Abstract

We extend the spectrally resolved two-beam coupling technique for measurements of cubic nonlinearities of materials to include higher-order nonlinearities. The governing equations of the extended technique are derived and applied to the analysis of the experimental measurements. We report the observation of saturation of the cubic optical nonlinearity of several glasses. Fifth- and seventh-order nonlinearities are required to account for the measured nonlinear phase shifts. The observation of saturable nonlinear indices accompanied by only moderate nonlinear absorption will be relevant to some applications.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, "'Robust' bistable solitons of the highly nonlinear Schrödinger equation," Phys. Rev. A 35, 466-469 (1987).
    [CrossRef] [PubMed]
  2. M. L. Quiroga-Teixeiro, A. Berntson, and H. Michinel, "Internal dynamics of nonlinear beams in their ground states: short- and long-lived excitation," J. Opt. Soc. Am. B 16, 1697-1704 (1999).
    [CrossRef]
  3. I. Kang, T. D. Krauss, F. W. Wise, B. G. Aitken, and N. F. Borrelli, "Femtosecond measurement of enhanced optical nonlinearities of sulfide glasses and heavy-metal-doped oxide glasses," J. Opt. Soc. Am. B 12, 2053-2059 (1995).
    [CrossRef]
  4. J. M. Harbold, F. Ö. Ilday, F. W. Wise, J. S. Sanghera, V. O. Nguyen, L. B. Shaw, and I. D. Aggarwal, "Highly nonlinear As S Se glasses for all-optical switching," Opt. Lett. 27, 119-121 (2002).
    [CrossRef]
  5. F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
    [CrossRef]
  6. I. Kang, T. Krauss, and F. W. Wise, "Sensitive measurement of nonlinear refraction and two-photon absorption by spectrally resolved two-beam coupling," Opt. Lett. 22, 1077-1079 (1997).
    [CrossRef] [PubMed]
  7. Y.-F. Chen, "Saturable nonlinearity and stable multi-dimensional optical solitons" Ph.D. thesis (Cornell University, 2005).
  8. D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
    [CrossRef]
  9. Y.-F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation of multidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 1-7 (2004).
    [CrossRef]
  10. A. Dubietis, G. Tamos̆auskas, G. Fibich, and B. Ilan, "Multiple filamentation induced by input-beam ellipticity," Opt. Lett. 29, 1126-1128 (2004).
    [CrossRef] [PubMed]

2004 (2)

Y.-F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation of multidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 1-7 (2004).
[CrossRef]

A. Dubietis, G. Tamos̆auskas, G. Fibich, and B. Ilan, "Multiple filamentation induced by input-beam ellipticity," Opt. Lett. 29, 1126-1128 (2004).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

1999 (1)

1997 (1)

1995 (1)

1989 (1)

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

1987 (1)

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, "'Robust' bistable solitons of the highly nonlinear Schrödinger equation," Phys. Rev. A 35, 466-469 (1987).
[CrossRef] [PubMed]

Aggarwal, I. D.

Aitken, B. G.

Barthélémy, A.

F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Beckwitt, K.

Y.-F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation of multidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 1-7 (2004).
[CrossRef]

Berntson, A.

Borrelli, N. F.

I. Kang, T. D. Krauss, F. W. Wise, B. G. Aitken, and N. F. Borrelli, "Femtosecond measurement of enhanced optical nonlinearities of sulfide glasses and heavy-metal-doped oxide glasses," J. Opt. Soc. Am. B 12, 2053-2059 (1995).
[CrossRef]

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

Chen, Y.-F.

Y.-F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation of multidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 1-7 (2004).
[CrossRef]

Y.-F. Chen, "Saturable nonlinearity and stable multi-dimensional optical solitons" Ph.D. thesis (Cornell University, 2005).

Couderc, V.

F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Dubietis, A.

Dumbaugh, W. H.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

Enns, R. H.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, "'Robust' bistable solitons of the highly nonlinear Schrödinger equation," Phys. Rev. A 35, 466-469 (1987).
[CrossRef] [PubMed]

Fibich, G.

Hall, D. W.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

Harbold, J. M.

Ilan, B.

Ilday, F. Ö.

Kang, I.

Kaplan, A. E.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, "'Robust' bistable solitons of the highly nonlinear Schrödinger equation," Phys. Rev. A 35, 466-469 (1987).
[CrossRef] [PubMed]

Krauss, T.

