Abstract

A dual-rotating-retarder polarimeter was used to determine the six measurable observables of the first hyperpolarizability tensor. Calibration of such an instrument requires a reference sample dedicated to wavelength conversion. We calibrated our experimental setup by using a quartz-plate sample in a two step procedure: at first the first retarder then the second one. The retardance and ellipticity angle of both retarders were estimated by minimizing a χ2 function. We estimated the standard deviation of each parameter from noise spreading and performed this calibration procedure for two experimental case studies, i.e., two angular positions of the quartz sample.

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References

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    [CrossRef]
  2. J. Giordmaine, "Nonlinear optical properties of liquid," Phys. Rev. 138, A1599-A1606 (1965).
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  3. V. Ostroverkhov, R. G. Petschek, K. D. Singer, L. Sukhomlinova, R. J. Twieg, S.-X. Wang, and L. C. Chien, "Measurements of the hyperpolarizability tensor by means of hyper-Rayleigh scattering," J. Opt. Soc. Am. B 17, 1531-1542 (2000).
    [CrossRef]
  4. M. Kauranen and A. Persoons, "Theory of polarization measurements of second-order non-linear light scattering," J. Chem. Phys. 104, 3445-3456 (1996).
    [CrossRef]
  5. S. F. Hubbard, R. G. Petschek, K. D. Singer, N. D'Sidocky, C. Hudson, L. C. Chien, C. C. Henderson, and P. A. Cahill, "Measurements of Kleinman-disallowed hyperpolarisability in conjugated chiral molecules," J. Opt. Soc. Am. B 15, 289-301 (1998).
    [CrossRef]
  6. B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
    [CrossRef]
  7. B. Le Jeune, J. Marie, P. Gerliguand, J. Cariou, and J. Lotrian, "Mueller matrix formalism in imagery: an experimental arrangement for noise reduction," Proc. SPIE 2265, 443-451 (1994).
    [CrossRef]
  8. B. Boulbry, B. Bousquet, B. Le Jeune, Y. Guern, and J. Lotrian, "Polarization errors associated with zero-order achromatic quarter-wave plates in the whole visible spectral range," Opt. Express 9, 225-235 (2001).
    [CrossRef] [PubMed]
  9. H. Rabin and P. P. Bey, "Phase matching in harmonic generation employing optical rotatory dispersion," Phys. Rev. 156, 1010-1016 (1967).
    [CrossRef]
  10. R. Boyd, Nonlinear Optics (Academic, 1992).
  11. H. J. Simon and N. Bloembergen, "Second-harmonic light generation in crystals with natural optical activity," Phys. Rev. 171, 1104-1114 (1968).
    [CrossRef]
  12. J. Delmas, Introduction aux Probabilités (Ellipses, 1993).
  13. J. Nougier, ,i.Méthodes de Calcul Numérique (Masson, 1981).
  14. P. Lemaillet, F. Pellen, S. Rivet, B. Le Jeune, and J. Cariou, "Optimization of a dual-rotating-retarder polarimeter designed for hyper-Rayleigh scattering," J. Opt. Soc. Am. B 24, 609-614 (2007).
    [CrossRef]
  15. G. Golub and C. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins U. Press, 1996).
  16. W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, 1992).
  17. G. Ghosh, "Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals," Opt. Commun. 163, 95-102 (1999).
    [CrossRef]
  18. L. I. Katzin, "The rotatory dispersion of quartz," J. Phys. Chem. 68, 2367-2370 (1964).
    [CrossRef]
  19. T. Bürer and L. I. Katzin, "Dipole strengths and rotational strengths from dispersion for quartz," J. Inorg. Nucl. Chem. 29, 2715-2722 (1967).
    [CrossRef]

2007 (1)

2002 (1)

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

G. Ghosh, "Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals," Opt. Commun. 163, 95-102 (1999).
[CrossRef]

1998 (1)

1996 (1)

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

1994 (2)

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

B. Le Jeune, J. Marie, P. Gerliguand, J. Cariou, and J. Lotrian, "Mueller matrix formalism in imagery: an experimental arrangement for noise reduction," Proc. SPIE 2265, 443-451 (1994).
[CrossRef]

