Abstract

The properties of wide-angle integrated optical Bragg deflectors which utilize slab-coupled optical waveguides, are analyzed. Specifically considered is the interaction that occurs, via the intermediary of a periodic waveguide perturbation, between an incident wave guide within the core region of the structure and a Bragg deflected beam guided within the slab region of the structure. The deflection efficiencies and far-field deflected-beam intensity patterns characteristic of this device configuration, which we term a distributed Bragg deflector or DBD, are derived for both T.E.-polarized and T.M.-polarized incident waves and for deflection angles between π/4 and 3π/4 rad. Following these derivations, discussion is made of the potential device applications of the basic DBD structure, including its use as an integrated optical modulator, polarizer/analyzer, beam divider, beam deflector, and intrawaveguide beam expander.

© 1978 Optical Society of America

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References

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  1. J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
    [CrossRef]
  2. J. N. Polky, J. H. Harris, Appl. Phys. Lett. 21, 307 (1972).
    [CrossRef]
  3. J. M. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
    [CrossRef]
  4. L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
    [CrossRef]
  5. C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
    [CrossRef]
  6. O. Mikami, Opt. Commun. 19, 42 (1976).
    [CrossRef]
  7. R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
    [CrossRef]
  8. G. F. Sauter, M. M. Hanson, D. L. Fleming, Appl. Phys. Lett. 30, 11 (1977).
    [CrossRef]
  9. E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1973).
  10. J. E. Goell, Appl. Opt. 12, 2797 (1973).
    [CrossRef] [PubMed]
  11. H. Furuta, H. Noda, A. Ihaya, Appl. Opt. 13, 322 (1974).
    [CrossRef] [PubMed]
  12. See, for example, p. 650 of Ref. 9.
  13. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  14. R. P. Kenan, J. Appl. Phys. 46, 4545 (1975).
    [CrossRef]
  15. T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
    [CrossRef]
  16. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  17. See, for example, H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974).
    [CrossRef]
  18. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  19. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  20. H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
    [CrossRef]
  21. S. Wang, IEEE J. Quantum Electron. QE-13, 176 (1977).
    [CrossRef]
  22. D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
    [CrossRef]

1977

G. F. Sauter, M. M. Hanson, D. L. Fleming, Appl. Phys. Lett. 30, 11 (1977).
[CrossRef]

S. Wang, IEEE J. Quantum Electron. QE-13, 176 (1977).
[CrossRef]

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
[CrossRef]

1976

C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
[CrossRef]

O. Mikami, Opt. Commun. 19, 42 (1976).
[CrossRef]

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

1975

R. P. Kenan, J. Appl. Phys. 46, 4545 (1975).
[CrossRef]

1974

H. Furuta, H. Noda, A. Ihaya, Appl. Opt. 13, 322 (1974).
[CrossRef] [PubMed]

See, for example, H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974).
[CrossRef]

1973

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1973).

J. E. Goell, Appl. Opt. 12, 2797 (1973).
[CrossRef] [PubMed]

J. M. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
[CrossRef]

1972

J. N. Polky, J. H. Harris, Appl. Phys. Lett. 21, 307 (1972).
[CrossRef]

1971

J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
[CrossRef]

1970

L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
[CrossRef]

1969

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Anderson, D. B.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
[CrossRef]

August, R. R.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
[CrossRef]

Boyd, J. T.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
[CrossRef]

Channin, D. J.

J. M. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
[CrossRef]

Dakss, M. L.

L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
[CrossRef]

Duffy, M. T.

J. M. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
[CrossRef]

Fleming, D. L.

G. F. Sauter, M. M. Hanson, D. L. Fleming, Appl. Phys. Lett. 30, 11 (1977).
[CrossRef]

Furuta, H.

Giallorenzi, T. G.

T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
[CrossRef]

Goell, J. E.

Hamilton, M. C.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
[CrossRef]

Hammer, J. M.

J. M. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
[CrossRef]

J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
[CrossRef]

Hanson, M. M.

G. F. Sauter, M. M. Hanson, D. L. Fleming, Appl. Phys. Lett. 30, 11 (1977).
[CrossRef]

Harris, J. H.

J. N. Polky, J. H. Harris, Appl. Phys. Lett. 21, 307 (1972).
[CrossRef]

Hartman, N. F.

C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
[CrossRef]

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

Heidrich, P. F.

L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
[CrossRef]

Ihaya, A.

Kenan, R. P.

