Abstract

A general analysis and experimental validation of transmission wavefront shearing interferometry for photoelastic materials are presented. These interferometers applied to optically isotropic materials produce a single interference pattern related to one phase term, but when applied to photoelastic materials, they produce the sum of two different interference patterns with phase terms that are the sum and difference, respectively, of two stress-related phase terms. The two stress-related phase terms may be separated using phase shifting and polarization optics. These concepts are experimentally demonstrated using coherent gradient sensing in full field for a compressed polycarbonate plate with a V-shaped notch with good agreement with theoretical data. The analysis may be applied to any wavefront shearing interferometer by modifying parameters describing the wavefront shearing distance.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. H. Tippur, S. Krishnaswamy, and A. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fracture 48, 193-204 (1991).
    [CrossRef]
  4. H. Tippur, S. Krishnaswamy, and A. Rosakis, “Optical mapping of crack tip deformations using the methods of trasmission and reflection coherent gradient sensing: a study of crack tip k-dominance,” Int. J. Fracture 52, 91-117 (1991).
  5. A. J. Rosakis, “Optical techniques sensitve to gradients of optical path difference: the method of caustics and the coherent gradient sensor (CGS),” in Experimental Techniques in Fracture, J. S. Epstein, ed. (Wiley, 1993), pp. 327-425.
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    [CrossRef]
  11. C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2003

T. Park, S. Suresh, A. Rosakis, and J. Ryu, “Measurement of full-field curvatue and geometrical instability of thin film-substrate systems through cgs interferometry,” J. Mech. Phys. Solids 51, 2191-2211 (2003).
[CrossRef]

2002

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Laser Eng. 37, 313-330(2002).
[CrossRef]

2001

H. Lee, A. Rosakis, and L. Freund, “Full-field optical measurement of curvatures in unltra-thin-film-substrate systems in the range of geometrically nonlinear deformations,” J. Appl. Phys. 89, 6116-6129 (2001).
[CrossRef]

1998

C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
[CrossRef]

K. Shimizu, M. Suetsugu, T. Nakamura, and S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. 41, 134-141 (1998).

1994

1991

H. Tippur, S. Krishnaswamy, and A. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fracture 48, 193-204 (1991).
[CrossRef]

H. Tippur, S. Krishnaswamy, and A. Rosakis, “Optical mapping of crack tip deformations using the methods of trasmission and reflection coherent gradient sensing: a study of crack tip k-dominance,” Int. J. Fracture 52, 91-117 (1991).

1964

1952

M. Williams, “Stress singularities resulting from various boundary conditions in angular corners of plates in extension,” J. Appl. Mech. 19, 526-528 (1952).

Baldi, A.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Laser Eng. 37, 313-330(2002).
[CrossRef]

Bertolino, F.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Laser Eng. 37, 313-330(2002).
[CrossRef]

Coker, E.

E. Coker and L. Filon, A Treatise on Photo-elasticity (Cambridge U. Press, 1993).

Ellis, R.

C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
[CrossRef]

Filon, L.

E. Coker and L. Filon, A Treatise on Photo-elasticity (Cambridge U. Press, 1993).

Freund, L.

H. Lee, A. Rosakis, and L. Freund, “Full-field optical measurement of curvatures in unltra-thin-film-substrate systems in the range of geometrically nonlinear deformations,” J. Appl. Phys. 89, 6116-6129 (2001).
[CrossRef]

Frocht, M.

M. Frocht, Photoelasticity (Wiley, 1941), Vol. 1.

Ghiglia, D.

Ginesu, F.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Laser Eng. 37, 313-330(2002).
[CrossRef]

Krishnaswamy, S.

H. Tippur, S. Krishnaswamy, and A. Rosakis, “Optical mapping of crack tip deformations using the methods of trasmission and reflection coherent gradient sensing: a study of crack tip k-dominance,” Int. J. Fracture 52, 91-117 (1991).

H. Tippur, S. Krishnaswamy, and A. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fracture 48, 193-204 (1991).
[CrossRef]

S. Krishnaswamy, “Photomechanics,” in Techniques for Non-Birefringnet Objects: Coherent Shearing Interferometry and Caustics, Vol. 77 of Topics in Applied Physics (Springer-Verlag, 2000), pp. 295-321.