1968 (1)

H. J. Simon and N. Bloembergen, "Second-harmonic light generation in crystals with natural optical activity," Phys. Rev. 171, 1104-1114 (1968).
[CrossRef]

1967 (2)

T. Bürer and L. I. Katzin, "Dipole strengths and rotational strengths from dispersion for quartz," J. Inorg. Nucl. Chem. 29, 2715-2722 (1967).
[CrossRef]

H. Rabin and P. P. Bey, "Phase matching in harmonic generation employing optical rotatory dispersion," Phys. Rev. 156, 1010-1016 (1967).
[CrossRef]

1965 (1)

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

1964 (1)

L. I. Katzin, "The rotatory dispersion of quartz," J. Phys. Chem. 68, 2367-2370 (1964).
[CrossRef]

Chem. Rev. (1)

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

J. Chem. Phys. (1)

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

J. Inorg. Nucl. Chem. (1)

T. Bürer and L. I. Katzin, "Dipole strengths and rotational strengths from dispersion for quartz," J. Inorg. Nucl. Chem. 29, 2715-2722 (1967).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. Chem. (1)

L. I. Katzin, "The rotatory dispersion of quartz," J. Phys. Chem. 68, 2367-2370 (1964).
[CrossRef]

Meas. Sci. Technol. (1)

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Opt. Commun. (1)

G. Ghosh, "Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals," Opt. Commun. 163, 95-102 (1999).
[CrossRef]

Opt. Express (1)

Phys. Rev. (3)

H. Rabin and P. P. Bey, "Phase matching in harmonic generation employing optical rotatory dispersion," Phys. Rev. 156, 1010-1016 (1967).
[CrossRef]

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

H. J. Simon and N. Bloembergen, "Second-harmonic light generation in crystals with natural optical activity," Phys. Rev. 171, 1104-1114 (1968).
[CrossRef]

Proc. SPIE (1)

B. Le Jeune, J. Marie, P. Gerliguand, J. Cariou, and J. Lotrian, "Mueller matrix formalism in imagery: an experimental arrangement for noise reduction," Proc. SPIE 2265, 443-451 (1994).
[CrossRef]

Other (5)

R. Boyd, Nonlinear Optics (Academic, 1992).

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

J. Nougier, ,i.Méthodes de Calcul Numérique (Masson, 1981).

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

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, 1992).

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Figures (7)

Fig. 1
Fig. 1

SHG polarimeter setup. L1 and L2 are the first and second quarter-wave plates. α i ( α o ) is the angle between the fast axis, f i ( f o ) , of the first (second) quarter-wave plate and X axis; α p is the angle of the rotating polarizer, P2.

Fig. 2
Fig. 2

Propagation of a Y linearly polarized incident wave E ω ( 0 ) through a rotatory dispersive α quartz of angular orientation θ. The incident wave is subject to a specific rotation, ρ ω , and generates a SHG wave, E 2 ω ( z ) , with an angular orientation, α 2 ω , that is dependent on the orientation of E ω ( z ) . E 2 ω ( z ) travels a distance ez and is subject to a specific rotation, ρ 2 ω .

Fig. 3
Fig. 3

| a | 2 and | b | 2 computed as a function of the quartz-plate length, e, for 1265   nm .

Fig. 4
Fig. 4

Experimental setup for SHG-polarimeter calibration by means of a quartz plate: P1 and P3: Glan–Taylor linear polarizers; P2: glass sheet linear polarizer; L1 and L2: achromatic quarter-wave plates; L3: achromatic half-wave plate; BS: beam splitter; l1: converging lense; l2: diverging lense; QP: quartz plate; F1, F2, and F3: filters; D: photodiode; PMT: photomultiplier tube.

Fig. 5
Fig. 5

Variations of (a) A 2 α p and (b) B 2 α p versus the angular position of the first quarter-wave plate when θ = θ 1 .