C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
[CrossRef]

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

R. P. Kenan, J. Appl. Phys. 46, 4545 (1975).
[CrossRef]

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Kuhn, L.

L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1973).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

Mikami, O.

O. Mikami, Opt. Commun. 19, 42 (1976).
[CrossRef]

Noda, H.

Polky, J. N.

J. N. Polky, J. H. Harris, Appl. Phys. Lett. 21, 307 (1972).
[CrossRef]

Sauter, G. F.

G. F. Sauter, M. M. Hanson, D. L. Fleming, Appl. Phys. Lett. 30, 11 (1977).
[CrossRef]

Scott, B. A.

L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
[CrossRef]

Stoll, H.

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

Taylor, H. F.

See, for example, H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974).
[CrossRef]

Vahey, D. W.

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

Verber, C. M.

C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
[CrossRef]

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

Wang, S.

S. Wang, IEEE J. Quantum Electron. QE-13, 176 (1977).
[CrossRef]

Wood, V. E.

C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
[CrossRef]

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

Yariv, A.

See, for example, H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974).
[CrossRef]

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. F. Sauter, M. M. Hanson, D. L. Fleming, Appl. Phys. Lett. 30, 11 (1977).
[CrossRef]

J. M. Hammer, Appl. Phys. Lett. 18, 147 (1971).
[CrossRef]

J. N. Polky, J. H. Harris, Appl. Phys. Lett. 21, 307 (1972).
[CrossRef]

J. M. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
[CrossRef]

L. Kuhn, M. L. Dakss, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 17, 265 (1970).
[CrossRef]

Bell Syst. Tech. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 53, 645 (1973).

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Ferroelectrics

C. M. Verber, V. E. Wood, R. P. Kenan, N. F. Hartman, Ferroelectrics 10, 253 (1976).
[CrossRef]

IEEE J. Quantum Electron.

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

S. Wang, IEEE J. Quantum Electron. QE-13, 176 (1977).
[CrossRef]

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, IEEE J. Quantum Electron. QE-13, 268 (1977).
[CrossRef]

J. Appl. Phys.

R. P. Kenan, J. Appl. Phys. 46, 4545 (1975).
[CrossRef]

T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
[CrossRef]

Opt. Commun.

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

O. Mikami, Opt. Commun. 19, 42 (1976).
[CrossRef]

Opt. Eng.

R. P. Kenan, D. W. Vahey, N. F. Hartman, V. E. Wood, C. M. Verber, Opt. Eng. 15, 12 (1976).
[CrossRef]

Proc. IEEE

See, for example, H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974).
[CrossRef]

Other

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

See, for example, p. 650 of Ref. 9.

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Figures (6)

Fig. 1
Fig. 1

Distributed Bragg deflector: (a) top view; (b) side cross-sectional view; and (c) end cross-sectional view. Small and large (solid) arrows in (a) represent incident and deflected beams respectively. Dashed sinusoid in (c) represents equivalent core field distribution in the y direction.

Fig. 2
Fig. 2

(a) End cross-sectional view of imbedded ridge DBD; (b) end cross-sectional view of strip-loaded DBD.

Fig. 3
Fig. 3

(a) Deflected beam far-field pattern factors [see Eqs. (26) and (27)]; (b) corresponding phase vector or momentum matching diagram for exact satisfaction of the Bragg condition.

Fig. 4
Fig. 4

Normalized T.E. and T.M. core-propagating attenuation coefficients for exact satisfaction of the Bragg condition.

Fig. 5
Fig. 5

Plot of factor by which αT.E.,T.M. must be reduced when Bragg condition is not exactly satisfied.

Fig. 6
Fig. 6

Cylindrical volume over which Eq. (A3) is integrated in order to determine the form of G, the scalar Green’s function.

Equations (62)

