Abstract

Optical harmonic generation from a multilayer sample consisting of an arbitrary number of parallel slabs of arbitrary thicknesses can be treated exactly (in the limit where the nonlinear depletion of the pump beam can be neglected) by using a transfer matrix technique. This approach is described, and the results are given in a computationally convenient form. The technique is illustrated by application to a typical case.

© 1989 Optical Society of America

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References

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  1. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
    [Crossref]
  2. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
    [Crossref]
  3. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  4. B. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A 32, 2352 (1985).
    [Crossref] [PubMed]
  5. B. B. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040 (1986).
    [Crossref]
  6. F. Abèles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies: application aux couches minces,” Ann. Phys. (Paris) 5, 598 (1950).
  7. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985), Chap. 2.
  8. O. S. Heavens, Optical Properties of Thin Films (Butterworth, London, 1955; Dover, New York, 1965).
  9. J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
    [Crossref]
  10. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical Analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. LT-5, 660 (1987).
    [Crossref]

1987 (1)

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical Analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

1986 (1)

B. B. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040 (1986).
[Crossref]

1985 (2)

B. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A 32, 2352 (1985).
[Crossref] [PubMed]

J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
[Crossref]

1962 (2)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[Crossref]

1950 (1)

F. Abèles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies: application aux couches minces,” Ann. Phys. (Paris) 5, 598 (1950).

Abèles, F.

F. Abèles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies: application aux couches minces,” Ann. Phys. (Paris) 5, 598 (1950).

Bloembergen, N.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[Crossref]

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical Analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

Giergiel, J.

J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
[Crossref]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Films (Butterworth, London, 1955; Dover, New York, 1965).

Hemminger, J. C.

J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
[Crossref]

Kajzar, B. B.

B. B. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040 (1986).
[Crossref]

Kajzar, B. F.

B. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A 32, 2352 (1985).
[Crossref] [PubMed]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Messier, J.

B. B. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040 (1986).
[Crossref]

B. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A 32, 2352 (1985).
[Crossref] [PubMed]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[Crossref]

Reed, C. E.

J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
[Crossref]

Rosilio, C.

B. B. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040 (1986).
[Crossref]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical Analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical Analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

Ushioda, S.

J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
[Crossref]

Ann. Phys. (Paris) (1)

F. Abèles, “Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies: application aux couches minces,” Ann. Phys. (Paris) 5, 598 (1950).

IEEE J. Lightwave Technol. (1)

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical Analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. LT-5, 660 (1987).
[Crossref]

J. Appl. Phys. (1)

B. B. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040 (1986).
[Crossref]

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[Crossref]

Phys. Rev. A (1)

B. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A 32, 2352 (1985).
[Crossref] [PubMed]

Phys. Rev. B (1)

J. Giergiel, C. E. Reed, S. Ushioda, and J. C. Hemminger, “Attenuated-total-reflection study of pyridine overlayers on silver films,” Phys. Rev. B 31, 3323 (1985).
[Crossref]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Other (3)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985), Chap. 2.

O. S. Heavens, Optical Properties of Thin Films (Butterworth, London, 1955; Dover, New York, 1965).

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Figures (1)

Fig. 1
Fig. 1

Schematic of a multilayer nonlinear medium, showing (in layer 1) the coordinate axes used and the orientation of the E-field basis vectors associated with k 1 + and k - 1 for p polarization and (in layer j) the components of P NL associated with a given source-wave k vector k s and the various k s vectors (dashed arrows) that arise in the case of THG when both rightward- and leftward-propagating pump waves are present.

Equations (43)