Fig. 6
Fig. 6

Variations of (a) A 2 α p and (b) B 2 α p versus the angular position of the second quarter-wave plate when θ = θ 1 .

Fig. 7
Fig. 7

Variations of retardance of the second quarter-wave plate versus wavelength as presented by Boulbry et al. [6].

Tables (5)

Tables Icon

Table 1 Correspondence Table between l Index and ( k , n ) Indices Related to the Calibration of the First Quarter-Wave Plate

Tables Icon

Table 2 Correspondence Table between l Index and ( k , n ) Indices Related to the Calibration of the Second Quarter-Wave Plate

Tables Icon

Table 3 Mean Values and Standard Deviations of the Quartz Parameters, ( a , b ), issued for 20 Identical Measurements for Two Angular Positions, θ1 and θ2, of the Quartz Plate (arb. units)

Tables Icon

Table 4 Retardance, δ, and Ellipticity Angle, ϵ, along with Their Standard Deviations at the Two Angular Positions of the Quartz Plate (θ1 and θ2) a

Tables Icon

Table 5 Coefficient of the Dispersion Equation for the Ordinary Refractive Index of α Quartz at Room Temperature by Ghosh a (λ = 0.2–2 μm)

Equations (63)

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E 2 ω x d 11 [ E ω x 2 E ω y 2 ] 2 d 14 E ω x E ω z ,
E 2 ω y 2 d 14 E ω x E ω z 2 d 11 E ω x E ω y ,
[ E ω x E ω y ] = [ R ( θ ) ] [ E ω X E ω Y ] = [ cos θ sin θ sin θ cos θ ] [ E ω X E ω Y ] .
E 2 ω x ( z ) d 11 { cos 2 θ [ E ω X ( z ) 2 E ω Y ( z ) 2 ] + 2 sin 2 θ E ω X ( z ) E ω Y ( z ) } ,
E 2 ω y ( z ) d 11 { sin 2 θ [ E ω X ( z ) 2 E ω Y ( z ) 2 ] 2 cos 2 θ E ω X ( z ) E ω Y ( z ) } .
E 2 ω X ( z ) d 11 { cos 3 θ [ E ω X ( z ) 2 E ω Y ( z ) 2 ] + 2 sin 3 θ E ω X ( z ) E ω Y ( z ) } ,
E 2 ω Y ( z ) d 11 { sin 3 θ [ E ω X ( z ) 2 E ω Y ( z ) 2 ] 2 cos 3 θ E ω X ( z ) E ω Y ( z ) } .
d E 2 ω X ( e , z ) d 11 exp ( i 4 π λ n 2 ω o e ) exp [ i 4 π λ ( n ω o n 2 ω o ) z ] × { cos [ 3 θ ρ 2 ω e + ( ρ 2 ω + 2 ρ ω ) z ] [ E ω X ( 0 ) 2 E ω Y ( 0 ) 2 ] + 2 sin [ 3 θ ρ 2 ω e + ( ρ 2 ω + 2 ρ ω ) z ] × E ω X ( 0 ) E ω Y ( 0 ) } d z ,
d E 2 ω Y ( e , z ) d 11 exp ( i 4 π λ n 2 ω o e ) exp [ i 4 π λ ( n ω o n 2 ω o ) z ] × { sin [ 3 θ ρ 2 ω e + ( ρ 2 ω + 2 ρ ω ) z ] [ E ω X ( 0 ) 2 E ω Y ( 0 ) 2 ] 2 cos [ 3 θ ρ 2 ω e + ( ρ 2 ω + 2 ρ ω ) z ] × E ω X ( 0 ) E ω Y ( 0 ) } d z ,