Krauss, T. D.

Malomed, B. A.

Y.-F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation of multidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 1-7 (2004).
[CrossRef]

Michinel, H.

Newhouse, M. A.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

Nguyen, V. O.

Quemard, C.

F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Quiroga-Teixeiro, M. L.

Rangnekar, S. S.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, "'Robust' bistable solitons of the highly nonlinear Schrödinger equation," Phys. Rev. A 35, 466-469 (1987).
[CrossRef] [PubMed]

Sanghera, J. S.

Shaw, L. B.

Smektala, F.

F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Tamos?auskas, G.

Weidman, D. L.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

Wise, F. W.

Appl. Phys. Lett. (1)

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293-1295 (1989).
[CrossRef]

J. Non-Cryst. Solids (1)

F. Smektala, C. Quemard, V. Couderc and A. Barthélémy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (3)

Phys. Rev. A (1)

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, "'Robust' bistable solitons of the highly nonlinear Schrödinger equation," Phys. Rev. A 35, 466-469 (1987).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Y.-F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation of multidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 1-7 (2004).
[CrossRef]

Other (1)

Y.-F. Chen, "Saturable nonlinearity and stable multi-dimensional optical solitons" Ph.D. thesis (Cornell University, 2005).

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 (4)

Fig. 1
Fig. 1

Numerical calculation is used to determine the dependence of the signal on pump-beam intensity in the presence of χ ( 3 ) alone. Inset, SRTBC signals calculated for the indicated nonlinear phase shifts. The time delay is in the units of the pulse duration.

Fig. 2
Fig. 2

The model including higher-order nonlinear effects is used to predict the SRTBC signal. Shown here is the effect of a self-defocusing χ ( 5 ) on a self-focusing χ ( 3 ) . The time delay is in the units of the pulse duration.

Fig. 3
Fig. 3

Intensity dependence of the SRTBC signal magnitude (peak–valley) for various values of self-focusing χ ( 3 ) and for a self-focusing χ ( 3 ) with a self-defocusing χ ( 5 ) .

Fig. 4
Fig. 4

Intensity dependence of (a) SRTBC signal magnitude (normalized peak–valley transmission difference) and (b) nonlinear absorption signal of As 2 S 3 . Insets show examples of SRTBC and nonlinear absorption traces (symbols) along with the best-fit theoretical curves. The time delay is given in units of the pulse duration.

Tables (1)

Tables Icon

Table 1 Summary of Measured Nonlinearities a

Equations (35)

