Abstract

Hyper-Rayleigh scattering is frequently used to determine all of the rotational invariants of the first hyperpolarizability tensor. It requires numerous polarization states of both incident and scattered light and thus justifies the use of a dual-rotating-retarder polarimeter. We optimized our experimental setup by reducing the condition number of the polarization processing matrix. Our numerical study showed that, on condition to make six measurements, the choice of the detector angle and of the angular steps of both retarders was paramount. Overspecifying the calculation through a much higher number of measurements allowed us to make broad optimal detector angles and retarder angular steps available. Numerical simulations are presented to optimize our experimental setup.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2002 (1)

2000 (3)

1998 (1)

1996 (1)

M. Kauranen and A. Persoons, "Theory of polarization measurements of second-order nonlinear light scattering," J. Chem. Phys. 104, 3445-3456 (1996).
[CrossRef]

1994 (1)

J. Zyss and I. Ledoux, "Nonlinear optics in multipolar media: theory and experiments," Chem. Rev. (Washington, D.C.) 94, 77-105 (1994).
[CrossRef]

1993 (1)

G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, "Determination of hyperpolarizability measurements by depolarized hyper-Rayleigh scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

1977 (1)

1975 (1)

B. Levine and C. Bethea, "Second and third order hyperpolarizabilities of organic molecules," J. Chem. Phys. 63, 2666-2682 (1975).
[CrossRef]

1965 (2)

J. Giordmaine, "Nonlinear optical properties of liquid," Phys. Rev. 138, A1599-A1606 (1965).
[CrossRef]

R. Terhune, P. Maker, and C. Savage, "Measurements of nonlinear light scattering," Phys. Rev. Lett. 14, 681-684 (1965).
[CrossRef]

1962 (1)

D. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977-1979 (1962).
[CrossRef]

Azzam, R.

Bethea, C.

B. Levine and C. Bethea, "Second and third order hyperpolarizabilities of organic molecules," J. Chem. Phys. 63, 2666-2682 (1975).
[CrossRef]

Bölger, B.

G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, "Determination of hyperpolarizability measurements by depolarized hyper-Rayleigh scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Cahill, P. A.

Chien, L. C.

Delmas, J.

J. Delmas, Introduction aux Probabilités (Ed. Ellipses, 1993).

Dereniak, E.

Descour, M.

D'Sidocky, N.

Giordmaine, J.

J. Giordmaine, "Nonlinear optical properties of liquid," Phys. Rev. 138, A1599-A1606 (1965).
[CrossRef]

Golub, G.

G. Golub and C. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Heesink, G.

G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, "Determination of hyperpolarizability measurements by depolarized hyper-Rayleigh scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Henderson, C. C.

Hubbard, S. F.

Hudson, C.

Issacson, E.

E. Issacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).

Kauranen, M.

M. Kauranen and A. Persoons, "Theory of polarization measurements of second-order nonlinear light scattering," J. Chem. Phys. 104, 3445-3456 (1996).
[CrossRef]

Keller, H. B.

E. Issacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).

Kemme, S.

Kleinman, D.

D. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977-1979 (1962).
[CrossRef]

Ledoux, I.

J. Zyss and I. Ledoux, "Nonlinear optics in multipolar media: theory and experiments," Chem. Rev. (Washington, D.C.) 94, 77-105 (1994).
[CrossRef]

Levine, B.

B. Levine and C. Bethea, "Second and third order hyperpolarizabilities of organic molecules," J. Chem. Phys. 63, 2666-2682 (1975).
[CrossRef]

Loan, C. V.

G. Golub and C. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Maker, P.

R. Terhune, P. Maker, and C. Savage, "Measurements of nonlinear light scattering," Phys. Rev. Lett. 14, 681-684 (1965).
[CrossRef]

Ostroverkhov, V.

Persoons, A.

M. Kauranen and A. Persoons, "Theory of polarization measurements of second-order nonlinear light scattering," J. Chem. Phys. 104, 3445-3456 (1996).
[CrossRef]

Petschek, R. G.

Phipps, G.

Ruiter, A.

G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, "Determination of hyperpolarizability measurements by depolarized hyper-Rayleigh scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Sabatke, D.

Savage, C.

R. Terhune, P. Maker, and C. Savage, "Measurements of nonlinear light scattering," Phys. Rev. Lett. 14, 681-684 (1965).
[CrossRef]

Singer, K. D.

Smith, M.

Sukhomlinova, L.

Sweatt, W.

Terhune, R.

R. Terhune, P. Maker, and C. Savage, "Measurements of nonlinear light scattering," Phys. Rev. Lett. 14, 681-684 (1965).
[CrossRef]

Twieg, R. J.

Tyo, J.

van Hulst, N.

G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, "Determination of hyperpolarizability measurements by depolarized hyper-Rayleigh scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Wang, S.-X.