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( 2 + n i 2 k 0 2 ) E = { Q T . E . : d z 0 0 : otherwise ,
( 2 + n i 2 k 0 2 ) H = { Q T . M . : d z 0 0 : otherwise ,
Q T . E . = k 0 2 Δ n 2 ( r ) E { E · log [ ε 0 n 2 2 + ε 0 Δ n 2 ( r ) ] } ,
Q T . M . = k 0 2 Δ n 2 ( r ) H log [ ε 0 n 2 2 + ε 0 n 2 ( r ) ] × × H .
Δ n 2 ( r ) = ( n 1 2 n 2 2 ) { 1 2 2 π l = 1 ( 1 ) l 1 ( 2 l 1 ) cos [ ( 2 l 1 ) K · r ] } ,
E = E c + E s = E c ( y , z ) · exp [ i ( k ˜ T . E . c ) x ] · a ˆ y + E s ,
H = H c + H s = H c ( y , z ) · exp [ i ( k ˜ T . M . c ) x ] · a ˆ y + H s ,
E c ( y , z ) = ( 2 ω μ 0 k T . E . c ) 1 / 2 W ( y ) · E ( z ) ,
H c ( y , z ) = ( 2 ω ε 0 k T . M . c ) 1 / 2 W ( y ) · H ( z ) .
W ( y ) = ( 2 w ) 1 / 2 cos ( π y w ) ,
{ d 2 d z 2 + [ n i 2 k 0 2 ( π / w ) 2 ( k T . E . c ) 2 ] } E ( z ) = 0 ; i = 1,2,3 ,
{ d 2 d z 2 + [ n i 2 k 0 2 ( π / w ) 2 ( k T . M . c ) 2 ] } H ( z ) = 0 ; i = 1,2,3 ,
E 2 ( z ) d z = H 2 ( z ) n i 2 d z = 1 ; i = 1,2,3.
Q T . E . = Q T . E . ( E c ) + Q T . E . ( E s ) = ( A x a ˆ x + A y a ˆ y ) E c ( y , z ) · exp { i [ k ˜ T . E . c · x + ( 2 l 1 ) K · r ] } + Q T . E . ( E s ) ,
Q T . M . = Q T . M . ( H c ) + Q T . M . ( H s ) = ( B x a ˆ x + B y a ˆ y ) H c ( y , z ) · exp { i [ k ˜ T . M . c · x + ( 2 l 1 ) K · r ] } + Q T . M . ( H s ) ,
A x = 2 ( 1 ) l π · log ( n 1 n 2 ) · [ ( 2 l 1 ) K x K y + k T . E . c K y ] ,
A y = ( 1 ) l π · [ k 0 2 ( n 1 2 n 2 2 ) ( 2 l 1 ) + 2 ( 2 l 1 ) K y 2 log ( n 1 n 2 ) ] ,
B x = 2 ( 1 ) l π K y · k T . M . c · log ( n 1 n 2 ) ,
B y = ( 1 ) l π · [ k 0 2 ( n 1 2 n 2 2 ) ( 2 l 1 ) + 2 · k T . M . c · K x log ( n 1 n 2 ) ] .
y E c ( y , c ) , z E c ( y , z ) K x , y .
( 2 + n i 2 k 0 2 ) G T . E . ( r | r ) = I · δ ( r r ) ; i = 1,2,3,4 ,
( 2 + n i 2 k 0 2 ) G T . M . ( r | r ) = I · δ ( r r ) ; i = 1,2,3,4 ,
G T . E . ( r | r ) = 1 4 [ H 0 ( 2 ) ( k T . E . c · R ) · E 0 ( z ) · E 0 ( z ) + ν H 0 ( 2 ) ( k T . E . ν · R ) · E ν ( z ) · E ν ( z ) d ν ] · ( a ˆ x a ˆ x + a ˆ y a ˆ y ) ,
G T . M . ( r | r ) = 1 4 [ H 0 ( 2 ) ( k T . M . c · R ) · H 0 ( z ) · H 0 ( z ) n j 2 + ν H 0 ( 2 ) ( k T . M . ν · R ) · H ν ( z ) · H ν ( z ) n j 2 d ν ] · ( a ˆ x a ˆ x + a ˆ y a ˆ y ) ,
E = E c + E s = G T . E . ( r | r ) · Q T . E . ( E c + E s ) d x d y d z ,
H = H c + H s = G T . M . ( r | r ) · Q T . M . ( H c + H s ) d x d y d z .
E s = G T . E . p ( r | r ) · Q T . E . ( E c ) d x d y d z ,
H s = G T . M . p ( r | r ) · Q T . M . ( H c ) d x d y d z ,
