. Soc. Am (1)

H. K. V. Lotsch, J. Opt. Soc. Am, 58, 551 (1968).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Commun. (1)

A. Otto, W. Sohler, Opt. Commun. 3, 254 (1971).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

V. A. Pruzhanovskii, Sov. Phys. Tech. Phys., 13, 784, (1968); see also: E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 763 (1972).
[CrossRef]

Other (9)

This numbering system is consistent with our convention that, as i2 → π/2, the real part of the phase shift approaches or for either polarization.

A MWVE calculation was recently performed by Chang and Loh (Ref. 24) in the case of Al electrodes @10.6 μ. At that wavelength the coefficient Δ ≃ 1.3, indicating that their results should be in excess by a factor of Δ + 1 ≃ 2.3. (The actual figure depends, on the thickness of the waveguide, and we found it to be always <2.4.)

R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966), Sec. 3.2.

The author is indebted to Eli Burstein for pointing out that the attenuation of this mode decreases rapidly in the ir.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965), Chap. 13.

D. Grey, Ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1963), Sec. 6g.

R. K. Williardson, A. Beer, Eds., Semiconductors and Semimetals (Academic Press, New York, 1967), Vol. 3, Chap. 12.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965) Sec. 4–4.

The author is indebted to an anonymous reviewer for suggesting this simple procedure.

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Figures (14)

Fig. 1
Fig. 1

Optical waveguide configuration. The coordinate system is the one used throughout this paper.

Fig. 2
Fig. 2

Geometry used in the ray-optics (RO) approximation. δ21 and δ23 are the Goos-Haenchen shifts at the interfaces 2–1 and 2–3, respectively.

Fig. 3
Fig. 3

Reflectivity (a) and phase shift (b) vs angle of incidence as given by Fresnel’s coefficient for TE polarization. The wave traveling in a medium of index 1.95 is reflected by a lossy dielectric of index 1.75 − j × |κ1|, with |κ1| variable.

Fig. 4
Fig. 4

Reflectivity vs angle of incidence as given by Fresnel’s coefficients for TE and TM waves traveling in a dielectric of index 1.95 and reflected by various conducting materials.

Fig. 5
Fig. 5

Phase shift vs angle of incidence for TE (a) and TM (b) waves under same conditions as Fig. 4.

Fig. 6
Fig. 6

Normalized attenuation coefficient of TE and TM modes vs film thickness @6328 Å. Referring to Fig. 1, the optical constants are n1 = 1.75 − j × 10−4, n2 = 1.95, n3 = 1. The normalizing factor is the attenuation coefficient of a pure plane wave propagating in the unbounded lossy substrate.

Fig. 7
Fig. 7

Accuracy of the DPRT approximation with increasing extinction coefficient |κ1| of the substrate in the following particular case: TE0 mode film thickness 5000 Å, n1 = 1.75 − j × |κ1|, n2 = 1.95, and n3 = 1. α is the attenuation coefficient, ng the guided index. Subscripts E and A refer to exact and approximate values, respectively.

Fig. 8
Fig. 8

Attenuation coefficient of TE0 mode vs thickness of film (n2 = 1.95) deposited on silver (a) and germanium (b). Optical constants @6328 Å are listed in Table I. The various approximations are compared to the exact solution.

Fig. 9
Fig. 9

Relative error of the MWVE approximation for TE modes plotted against the coefficient ξ defined as the ratio of the displacement over the conduction current. Semiconductors usually have ξ < 0 at optical frequencies, leading to a smaller error than for metals.

Fig. 10
Fig. 10

Attenuation coefficient of the first few TE and TM modes vs thickness of film (n2 = 1.95) on aluminum surrounded by air.

Fig. 11
Fig. 11

Normalized field distribution of the TM−1 (a) and TM0 (b) modes for the conditions of Fig. 10 and a film thickness of 5000 Å. Crosses (+) show the accuracy of the DCPL0 approximation.

Fig. 12
Fig. 12

Low index dielectric buffer between guiding film and lossy substrate.

Fig. 13
Fig. 13

Attenuation coefficient vs buffer thickness. Referring to Fig. 12, the parameters are aluminum substrate, nB = 1.45, n2 = 1.95, t2 = 5000 Å, n3 = 1. As tB → 0, the TM1 and TM0 modes gradually change to become TM0 and TM−1, respectively. The inset shows the corresponding change in guided index.

Fig. 14
Fig. 14

Reflectivity vs buffer thickness for a TE and TM wave incident at an angle of 80 degrees. The parameters are the same as in Fig. 13.

Tables (3)

Tables Icon

Table I Optical Constants of a Few Materials at 6328 Å

Tables Icon

Table II Accuracy of the MWVE Approximation for Various Conducting Substratesa

Tables Icon

Table III Various approximations Compared to the Exact Solutiona

Equations (41)