E 2 ω X ( e ) = 0 e d E 2 ω X ( e , z ) d 11 exp [ i 4 π λ n 2 ω o e ] × { a [ E ω X ( 0 ) i E ω Y ( 0 ) ] 2 + b [ E ω X ( 0 ) + i E ω Y ( 0 ) ] 2 } ,
E 2 ω Y ( e ) = 0 e d E 2 ω Y ( e , z ) i d 11 exp [ i 4 π λ n 2 ω o e ] × { a [ E ω X ( 0 ) i E ω Y ( 0 ) ] 2 b [ E ω X ( 0 ) + i E ω Y ( 0 ) ] 2 } ,
E 2 ω X ( e ) + i E 2 ω Y ( e ) d 11 exp [ i 4 π λ n 2 ω o e ] × 2 a [ E ω X ( 0 ) i E ω Y ( 0 ) ] 2 ,
E 2 ω X ( e ) i E 2 ω Y ( e ) d 11 exp [ i 4 π λ n 2 ω o e ] × 2 b [ E ω X ( 0 ) + i E ω Y ( 0 ) ] 2 .
I = | E 2 ω X | 2 + | E 2 ω Y | 2 = A 0 + A 2 α p cos 2 α p + B 2 α p sin 2 α p = 1 2 { [ | E 2 ω X ( e ) | 2 + | E 2 ω Y ( e ) | 2 ] + [ | E 2 ω X ( e ) | 2 | E 2 ω Y ( e ) | 2 ] cos 2 α p + sin 2 α p [ E 2 ω X ( e ) E 2 ω Y * ( e ) + E 2 ω X * ( e ) E 2 ω Y ( e ) ] } ,
A 0 = ( | a | 2 + | b | 2 ) d 11 2 | E ω Y | 4 ,
A 2 α p = ( a b * + a * b ) d 11 2 | E ω Y | 4 ,
B 2 α p = i ( a b * a * b ) d 11 2 | E ω Y | 4 .
J i = [ A i * B i * B i A i ] ,
A i = cos δ i 2 + i sin δ i 2 cos 2 ϵ i cos 2 α i ,
B i = sin δ i 2 ( sin 2 ϵ i i cos 2 ϵ i sin 2 α i ) ,
[ E ω X ( 0 ) E ω Y ( 0 ) ] = [ A i * B i * B i A i ] [ 0 E ω Y ] = [ B i * E ω Y A i E ω Y ] .
I ( α i , α p , δ i , ϵ i , a , b ) = A 0 + A 2 α p cos 2 α p + B 2 α p sin 2 α p = [ n = 0 , 2 , 4 0 , n ( δ i , ϵ i , a , b ) cos ( n α i ) + 1 , n ( δ i , ϵ i , a , b ) sin ( n α i ) ] + [ n = 0 , 4 , 8 2 , n ( δ i , ϵ i , a , b ) cos ( n α i ) + 3 , n ( δ i , ϵ i , a , b ) sin ( n α i ) ] cos ( 2 α p ) + [ n = 0 , 4 , 8 4 , n ( δ i , ϵ i , a , b ) cos ( n α i ) + 5 , n ( δ i , ϵ i , a , b ) sin ( n α i ) ] sin ( 2 α p ) ,
[ I ( α i , α p , δ i , ϵ i , a , b ) ] = [ M ( α i , α p ) ] [ ( δ i , ϵ i , a , b ) ] .
[ ( δ i , ϵ i , a , b ) ] = ( [ M ] T [ M ] ) 1 [ M ] T [ I ] = [ M p ( α i , α p ) 1 ] [ I ( α i , α p , δ i , ϵ i , a , b ) ] .
χ 2 = l = 1 15 [ l m e a s l ( δ i , ϵ i , a , b ) σ s t a t ( l m e a s ) ] 2 .
[ Cov s t a t ( ) ] = [ M p 1 ] [ Cov s t a t ( I ) ] [ M p 1 ] T .
σ t o t ( I l ) | I l m e a s I l c a l c | ,