E s = i 2 k ( ω μ 0 π ) 1 / 2 · exp ( + i π 4 α L 2 ) · [ d 0 E 0 2 ( z ) d z ] · E 0 ( z ) · exp ( i k · r ) ( r ) 1 / 2 · f ( θ ) × [ u ˆ r ( A x cos θ + A y sin θ ) + u ˆ θ ( A x sin θ + A y cos θ ) ] ; j = 1,2,3 ,
H s = i 2 k ( ω ε 0 π ) 1 / 2 · exp ( + i π 4 α L 2 ) · [ d 0 H 0 2 ( z ) n j 2 ( z ) d z ] · H 0 ( z ) · exp ( i k · r ) ( r ) 1 / 2 · f ( θ ) × [ u ˆ r ( B x cos θ + B y sin θ ) + u ˆ θ ( B x sin θ + B y cos θ ) ] ; j = 1,2,3 ,
H 0 ( 2 ) ( x ) ( 2 π x ) 1 / 2 · exp ( i ( x π / 4 ) ] ;
f ( θ ) = W ( y ) · exp [ i u ( θ ) y ] · exp [ α x i υ ( θ ) x ] d x d y .
u ( θ ) = ( 2 l 1 ) K y k sin θ ,
υ ( θ ) = k ( 1 cos θ ) + ( 2 l 1 ) K x ,
1 2 Re ( E × H * ) r d θ d z = 1 exp ( 2 α L )
Γ T . E . 2 8 π k T . E . ( A x sin θ + A y cos θ ) 2 · | f ( θ ) | 2 · d θ = 1 exp ( 2 α T . E . · L ) ,
Γ T . M . 2 8 π k T . M . ( B x sin θ + B y cos θ ) 2 · | f ( θ ) | 2 · d θ = 1 exp ( 2 α T . M . · L ) ,
Γ T . E . d 0 E 0 2 ( z ) d z , Γ T . M . d 0 H 0 2 ( z ) n j 2 d z ( j = 1,2,3 ) , | f ( θ ) | 2 = 8 w π 2 · g x ( θ ) · g y ( θ )
g x ( θ ) = 2 · exp ( α L ) α 2 + υ 2 ( θ ) { cosh ( α L ) cos [ υ ( θ ) · L ] } ,
g y ( θ ) = { cos [ u ( θ ) · w / 2 ] 1 [ u ( θ ) · w / π ] 2 } 2 .
Δ θ x 2 α k sin ( θ B ) ,
Δ θ y { 2 · [ 2 π w · k · sin ( θ B ) ] 1 / 2 : θ B near π / 2 2 π w · k · cos ( θ B ) : otherwise ,
u ( θ B ) · w 2 3 π 2 or that w 3 π 2 k sin ( θ B ) .
α T . E . = w [ k 0 ( n 1 2 n 2 2 ) π 2 ( 2 l 1 ) n 2 · Γ T . E . ] 2 cos 2 ( θ B ) ,
α T . M . = w [ k 0 ( n 1 2 n 2 2 ) π 2 ( 2 l 1 ) n 2 · Γ T . M . ] 2 × { 2 n 2 2 ( n 1 2 n 2 2 ) · log ( n 1 n 2 ) · [ cos ( θ B ) 1 ] cos ( θ B ) } 2 ,
ν = ( 2 l 1 ) 2 w ( t d ) 6
η T . E ., T . M . = 1 exp ( 2 · L · α T . E ., T . M . ) .
I T . E ., T . M . = g x ( θ ) = 1 ( α T . E ., T . M . ) 2 + υ 2 ( θ ) ,
Δ θ T . E ., T . M . = Δ θ x = 2 α T . E ., T . M . 2 k sin ( θ B ) ,
K = 2 k sin ( θ B / 2 ) 2 k cos ( φ )
α T . E ., T . M . α T . E ., T . M . · g y ( u · w ) ,
u [ ( 2 l 1 ) sin ( 3 θ B / 2 ) sin ( θ B ) ] · Δ K k · Δ φ + [ cos ( θ B ) 1 sin ( θ B ) ] · Δ k .
M λ 0 n 2 · w / Δ θ x .
2 G + n i 2 k 0 2 G = I · δ ( r r ) ; i = 1,2,3 ,
G = m = 1 3 e ˆ m · e ˆ m G ,
1 R R ( R G R ) + 2 G z 2 + n i 2 k 0 2 G = δ ( R ) · δ ( θ 0 ) · δ ( z z ) R .
1 R d d R ( R d f d R ) + k 2 · f = 0 ,
d 2 h d z 2 + [ n i 2 k 0 2 k 2 ] h = 0 ,
2 π ε z = z z = z + G R | R = ε · d z = 1 ,
2 π ε G R | R = ε = δ ( z z )
2 π ε G T . E . R | R = ε = E 0 ( z ) · E 0 ( z ) + ν E ν ( z ) · E ν ( z ) d ν
2 π ε G T . M . R | R = ε = H 0 ( z ) · H 0 ( z ) n i 2 + ν H ν ( z ) · H ν ( z ) n i 2 d ν

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