Zyss, J.

J. Zyss and I. Ledoux, "Nonlinear optics in multipolar media: theory and experiments," Chem. Rev. (Washington, D.C.) 94, 77-105 (1994).
[CrossRef]

Appl. Opt. (1)

Chem. Rev. (Washington, D.C.) (1)

J. Zyss and I. Ledoux, "Nonlinear optics in multipolar media: theory and experiments," Chem. Rev. (Washington, D.C.) 94, 77-105 (1994).
[CrossRef]

J. Chem. Phys. (2)

B. Levine and C. Bethea, "Second and third order hyperpolarizabilities of organic molecules," J. Chem. Phys. 63, 2666-2682 (1975).
[CrossRef]

M. Kauranen and A. Persoons, "Theory of polarization measurements of second-order nonlinear light scattering," J. Chem. Phys. 104, 3445-3456 (1996).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. (2)

D. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977-1979 (1962).
[CrossRef]

J. Giordmaine, "Nonlinear optical properties of liquid," Phys. Rev. 138, A1599-A1606 (1965).
[CrossRef]

Phys. Rev. Lett. (2)

R. Terhune, P. Maker, and C. Savage, "Measurements of nonlinear light scattering," Phys. Rev. Lett. 14, 681-684 (1965).
[CrossRef]

G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, "Determination of hyperpolarizability measurements by depolarized hyper-Rayleigh scattering," Phys. Rev. Lett. 71, 999-1002 (1993).
[CrossRef] [PubMed]

Other (3)

E. Issacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).

G. Golub and C. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

J. Delmas, Introduction aux Probabilités (Ed. Ellipses, 1993).

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Figures (7)

Fig. 1
Fig. 1

HRS polarimeter setup. α ( α ) is the angle between the fast axis, f ( f ) , of the first (second) quarter-wave plate and x axis, and Γ is the scattering angle.

Fig. 2
Fig. 2

Condition number of [ A ] with six evenly spaced angular increments and a detector angle of 45°. Minimum condition number is 13.5 for a generator angular step equal to 69.2° and an analyzer angular step of 42.7° (∗).

Fig. 3
Fig. 3

Absolute minimum condition number of [ A ] versus scattering angle with six evenly-spaced angular increments.

Fig. 4
Fig. 4

Absolute minimum condition number of [ A ] versus scattering angle with 64 evenly spaced angular increments.

Fig. 5
Fig. 5

σ β 2 β 1 s s 2 versus condition number for pNA (•, 6 measurements; +, 64 measurements). Each dot corresponds to 64 6 × 5000 realizations (6 measurements) and 5000 realizations (64 measurements).

Fig. 6
Fig. 6

σ β 2 β 1 s s 2 versus condition number for CV (•, 6 measurements; +, 64 measurements). Each dot corresponds to 64 6 × 5000 realizations (6 measurements) and 5000 realizations (64 measurements).

Fig. 7
Fig. 7

Condition number of [ A ] with 64 evenly spaced angular increments for Γ = 75 ° . Minimum condition number is 8.7 for a generator angular step of 80.6° and an analyzer angular step of 45.2° (∗). White regions plotted on the map correspond to condition numbers in the ranges (a) 8.7–11.7 (CV) and (b) 8.7–24.7 (pNA), these maxima were found at σ β 2 β 1 s s 2 = 0.25 for β 1 s m 2 .

Tables (3)

Tables Icon

Table 1 Rotational Invariants Measured by Ostroverkov et al.[10]

Tables Icon

Table 2 σ β 2 β 1 s s 2 for Each Rotational Invariant at 1064 nm for pN A

Tables Icon

Table 3 σ β 2 β 1 s s 2 of Each Rotational Invariant at 1064 nm for CV

Equations (9)

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

[ I ( β 2 , Γ , α , α ) ] = [ A ( Γ , α , α ) ] [ β 2 ] ,
[ β 2 ] = [ A ( Γ , α , α ) ] 1 [ I ( β 2 , Γ , α , α ) ] .
[ β 2 ] = ( [ A ] T [ A ] ) 1 [ A ] T [ I ] = [ A p ( Γ , α , α ) 1 ] [ I ( β 2 , Γ , α , α ) ] .
Δ β 2 β 2 Cond ( A ) 1 Δ A A Cond ( A ) ( Δ I I + Δ A A ) ,
Cond ( A ) = A A 1 ,
A 2 = sup x 0 A x 2 x 2 ,
[ Cov ( β 2 ) ] = [ A p 1 ] [ Cov ( I ) ] [ A p 1 ] T .
Var ( β i 2 ) = j Var ( I ) j ( A p 1 ) i j 2 .
Var ( β i 2 ) = Var ( I ) ( A p 1 ) i 2 2 ,

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