Abstract

An extended theory of Maker fringes, applicable to extraordinary waves, is presented for the exact estimation of nonlinear-optical coefficients d31 and d33 of anisotropic films with Cυ symmetry, such as poled and Langmuir-Blodgett films. Numerical data show that, while the value of d31 decreases with increasing anisotropy of the refractive index at the harmonic frequency, the value of d33 increases remarkably. Therefore, if anisotropy of a few percent is neglected, as in conventional estimations, the values of d31 and d33 are overestimated and underestimated, respectively, and large errors occur. Moreover, d33 strongly depends on the anisotropy at the fundamental frequency.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
    [CrossRef]
  2. J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
    [CrossRef]
  3. G. R. Meredith, J. G. Van Dusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385–1389 (1982).
    [CrossRef]
  4. D. J. Williams, ed., Nonlinear Optical Properties of Organic and Polymeric Materials, ACS Symp. Ser. 233, 109–134 (1983).
    [CrossRef]
  5. K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
    [CrossRef]
  6. B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
    [CrossRef]
  7. M. Eich, H. Looser, D. Y. Yoon, R. Twieg, G. Bjorklund, J. C. Baumert, “Second-harmonic generation in poled organic monomeric glasses,” J. Opt. Soc. Am. B 6, 1590–1597 (1989).
    [CrossRef]
  8. J. Messier, F. Kajzar, P. Prasad, eds., Organic Molecules for Nonlinear Optics and Photonics (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 497–511.

1989 (1)

1986 (1)

K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

1983 (1)

D. J. Williams, ed., Nonlinear Optical Properties of Organic and Polymeric Materials, ACS Symp. Ser. 233, 109–134 (1983).
[CrossRef]

1982 (1)

G. R. Meredith, J. G. Van Dusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385–1389 (1982).
[CrossRef]

1970 (1)

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

1962 (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Baumert, J. C.

Bjorklund, G.

Bjorklund, G. C.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Eich, M.

M. Eich, H. Looser, D. Y. Yoon, R. Twieg, G. Bjorklund, J. C. Baumert, “Second-harmonic generation in poled organic monomeric glasses,” J. Opt. Soc. Am. B 6, 1590–1597 (1989).
[CrossRef]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Jerphagnon, J.

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Jungbauer, D.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Kurtz, S. K.

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Lalama, S. J.

K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

Looser, H.

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Meredith, G. R.

G. R. Meredith, J. G. Van Dusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385–1389 (1982).
[CrossRef]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Reck, B.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Singer, K. D.

K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

Sohn, J. E.

K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Twieg, R.

Twieg, R. J.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Van Dusen, J. G.

G. R. Meredith, J. G. Van Dusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385–1389 (1982).
[CrossRef]

Williams, D. J.

G. R. Meredith, J. G. Van Dusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385–1389 (1982).
[CrossRef]

Willson, C. G.

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Yoon, D. Y.

M. Eich, H. Looser, D. Y. Yoon, R. Twieg, G. Bjorklund, J. C. Baumert, “Second-harmonic generation in poled organic monomeric glasses,” J. Opt. Soc. Am. B 6, 1590–1597 (1989).
[CrossRef]

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

Appl. Phys. Lett. (1)

K. D. Singer, J. E. Sohn, S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

J. Appl. Phys. (1)

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

J. Opt. Soc. Am. B (1)

Macromolecules (1)

G. R. Meredith, J. G. Van Dusen, D. J. Williams, “Optical and nonlinear optical characterization of molecularly doped thermotropic liquid crystalline polymers,” Macromolecules 15, 1385–1389 (1982).
[CrossRef]

Nonlinear Optical Properties of Organic and Polymeric Materials (1)

D. J. Williams, ed., Nonlinear Optical Properties of Organic and Polymeric Materials, ACS Symp. Ser. 233, 109–134 (1983).
[CrossRef]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Other (2)

B. Reck, M. Eich, D. Jungbauer, R. J. Twieg, C. G. Willson, D. Y. Yoon, G. C. Bjorklund, “Crosslinked epoxy polymers with large and stable nonlinear optical susceptibilities,” in Nonlinear Optical Properties of Materials II, G. Kharian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1147, 74–83 (1989).
[CrossRef]

J. Messier, F. Kajzar, P. Prasad, eds., Organic Molecules for Nonlinear Optics and Photonics (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 497–511.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Sample cell including nonlinear material with Cυ symmetry and beam geometry; pol., polarization.