Lee, H.

H. Lee, A. Rosakis, and L. Freund, “Full-field optical measurement of curvatures in unltra-thin-film-substrate systems in the range of geometrically nonlinear deformations,” J. Appl. Phys. 89, 6116-6129 (2001).
[CrossRef]

Liu, C.

C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
[CrossRef]

Murty, M.

Nakamura, T.

K. Shimizu, M. Suetsugu, T. Nakamura, and S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. 41, 134-141 (1998).

Narasimhamurty, T.

T. Narasimhamurty, Photoelastic and Electro-optic Properties of Crystals (Plenum, 1981).

Papadopoulos, G. A.

G. A. Papadopoulos, Fracture Mechanics: the Experimental Method of Caustics and the Det. Criterion of Fracture (Springer-Verlag, 1993).

Park, T.

T. Park, S. Suresh, A. Rosakis, and J. Ryu, “Measurement of full-field curvatue and geometrical instability of thin film-substrate systems through cgs interferometry,” J. Mech. Phys. Solids 51, 2191-2211 (2003).
[CrossRef]

Romero, L.

Rosakis, A.

T. Park, S. Suresh, A. Rosakis, and J. Ryu, “Measurement of full-field curvatue and geometrical instability of thin film-substrate systems through cgs interferometry,” J. Mech. Phys. Solids 51, 2191-2211 (2003).
[CrossRef]

H. Lee, A. Rosakis, and L. Freund, “Full-field optical measurement of curvatures in unltra-thin-film-substrate systems in the range of geometrically nonlinear deformations,” J. Appl. Phys. 89, 6116-6129 (2001).
[CrossRef]

C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
[CrossRef]

H. Tippur, S. Krishnaswamy, and A. Rosakis, “Optical mapping of crack tip deformations using the methods of trasmission and reflection coherent gradient sensing: a study of crack tip k-dominance,” Int. J. Fracture 52, 91-117 (1991).

H. Tippur, S. Krishnaswamy, and A. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fracture 48, 193-204 (1991).
[CrossRef]

Rosakis, A. J.

A. J. Rosakis, “Optical techniques sensitve to gradients of optical path difference: the method of caustics and the coherent gradient sensor (CGS),” in Experimental Techniques in Fracture, J. S. Epstein, ed. (Wiley, 1993), pp. 327-425.

Ryu, J.

T. Park, S. Suresh, A. Rosakis, and J. Ryu, “Measurement of full-field curvatue and geometrical instability of thin film-substrate systems through cgs interferometry,” J. Mech. Phys. Solids 51, 2191-2211 (2003).
[CrossRef]

Shimizu, K.

K. Shimizu, M. Suetsugu, T. Nakamura, and S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. 41, 134-141 (1998).

Stout, M.

C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
[CrossRef]

Suetsugu, M.

K. Shimizu, M. Suetsugu, T. Nakamura, and S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. 41, 134-141 (1998).

Suresh, S.

T. Park, S. Suresh, A. Rosakis, and J. Ryu, “Measurement of full-field curvatue and geometrical instability of thin film-substrate systems through cgs interferometry,” J. Mech. Phys. Solids 51, 2191-2211 (2003).
[CrossRef]

Takahashi, S.

K. Shimizu, M. Suetsugu, T. Nakamura, and S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. 41, 134-141 (1998).

Tippur, H.

H. Tippur, S. Krishnaswamy, and A. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fracture 48, 193-204 (1991).
[CrossRef]

H. Tippur, S. Krishnaswamy, and A. Rosakis, “Optical mapping of crack tip deformations using the methods of trasmission and reflection coherent gradient sensing: a study of crack tip k-dominance,” Int. J. Fracture 52, 91-117 (1991).

Williams, M.

M. Williams, “Stress singularities resulting from various boundary conditions in angular corners of plates in extension,” J. Appl. Mech. 19, 526-528 (1952).

Appl. Opt.