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q 2 t 2 = tan - 1 γ 21 ( p 1 / q 2 ) + tan - 1 γ 23 ( p 3 / q 2 ) + N π ,
p 1 2 = β 2 - n 1 2 k 0 2 ,
p 3 2 = β 2 - n 3 2 k 0 2 ,
q 2 2 = n 2 2 k 0 2 - β 2 .
E y , i H y , i { = A 0 u i , N ( x ) exp [ j ( ω t - β z ) ] { for TE N modes , for TM N modes .
α = - 2 β with β < 0.
[ 1 / u i ( x ) ] ( 2 u i / x 2 ) = β ˜ 2 - k ˜ i 2 ,
u i ( x ) = u i ( x ) exp [ j ϕ i ( x ) ] .
α = 1 W Δ W Δ z = 1 - ρ 21 2 ( i 2 ) ρ 23 2 ( i 2 ) 2 t 2 tan i 2 + δ 21 ( i 2 ) + δ 23 ( i 2 )
ϕ total = - 2 n 2 k 0 t 2 cos i 2 + ϕ 21 ( i 2 ) + ϕ 23 ( i 2 ) = 2 N π .
α = ( 0 ω / 2 ) L s L E · E * d S / ( 1 / 2 ) Re { - E × H * d S } ,
α TE = 2 n L κ L k 0 n g { [ s L u L 2 ( x ) d x ] / [ i = 1 3 region i u i 2 ( x ) d x ] } .
α TM = 2 κ L k 0 n g n L 3 s L u L 2 ( x ) d x + 1 ( k 0 n g ) 2 s L [ u L x ] 2 d x i = 1 3 region i ( u i 2 ( x ) / n i 2 ) d x .
α L = 2 k 0 κ L = k 0 L / n L .
α TE = ( μ ω / σ δ ) ( 1 / β ) { H s 2 / [ i = 2 3 0 u i 2 ( x ) d x ] } .
ξ = = 0 ω σ = n 1 2 - κ 1 2 2 n 1 κ 1 1.
u ˜ 1 ( x ) = A exp ( p ˜ 1 x ) , u ˜ 2 ( x ) = A [ cos q 2 x + γ ( p ˜ 1 / q 2 ) sin q 2 x ] , u ˜ 3 ( x ) = A [ cos q 2 t 2 + γ ( p ˜ 1 / q 2 ) sin q 2 t 2 ] exp [ - p 3 ( x + t 2 ) ] ,
α TE = 2 n 1 κ 1 ( k 0 / n g ) [ I 1 / ( I 1 + I 2 + I 3 ) ] ,
I i = region i u ˜ i ( x ) 2 d x .
α TM = 2 n 1 κ 1 ( n 1 2 + κ 1 2 ) 2 k 0 n g × I 1 + J 1 / ( k 0 n g ) 2 { ( n 1 2 - κ 1 2 ) / [ ( n 1 2 + κ 1 2 ) 2 ] } I 1 + ( I 2 / n 2 2 ) + ( I 3 / n 3 2 ) ,
J 1 = - 0 | u ˜ 1 x | 2 d x .
Δ = { ( ξ 2 + 1 ) [ ξ + ( ξ 2 + 1 ) 1 / 2 ] } 1 / 2 - 1 ,
α TE 1 = 4 α TE 0 and α TE 2 = 9 α TE 0 .
α = ( 1 - ρ 21 2 ) / ( 2 t 2 tan i 2 ) .
n 2 k 0 t 2 cos i 2 ( 1 + N ) π ,
cos i 2 = ( 1 / 2 n 2 ) [ ( N + 1 ) λ 0 ] / t 2 1 / ( tan i 2 ) .
α TE , TM = Γ TE , TM [ ( N + 1 ) 2 / t 2 3 ] λ 0 2 .
q 2 t 2 = tan - 1 γ 21 ( p 1 / q 2 ) + tan - 1 γ 23 ( p 3 / q 2 ) + N π ,
p 1 = p 1 γ B 1 ( p 1 / p B ) + tanh p B t B 1 + γ B 1 ( p 1 / p B ) tanh p B t B , γ 21 = γ B 1 = γ 23 = 1 for TE modes , γ 21 = 2 / ˜ 1 ; γ B 1 = B / ˜ 1 ; γ 23 = 2 / 3 for TM modes , t B = buffer thickness , p B = β 2 - n B 2 k 0 2 .
r 2 B = ρ 2 B exp ( j Δ ϕ 2 B ) = r 2 B + r B 1 exp ( - 2 p B t B ) 1 + r 2 B r B 1 exp ( - 2 p B t B ) .
tan - 1 z = 1 / ( 2 j ) ln [ ( 1 + j z ) / ( 1 - j z ) ] .
α 0 TE = 1 2 1 β μ 0 ω σ d [ E s 2 / ( - + E × H * d x ) ] ,
α 1 TE = 1 2 1 β μ 0 ω σ δ [ e s 2 / ( 0 e × h * d x ) ] .
e s 2 = Z s 2 h s 2 = [ 2 / ( σ δ ) 2 ] h s 2 .
α 1 TE = 1 β μ 0 ω σ δ h s 2 0 e × h * d x
α 1 TE / α 0 TE = ( e s 2 / E s 2 ) × ( δ / d ) .
e s 2 E s 2 Z s 2 ω 2 μ 0 2 / p ˜ 1 2 = U 1 2 + V 1 2 2 n 1 κ 1 .
n ˜ 1 cos ˜ i 1 = U 1 + j V 1
δ / d = V 1 / [ ( n 1 κ 1 ) 1 / 2 ] ,
α 1 TE / α 0 TE = [ V 1 ( U 1 2 + V 1 2 ) ] / [ 2 ( n 1 κ 1 ) 3 / 2 ] .
α 1 TE / α 0 TE = { ( ξ 2 + 1 ) [ ξ + ( ξ 2 + 1 ) 1 / 2 ] } 1 / 2 .

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