[ Cov ( δ i , ϵ i ) ] = ( [ Jac ] T [ Jac ] ) 1 [ Jac ] T [ Cov t o t ( ) ] [ Jac ] × ( [ Jac ] T [ Jac ] ) 1 ,
Q ( χ 2 | ν ) = Q ( ν 2 , χ 2 2 ) = Γ ( ν 2 , χ 2 2 ) Γ ( ν 2 ) ,
[ E 2 ω X E 2 ω Y ] = [ A o * B o * B o A o ] [ E 2 ω X ( e ) E 2 ω Y ( e ) ] = [ A o * E 2 ω X ( e ) B o * E 2 ω Y ( e ) B o E 2 ω X ( e ) + A s E 2 ω Y ( e ) ] .
I ( α i , α o , α p , δ i , ϵ i , δ o , ϵ o , a , b ) = A 0 + A 2 α p cos 2 α p + B 2 α p sin 2 α p = G 0 ( α i , δ i , ϵ i , a , b ) + n = 0 , 2 , 4 [ G 1 , n ( α i , δ i , ϵ i , δ o , ϵ o , a , b ) cos ( n α o ) + G 2 , n ( α i , δ i , ϵ i , δ o , ϵ o , a , b ) sin ( n α o ) ] × cos ( 2 α p ) + n = 0 , 2 , 4 [ G 3 , n ( α i , δ i , ϵ i , δ o , ϵ o , a , b ) cos ( n α o ) + G 4 , n ( α i , δ i , ϵ i , δ o , ϵ o , a , b ) sin ( n α o ) ] × sin ( 2 α p ) ,
a = exp ( i A 1 ) 1 B + A 2 exp [ i ( B + A 2 ) e 2 ] sin [ ( B + A 2 ) e 2 ] ,
b = exp ( i A 1 ) 1 B A 2 exp [ i ( B A 2 ) e 2 ] × sin [ ( B A 2 ) e 2 ] ,
n o , e = A o , e + B o , e λ 2 λ 2 C o , e + D o , e λ 2 λ 2 F o , e ,
ρ ( λ ) = 127.02476 λ 2 0.0979 2 119.77145 λ 2 0.0958 2 0.1879 ,
1 ( δ i , ϵ i ) = K ( | a | 2 + | b | 2 ) [ 1 + 2 sin 2 δ i 2 cos 2 2 ϵ i × ( sin 2 δ i 2 sin 2 2 ϵ i + cos 2 δ i 2 ) ] ,
2 ( δ i , ϵ i ) = 4 K ( | a | 2 | b | 2 ) sin δ i 2 cos 2 ϵ i sin δ i 2 × sin 2 ϵ i ,
3 ( δ i , ϵ i ) = 4 K ( | a | 2 | b | 2 ) sin δ i 2 cos 2 ϵ i cos δ i 2 ,
4 ( δ i , ϵ i ) = 2 K ( | a | 2 + | b | 2 ) sin 2 δ i 2 cos 2 2 ϵ i × ( sin 2 δ i 2 sin 2 2 ϵ i cos 2 δ i 2 ) ,
5 ( δ i , ϵ i ) = K ( | a | 2 + | b | 2 ) sin 2 δ i 2 cos 2 2 ϵ i sin δ i × sin 2 ϵ i ,
6 ( δ i , ϵ i ) = K { ( a b * + a * b ) ( cos 2 δ i + 2 cos δ i sin 2 δ i 2 × cos 2 2 ϵ i + sin 4 δ i 2 cos 4 2 ϵ i sin 2 δ i sin 2 2 ϵ i ) 2 i ( a b * a * b ) [ sin δ i sin 2 ϵ i ( cos δ i + sin 2 δ i 2 cos 2 2 ϵ i ) ] } ,
7 ( δ i , ϵ i ) = 2 K { ( a b * + a * b ) [ sin 2 δ i 2 cos 2 2 ϵ i ( cos δ i + sin 2 δ i 2 cos 2 2 ϵ i ) ] i ( a b * a * b ) × ( sin δ i sin 2 ϵ i sin 2 δ i 2 cos 2 2 ϵ i ) } ,
8 ( δ i , ϵ i ) = 2 K { ( a b * + a * b ) × ( sin δ i sin 2 δ i 2 sin 2 ϵ i cos 2 2 ϵ i ) + i ( a b * a * b ) ( cos δ i sin 2 δ i 2 cos 2 2 ϵ i + sin 4 δ i 2 cos 4 2 ϵ i ) } ,