Int. J. Fracture

H. Tippur, S. Krishnaswamy, and A. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fracture 48, 193-204 (1991).
[CrossRef]

H. Tippur, S. Krishnaswamy, and A. Rosakis, “Optical mapping of crack tip deformations using the methods of trasmission and reflection coherent gradient sensing: a study of crack tip k-dominance,” Int. J. Fracture 52, 91-117 (1991).

C. Liu, A. Rosakis, R. Ellis, and M. Stout, “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” Int. J. Fracture 90, 355-382 (1998).
[CrossRef]

J. Appl. Mech.

M. Williams, “Stress singularities resulting from various boundary conditions in angular corners of plates in extension,” J. Appl. Mech. 19, 526-528 (1952).

J. Appl. Phys.

H. Lee, A. Rosakis, and L. Freund, “Full-field optical measurement of curvatures in unltra-thin-film-substrate systems in the range of geometrically nonlinear deformations,” J. Appl. Phys. 89, 6116-6129 (2001).
[CrossRef]

J. Mech. Phys. Solids

T. Park, S. Suresh, A. Rosakis, and J. Ryu, “Measurement of full-field curvatue and geometrical instability of thin film-substrate systems through cgs interferometry,” J. Mech. Phys. Solids 51, 2191-2211 (2003).
[CrossRef]

J. Opt. Soc. Am. A

JSME Int. J.

K. Shimizu, M. Suetsugu, T. Nakamura, and S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. 41, 134-141 (1998).

Opt. Laser Eng.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Laser Eng. 37, 313-330(2002).
[CrossRef]

Other

E. Coker and L. Filon, A Treatise on Photo-elasticity (Cambridge U. Press, 1993).

M. Frocht, Photoelasticity (Wiley, 1941), Vol. 1.

T. Narasimhamurty, Photoelastic and Electro-optic Properties of Crystals (Plenum, 1981).

A. J. Rosakis, “Optical techniques sensitve to gradients of optical path difference: the method of caustics and the coherent gradient sensor (CGS),” in Experimental Techniques in Fracture, J. S. Epstein, ed. (Wiley, 1993), pp. 327-425.

S. Krishnaswamy, “Photomechanics,” in Techniques for Non-Birefringnet Objects: Coherent Shearing Interferometry and Caustics, Vol. 77 of Topics in Applied Physics (Springer-Verlag, 2000), pp. 295-321.

G. A. Papadopoulos, Fracture Mechanics: the Experimental Method of Caustics and the Det. Criterion of Fracture (Springer-Verlag, 1993).

A.Kobayashi, ed., Handbook of Experimental Mechanics (Wiley, 1993).

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Figures (9)

Fig. 1
Fig. 1

Working principle for horizontal shearing transmission coherent gradient sensing.

Fig. 2
Fig. 2

Polarization optics before the transparent sample: two configurations with either a λ / 4 or λ / 2 plate before the sample.

Fig. 3
Fig. 3

Schematic of a compressed polycarbonate plate with a side V-notch.

Fig. 4
Fig. 4

Experimental and theoretical images for horizontal shear with good comparison: (a) experimental I E x = I o E x + I c E x cos [ φ sum + φ α d ] , (b) theoretical I E x = I o E x + I c E x cos [ φ sum + φ α d ] , (c) experimental I E y = I o E y + I c E y cos [ φ sum φ α d ] , (d) theoretical I E y = I o E y + I c E y cos [ φ sum φ α d ] , (e) experimental I circ = I o circ + I o circ cos [ φ diff ] cos [ φ sum ] , and (f) theoretical I circ = I o circ + I o circ cos [ φ diff ] cos [ φ sum ] (note: the V-notch region is masked in white in the theoretical images).

Fig. 5
Fig. 5

Experimental and theoretical wrapped phase maps (in radians) with the V-notch masked in white: (a) experimental φ E x = φ sum + φ α d , (b) theoretical φ E x = φ sum + φ α d , (c) experimental φ E y = φ sum - φ α d , and (d) theoretical φ E y = φ sum - φ α d .

Fig. 6
Fig. 6

Wrapped phase maps from λ / 4 plate method (in radians) with the V-notch masked in white: (a) experimental φ sum for cos ( φ diff ) 0 , (b) theoretical φ sum , and (c) theoretical cos ( φ diff ) field with its four-lobed clover leaf pattern.