Fig. 2
Fig. 2

Ratios d33,r = d33/d33,i and d31,r = d31/d31,i for n ω o = 1.50 and n 2 ω o = 1.525 versus anisotropy Δn2ω, where d33,i and d31,i are for the isotropic case of Δnω = Δn2ω = 0.

Fig. 3
Fig. 3

Ratios d33,r = d33/d33,i and d31,i = d31/d31,i for n ω o = 1.50 and n 2 ω o = 1.575 versus anisotropy Δn2ω.

Fig. 4
Fig. 4

Ratio dr = d33/d31 versus anisotropy Δn2ω where n ω o = 1.50 and n 2 ω o = 1.55.

Fig. 5
Fig. 5

Ratios d33,rd31,r and dr versus n ω o for cases in which dispersion n 2 ω o n ω o equals 0.05 (solid lines) and 0.1 (dashed lines), where Δn2ω = 4% and Δnωn2ω = 0.5.

Fig. 6
Fig. 6

Maker fringes and envelope for anisotropic film (solid curves), with Δn2ω = 4%, Δnωn2ω = 0.5, and L = 0.1 mm, which is illuminated by a p-polarized incident beam, compared with isotropic film (dashed curves).

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

P 2 ω = [ 0 0 0 0 d 31 0 0 0 0 d 31 0 0 d 31 d 31 d 33 0 0 0 ] [ ( E ω x S ) 2 ( E ω y S ) 2 ( E ω z S ) 2 2 E ω y S E ω z S 2 E ω z S E ω x S 2 E ω x S E ω y S ] ,
E ω = i ω E ω exp [ i ( k ω · r ω t ) ] ,
E ω S = i ω S E ω t ω exp [ i ( k ω S · r ω t ) ] .
n ω = n ω 0 , s polarized , n ω = ñ ω e ( θ ω S ) = [ ( cos θ ω S / n ω 0 ) 2 + ( sin θ ω S / n ω e ) 2 ] 1 / 2 , p polarized ,
i ω S = ( 0 , 1 , 0 ) , s polarized , i ω S = { ( ñ ω e ( θ ω S ) / n ω 0 ) 2 cos θ ω S , 0 , [ ñ ω e ( θ ω S ) / n ω e ] 2 sin θ ω S } , p polarized ,
t ω = 4 n ω g cos θ cos θ ω g ( cos θ + n ω g cos θ ω g ) ( n ω 0 cos θ ω S + n ω g cos θ ω g ) , s polarized t ω = 4 n ω g cos θ cos θ ω g ( n ω g cos θ + cos θ ω g ) { n ω g [ ñ ω e ( θ ω S ) / n ω 0 ] 2 cos θ ω S + ñ ω e ( θ ω S ) cos θ ω g } , p polarized .
n ω sin θ ω S = n ω g sin θ ω g = sin θ .
P 2 ω = p ̂ d 31 ( E ω t ω ) 2 exp [ i ( k b · r 2 ω t ) ] ,
p ̂ = ( p x , 0 , p z ) = ( 0 , 0 , 1 ) , s polarized , p ̂ = ( p x , 0 , p z ) = ( ñ ω e ( θ ω S ) ) 4 [ sin ( 2 θ ω S ) / ( n ω 0 n ω e ) 2 , 0 , cos 2 θ ω S / ( n ω 0 ) 4 + d r sin 2 θ ω S / ( n ω e ) 4 ] , p polarized ,
× × E ( 2 ω ) 2 μ 0 D = { ( 2 ω ) 2 μ 0 P 2 ω bound wave 0 free wave ,
D = 0 [ ( n 2 ω o ) 2 0 0 0 ( n 2 ω o ) 2 0 0 0 ( n 2 ω e ) 2 ] E = 0 2 ω E .