9 ( δ i , ϵ i ) = K ( a b * + a * b ) ( sin 4 δ i 2 cos 4 2 ϵ i ) ,
10 ( δ i , ϵ i ) = i K ( a b * a * b ) ( sin 4 δ i 2 cos 4 2 ϵ i ) .
11 ( δ i , ϵ i ) = i K { ( a b * a * b ) ( cos 2 δ i + 2 cos δ i sin 2 δ i 2 cos 2 2 ϵ i + sin 4 δ i 2 cos 4 2 ϵ i sin 2 δ i sin 2 2 ϵ i ) 2 i ( a b * + a * b ) [ sin δ i sin 2 ϵ i ( cos δ i + sin 2 δ i 2 cos 2 2 ϵ i ) ] } ,
12 ( δ i , ϵ i ) = 2 i K { ( a b * a * b ) × [ sin 2 δ i 2 cos 2 2 ϵ i ( cos δ i + sin 2 δ i 2 cos 2 2 ϵ i ) ] i ( a b * + a * b ) × ( sin δ i sin 2 ϵ i sin 2 δ i 2 cos 2 2 ϵ i ) } ,
13 ( δ i , ϵ i ) = 2 i K { ( a b * a * b ) × ( sin δ i sin 2 δ i 2 sin 2 ϵ i cos 2 2 ϵ i ) + i ( a b * + a * b ) ( cos δ i sin 2 δ i 2 cos 2 2 ϵ i + sin 4 δ i 2 cos 4 2 ϵ i ) } ,
14 ( δ i , ϵ i ) = i K ( a b * a * b ) ( sin 4 δ i 2 cos 4 2 ϵ i ) ,
15 ( δ i , ϵ i ) = K ( a b * + a * b ) ( sin 4 δ i 2 cos 4 2 ϵ i ) .
a i = a ( B i * + i A i ) 2 ,
b i = b ( B i * + i A i ) 2 ,
G 0 ( α i , δ i , ϵ i ) = K ( | a i | 2 + | b i | 2 ) ,
G 1 ( α i , δ i , ϵ i , δ o , ϵ o ) = K [ ( a i b i * + a i * b i ) × ( cos 2 δ o 2 sin 2 δ o 2 sin 2 2 ϵ o ) + i ( a i b i * a i * b i ) sin δ o sin 2 ϵ o ] ,
G 2 ( α i , δ i , ϵ i , δ o , ϵ o ) = K ( | a i | 2 | b i | 2 ) sin 2 δ o 2 sin 4 ϵ o ,
G 3 ( α i , δ i , ϵ i , δ o , ϵ o ) = K ( | a i | 2 | b i | 2 ) sin δ o cos 2 ϵ o ,
G 4 ( α i , δ i , ϵ i , δ o , ϵ o ) = K ( a i b i * + a i * b i ) sin 2 δ o 2 cos 2 2 ϵ o ,
G 5 ( α i , δ i , ϵ i , δ o , ϵ o ) = i K ( a i b i * a i * b i ) sin 2 δ o 2 × cos 2 2 ϵ o .
G 6 ( α i , δ i , ϵ i , δ o , ϵ o ) = K [ ( a i b i * + a i * b i ) sin δ o sin 2 ϵ o i ( a i b i * a i * b i ) ( cos 2 δ o 2 sin 2 δ o 2 sin 2 2 ϵ o ) ] ,
G 7 ( α i , δ i , ϵ i , δ o , ϵ o ) = K ( | a i | 2 | b i | 2 ) sin δ o cos 2 ϵ o ,
G 8 ( α i , δ i , ϵ i , δ o , ϵ o ) = K ( | a i | 2 | b i | 2 ) sin 2 δ o 2 sin 4 ϵ o ,
G 9 ( α i , δ i , ϵ i , δ o , ϵ o ) = i K ( a i b i * a i * b i ) sin 2 δ o 2 cos 2 2 ϵ o ,
G 10 ( α i , δ i , ϵ i , δ o , ϵ o ) = K ( a i b i * + a i * b i ) sin 2 δ o 2 cos 2 2 ϵ o .

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