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

E p ( z , t ) = 1 2 { A p ( z , t ) exp [ i ( β p z ω p t ) ] + c.c. } ,
E s ( z , t ) = 1 2 { A s ( z , t ) exp [ i ( β s z ω s t ) ] + c.c. } ,
E ( z , t ) = E p ( z , t ) + E s ( z , t ) .
β p = ω p n 0 R ( ω p ) c ,
β s = ω s n 0 R ( ω s ) c ,
E ̃ p ( z , ω ) = 1 2 [ a p ( z , ω ) + a p * ( z , ω ) ] ,
E ̃ s ( z , ω ) = 1 2 [ a s ( z , ω ) + a s * ( z , ω ) ] ,
a p ( z , ω ) = F { A p ( z , t ) exp [ i ( β p z ω p t ) ] } ,
a s ( z , ω ) = F { A s ( z , t ) exp [ i ( β s z ω s t ) ] } ,
P ̃ L ( z , ω ) = ϵ 0 χ ( 1 ) ( ω ) E ̃ ( z , ω ) ,
P ̃ NL ( n ) ( z , ω 1 + ω 2 + + ω n ) = ϵ 0 χ ( n ) ( ω 1 , ω 2 , , ω n ) E ̃ ( z , ω 1 ) E ̃ ( z , ω 2 ) E ̃ ( z , ω n ) .
P L ( z , t ) = F 1 { P ̃ L ( z , ω ) } ,
P NL ( n ) ( z , t ) = F 1 { P ̃ NL ( n ) ( z , ω ) } .
P L ( z , t ) = 1 2 ϵ 0 { χ ( 1 ) ( ω p ) A p ( z , t ) exp [ i ( β p z ω p t ) ] + χ ( 1 ) ( ω s ) A s ( z , t ) exp [ i ( β s z ω s t ) ] } + c.c. ,
P NL ( n ) ( z , t ) = ( 1 2 ) n ω ̂ 1 ; g 1 ω ̂ n ; g n ϵ 0 χ ( n ) ( ω ̂ 1 , , ω ̂ n ) g 1 ( z , t ) g n ( z , t ) × exp [ i ( ω ̂ 1 + + ω ̂ n ) t ] ,
ω ̂ k = { ω p , ω p , ω s , ω s }
g k ( z , t ) = { A p ( z , t ) exp [ i β p z ] , A p * ( z , t ) exp [ i β p z ] A s ( z , t ) exp [ i β s z ] , A s * ( z , t ) exp [ i β s z ] }
for k = 1 , 2 , , n .
2 z 2 E ( z , t ) 1 c 2 2 t 2 E ( z , t ) μ 0 2 t 2 P ( z , t ) = 0 ,
i β p A p z = 1 2 { β p 2 ω p 2 c 2 [ 1 + χ ( 1 ) ( ω p ) + m = 1 ( 1 2 ) 2 m ( 2 m + 1 ) ! ( m + 1 ) ! m ! χ ( 2 m + 1 ) ( ω p ) A p 2 m ] } A p i ω p c 2 t { [ 1 + χ ( 1 ) ( ω p ) + m = 1 ( 1 2 ) 2 m ( 2 m + 1 ) ! ( m + 1 ) ! m ! χ ( 2 m + 1 ) ( ω p ) A p 2 m ] A p } ,
i β s A s z = 1 2 { β s 2 ω s 2 c 2 [ 1 + χ ( 1 ) ( ω s ) + m = 1 ( 1 2 ) 2 m ( 2 m + 1 ) ! m ! m ! χ ( 2 m + 1 ) ( ω s ) A p 2 m ] } A s i ω s c 2 t { [ 1 + χ ( 1 ) ( ω s ) + m = 1 ( 1 2 ) 2 m ( 2 m + 1 ) ! m ! m ! χ ( 2 m + 1 ) ( ω s ) A p 2 m ] A s } ,
I p z + 1 v ( ω p ) I p t = [ α 0 ( ω p ) + m = 1 α 2 m ( ω p ) I p m ] I p ,
ϕ p z + 1 v ( ω p ) ϕ p t = ω p c m = 1 n 2 m ( ω p ) I p m ,
I s z + 1 v ( ω s ) I s t = [ α 0 ( ω s ) + m = 1 ( m + 1 ) α 2 m ( ω s ) I p m ] I s ,
ϕ s z + 1 v ( ω s ) ϕ s t = ω s c m = 1 ( m + 1 ) n 2 m ( ω s ) I p m ,
n 2 m ( ω p , s ) = ( μ 0 ϵ 0 ) ( m 2 ) ( 2 m + 1 ) ! ( 2 n 0 ) m + 1 ( m + 1 ) ! m ! Re [ χ ( 2 m + 1 ) ( ω p , s ) ] ,
α 2 m ( ω p , s ) = ω p , s c ( μ 0 ϵ 0 ) ( m 2 ) ( 2 m + 1 ) ! ( 2 n 0 ) m + 1 ( m + 1 ) ! m ! Im [ χ ( 2 m + 1 ) ( ω p , s ) ] .
I p z + 1 v I p t = ( α 0 + m = 1 α 2 m I p m ) I p ,
ϕ p z + 1 v ϕ p t = ω 0 c m = 1 n 2 m I p m ,
I s z + 1 v I s t = [ α 0 + m = 1 ( m + 1 ) α 2 m I p m ] I s ,
ϕ s z + 1 v ϕ s t = ω 0 c m = 1 ( m + 1 ) n 2 m I p m ,
n 2 m = ( μ 0 ϵ 0 ) ( m 2 ) ( 2 m + 1 ) ! ( 2 n 0 ) m + 1 ( m + 1 ) ! m ! Re [ χ ( 2 m + 1 ) ] ,
α 2 m = ω 0 c ( μ 0 ϵ 0 ) ( m 2 ) ( 2 m + 1 ) ! ( 2 n 0 ) m + 1 ( m + 1 ) ! m ! Im [ χ ( 2 m + 1 ) ] .
Δ n ( I ) = n 2 I ( 1 + I I sat ) ,
Δ n ( I ) = n 2 I n 2 I sat I 2 + n 2 ( I sat ) 2 I 3 n 2 I + n 4 I 2 + n 6 I 3 + ,

Metrics