D b = d ̂ b 0 [ ñ 2 ω e ( θ ω S ) ] 2 exp [ i ( k b · r 2 ω t ) ] ,
ñ 2 ω e ( θ ω S ) = [ ( cos θ ω S / n 2 ω o ) 2 + ( sin θ ω S / n 2 ω e ) 2 ] 1 / 2 .
d x = { p x [ 1 ( n ω sin θ ω S n 2 ω e ) 2 ] p z ( n ω / n 2 ω e ) 2 sin θ ω S cos θ ω S } Q ,
d z = { p x ( n ω / n 2 ω o ) 2 sin θ ω S cos θ ω S + p z [ 1 ( n ω cos θ ω S / n 2 ω o ) 2 ] } Q ,
Q = d 31 ( E ω t ω ) 2 0 { n ω 2 [ ñ 2 ω e ( θ ω S ) ] 2 } .
E b = ê b Q exp [ i ( k b · r 2 ω t ) ] ,
ê b = [ ñ 2 ω e ( θ ω S ) ] 2 ( 2 ω ) 1 p ̂ k b ( k b · p ̂ ) ( n 2 ω o n 2 ω e · 2 ω / c ) 2 .
H b = ( 0 , 1 , 0 ) Q n ω [ ñ 2 ω e ( θ ω S ) ] 2 [ p x cos θ ω S / ( n 2 ω o ) 2 p z sin θ ω S / ( n 2 ω e ) 2 ] ζ 0 exp [ i ( k b · r 2 ω t ) ] ,
E f = [ cos θ 2 ω S / ( n 2 ω o ) 2 , 0 , sin θ 2 ω S / ( n 2 ω e ) 2 ] E f [ ñ 2 ω e ( θ 2 ω S ) ] 2 × exp [ i ( k f · r 2 ω t ) ] ,
H f = ( 0 , 1 , 0 ) E f ñ 2 ω e ( θ 2 ω S ) ζ 0 exp [ i ( k f · r 2 ω t ) ] ,
E 2 ω g R = ( cos θ 2 ω g , 0 , sin θ 2 ω g ) E R exp [ i ( k R · r 2 ω t ) ] ,
n ω sin θ ω S = ñ 2 ω e ( θ 2 ω S ) sin θ 2 ω S = n 2 ω g sin θ 2 ω g .
E f = Q F ,
F = ( f cos θ 2 ω g + n 2 ω g g ) / [ ñ 2 ω e ( θ 2 ω S ) h ] ,
f = n ω [ ñ 2 ω e ( θ ω S ) ] 2 [ p x cos θ ω S / ( n 2 ω 0 ) 2 p z sin θ ω S / ( n 2 ω e ) 2 ] ,
g = p x [ ñ 2 ω e ( θ ω S ) / n 2 ω o ] 2 ( p x sin θ ω S + p z cos θ ω S ) × [ n ω ñ 2 ω e ( θ ω S ) / n 2 ω o n 2 ω e ] 2 sin θ ω S ,
h = cos θ 2 ω g + n 2 ω g ñ 2 ω e ( θ 2 ω S ) cos θ 2 ω S / ( n 2 ω o ) 2 .
E 2 ω g = ( cos θ 2 ω g , 0 , sin θ 2 ω g ) E 2 ω g exp [ i ( k 2 ω g · r 2 ω t ) ] ,
E f R = [ cos θ 2 ω S / ( n 2 ω o ) 2 , 0 , sin θ 2 ω S / ( n 2 ω e ) 2 ] E f R [ ñ 2 ω e ( θ 2 ω S ) ] 2 × exp [ i ( k f R · r 2 ω t ) ] ,
| E 2 ω g | 2 = 4 | Q | 2 T 2 ω sin 2 Ψ .
T 2 ω = 2 F [ ñ 2 ω e ( θ 2 ω S ) / n 2 ω o ] 2 × cos θ 2 ω S [ f ñ 2 ω e ( θ 2 ω S ) cos θ 2 ω S / ( n 2 ω o ) 2 + g ] / h 2 ,
Ψ = ( k b z k f z ) L / 2 = ( ω / c ) L [ n ω cos θ ω S ñ 2 ω e ( θ 2 ω S ) cos θ 2 ω S ] ,
I 2 ω = ζ 0 | t 2 ω E 2 ω g | 2 / 2 ,
t 2 ω = 2 n 2 ω g cos θ 2 ω g / ( cos θ 2 ω g + n 2 ω g cos θ ) .
P 2 ω = 8 d 31 2 t ω 4 T 2 ω t 2 ω 2 P ω 2 sin 2 Ψ A ζ 0 0 2 { n ω 2 [ ñ 2 ω e ( θ 2 ω s ) ] 2 } 2 ,
l c ( θ ) = λ [ 4 | n ω cos θ ω S ñ 2 ω e ( θ 2 ω S ) cos θ 2 ω S | ] ,

Metrics