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E i ± ( r , t ) = E i ± exp [ ( ± i N i k 0 z ) + ( i k x x - i ω t ) ] .
E i = M i j E j ,
E i = [ E i + E i - ] ,
M i j = 1 t i j [ 1 r i j r i j 1 ] .
E i ( z i + d i ) = Φ i E i ( z i ) ,
Φ i [ ϕ i 0 0 ϕ ¯ i ] ,
T ( ω ) M f ( f - 1 ) Φ f - 1 M ( f - 1 ) ( f - 2 ) Φ f - 2 M 21 .
r = - T 21 / T 22 ,
E j = M j ( j - 1 ) Φ ( j - 1 ) M ( j - 1 ) ( j - 2 ) Φ 2 M 21 [ 1 r ] .
P j NL ( ω s ) = χ j ( 3 ) [ E j + ( ω ) exp ( i N j k 0 z ) + E j - ( ω ) exp ( - i N j k 0 z ) ] 3 × exp [ i ( k s x x - ω s t ) ] ,
P NL = P y NL + P NL + P NL = P NL · ( e ^ y e ^ y + e ^ e ^ + e ^ e ^ ) ,
E s = E s y e ^ y + E s e ^ + E s e ^ = 4 π s - j ( P y NL + P NL ) - 4 π j P NL ,
B s = ( k s / k s 0 ) × E s ,
E s y = [ E s y + E s y - ] ,
E i = M i j E j + M i s E s ,
M k j Φ j E j + M k s Φ s E s = E k .
r j s 1 ,             t j s 2 n s N j ( 3 ω ) κ n j ( 3 ω ) .
E k = M k j Φ j ( M j i E i + S j ) ,
S j ( Φ ¯ j M j s Φ s - M j s ) E s ,
R j f [ E f + ( j ) 0 ] - L j 1 [ 0 E 1 - ( j ) ] = S j ,
T ¯ [ E f + ( j ) 0 ] - [ 0 E 1 - ( j ) ] = S ˜ j ,
[ E f + ( j ) E 1 - ( j ) ] = 1 ( R 11 L 22 - R 21 L 12 ) [ L 22 - L 12 R 21 - R 11 ] S j ,
[ E f + ( j ) E 1 - ( j ) ] = 1 T ¯ 11 [ 1 0 T ¯ 21 - T ¯ 11 ] S j ,
Φ 1 = L 11 = R f f I ,             S 1 M 1 s E 1 s ,             S f - M f s E f s ,
S j y = [ i k 0 d j - ( ϕ ¯ 2 - 1 ) / 2 N - ( ϕ 2 - 1 ) / 2 N - i k 0 d j ] ( Δ 2 N ) E s y ,
S j = [ i k 0 d j ( ϕ ¯ 2 - 1 ) ( - 2 N 2 ) / 2 N ( ϕ 2 - 1 ) ( - 2 N 2 ) / 2 N - i k 0 d j ] × ( Δ 2 N ) E s ,
S j = [ i k 0 d j Δ / 2 N ( ϕ ¯ 2 - 1 ) ( ϕ 2 - 1 ) - i k 0 d j Δ / 2 N ] ( κ 2 N ) E s ,
T ( ω ) = M 32 Φ 2 M 21 = ϕ ¯ 2 t 32 t 21 [ ( ϕ 2 2 + r 12 r 23 ) ( ϕ 2 2 r 21 + r 32 ) ( ϕ 2 2 r 32 + r 21 ) ( 1 + ϕ 2 2 r 12 r 23 ) ] .
r = [ r 12 + ϕ 2 2 r 23 1 + ϕ 2 2 r 12 r 23 ] ,             E 2 ( ω ) = 1 t 21 [ ( 1 + r 21 r ) ( r 21 + r ) ]
S = 1 t 2 s [ ( ϕ ¯ 2 ϕ s - 1 ) ( ϕ ¯ 2 ϕ ¯ s - 1 ) r 2 s ( ϕ 2 ϕ s - 1 ) r 2 s ( ϕ 2 ϕ ¯ s - 1 ) ] E 2 s , S = 1 t 12 [ S + + r 12 S - r 12 S + + S - ] ,
[ E 3 + E 1 - ] = 1 ( 1 + ϕ 2 2 r 12 r 23 ) [ ϕ 2 t 23 ( S + + r 12 S - ) t 21 ( ϕ 2 2 r 23 S + - S - ) ] ,
[ E 3 + E 1 - ] = 1 t 2 s ( 1 + ϕ 2 2 r 12 r 23 ) [ ϕ 2 t 23 [ ( ϕ ¯ 2 ϕ s - 1 ) + ( ϕ 2 ϕ s - 1 ) r 12 r 2 s ] ϕ 2 t 23 [ ( ϕ ¯ 2 ϕ ¯ s - 1 ) r 2 s + ( ϕ 2 ϕ ¯ s - 1 ) r 12 ] t 21 [ ϕ 2 2 ( ϕ ¯ 2 ϕ s - 1 ) r 23 - ( ϕ 2 ϕ s - 1 ) r 2 s ] t 21 [ ϕ 2 2 ( ϕ ¯ 2 ϕ ¯ 2 - 1 ) r 23 r 2 s - ( ϕ 2 ϕ ¯ s - 1 ) ] ] E 2 s .
[ E 3 + ( 1 ) E 1 - ( 1 ) ] = 1 t 1 s R 11 [ 1 r 1 s ( R 21 - r 1 s R 11 ) ( r 1 s R 21 - R 11 ) ] E 1 s ,
[ E 3 + ( 3 ) E 1 - ( 3 ) ] = 1 t 3 s L 22 [ ( r 3 s L 12 - L 22 ) ( L 12 - r 3 s L 22 ) r 3 s 1 ] E 3 s ,
L 12 L 22 = r 32 + ϕ 2 2 r 21 1 + ϕ 2 2 r 32 r 21 , L 22 - 1 = ϕ 2 t 32 t 21 1 + ϕ 2 2 r 32 r 21 , R 21 R 11 = r 12 + ϕ 2 2 r 23 1 + ϕ 2 2 r 12 r 23 , R 11 - 1 = ϕ 2 t 12 t 23 1 + ϕ 2 2 r 12 r 23 ,
r i j = N i - N j N i + N j ,             t i j = 2 N i N i + N j .
r i j = i N j - j N i i N j + j N i ,             t i j = 2 n i n j N i i N j + j N i .
r i j = - r j i ,             t i j t j k = t i k ( 1 + r i j r j k ) ,             t i j t j i = ( 1 - r i j 2 ) ,             r i k = r i j + r j k 1 + r i j r j k .
M i j = 1 t i j [ 1 r i j r i j 1 ] .
M i j ( s ) = 1 2 N i [ N i + N j N i - N j N i - N j N i + N j ] , M i j ( p ) = 1 2 n i n j N i [ i N j + j N i i N j - j N i i N j - j N i i N j + j N i ] .
R 22 = I ,             R 23 = Φ ¯ 2 M 23 = 1 t 23 [ ϕ ¯ 2 ϕ ¯ 2 r 23 ϕ 2 r 23 ϕ 2 ] , R 24 = Φ ¯ 2 M 23 Φ ¯ 3 M 34 = 1 t 23 t 34 [ ϕ ¯ 2 ϕ 3 ( 1 + ϕ 3 2 r 23 r 34 ) ϕ ¯ 2 ϕ 3 ( ϕ 3 2 r 23 + r 34 ) ϕ 2 ϕ 3 ( ϕ ¯ 3 2 r 23 + r 34 ) ϕ 2 ϕ 3 ( 1 + ϕ ¯ 3 2 r 23 r 34 ) ] , R 25 = Φ ¯ 2 M 23 Φ ¯ 3 M 34 Φ ¯ 4 M 45 = 1 t 23 t 34 t 45 [ ϕ ¯ 2 ϕ ¯ 3 [ ϕ ¯ 4 ( 1 + ϕ 3 2 r 23 r 34 ) + ϕ 4 r 45 ( ϕ 3 2 r 23 + r 34 ) ] ϕ ¯ 2 ϕ ¯ 3 [ ϕ ¯ 4 r 45 ( 1 + ϕ 3 2 r 23 r 34 ) + ϕ 4 ( ϕ 3 2 r 23 + r 34 ) ] ϕ 2 ϕ 3 [ ϕ ¯ 4 ( ϕ ¯ 3 2 r 23 + r 34 ) + ϕ 4 r 45 ( 1 + ϕ ¯ 3 2 r 23 r 34 ) ] ϕ 2 ϕ 3 [ ϕ ¯ 4 r 45 ( ϕ ¯ 3 2 r 23 + r 34 ) + ϕ 4 ( 1 + ϕ ¯ 3 2 r 23 r 34 ) ] ] .
E j = M j ( j - 1 ) Φ ( j - 1 ) E ( j - 1 ) , with Φ 1 I and E 1 = [ 1 r ] E 0 ,
S 1 M 1 s E 1 s and S f - M f s E f s .

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