Fig. 7
Fig. 7

Experimental and theoretical unwrapped phase term from the pure E x î and pure E y ĵ fields (in radians) with the V-notch masked in white: (a) experimental φ E x , (b) theoretical φ E x , (c) experimental φ E y , and (d) theoretical φ E y .

Fig. 8
Fig. 8

Experimental and theoretical phase maps of φ sum and φ diff (in radians) with the V-notch masked in white: (a) experimental φ sum = ( φ E x + φ E y ) / 2 , (b) experimental φ sum from the λ / 4 method, (c) theoretical φ sum , (d) experimental φ α d = ( φ E x - φ E y ) / 2 , and (e) theoretical φ α d .

Fig. 9
Fig. 9

Difference between theoretical and experimental φ sum and φ α d (in radians) with the V-notch masked in white: (a) comparison for φ sum = ( φ E x + φ E y ) / 2 , (b) comparison for φ sum from the λ / 4 method, and (c) comparison for φ α d = ( φ E x - φ E y ) / 2 .

Tables (2)

Tables Icon

Table 1 Polarization Optic Configurations Used in This Study

Tables Icon

Table 2 Error Analysis for Various Experimental Fields for Horizontal Shear

Equations (61)

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E in ( x , y , t ) = E x ( x , y , t ) î + E y ( x , y , t ) ĵ ,
E x ( x , y , t ) = A x exp [ j ( k z o ω t + ϕ x ) ] ,
E y ( x , y , t ) = A y exp [ j ( k z o ω t + ϕ y ) ] ,
E x sample ( x , y , t ) = A x exp [ j ( k z ω t + ϕ x + k ( n o 1 ) h + k Δ S x ( x , y ) ) ] ,
E y sample ( x , y , t ) = A y exp [ j ( k z ω t + ϕ y + k ( n o 1 ) h + k Δ S y ( x , y ) ) ] ,
Δ S a ( x , y ) = h Δ n a ( x , y ) + ( n o 1 ) Δ h ( x , y ) .
Δ n 1 = n 1 n o = A σ 1 + B ( σ 2 + σ 3 ) ,
Δ n 2 = n 2 n o = A σ 2 + B ( σ 1 + σ 3 ) ,
Δ n 3 = n 3 n o = A σ 3 + B ( σ 1 + σ 2 ) ,
Δ h = [ σ 3 E ν E ( σ 1 + σ 2 ) ] h ,
Δ S 1 ( x , y ) = C h [ ( σ 1 + σ 2 ) + g ( σ 1 σ 2 ) ] ,
Δ S 2 ( x , y ) = C h [ ( σ 1 + σ 2 ) g ( σ 1 σ 2 ) ] ,
E p in ( x , y , t ) = E 1 ( x , y , t ) p 1 ^ + E 2 ( x , y , t ) p 2 ^ ,
E 1 ( x , y , t ) = E x ( x , y , t ) cos ( α ) + E y ( x , y , t ) sin ( α ) ,
E 2 ( x , y , t ) = E x ( x , y , t ) sin ( α ) + E y ( x , y , t ) cos ( α ) ,
p 1 ^ = cos ( α ) i ^ + sin ( α ) ĵ ,
p 2 ^ = sin ( α ) i ^ + cos ( α ) ĵ ,
E p sample ( x , y , t ) = E 1 ( x , y , t ) exp [ j k Δ S 1 ( x , y ) ] p 1 ^ + E 2 ( x , y , t ) exp [ j k Δ S 2 ( x , y ) ] p 2 ^ .
E p image ( x , y ) = E 1 image ( x , y ) p 1 ^ + E 2 image ( x , y ) p 2 ^ ,
E 1 image ( x , y ) = E 1 ± 1 ( x , y ) + E 1 ± 1 ( x + d shear , y ) ,
E 2 image ( x , y ) = E 2 ± 1 ( x , y ) + E 2 ± 1 ( x + d shear , y ) ,
E 1 ± 1 ( x , y ) = A x ± 1 cos ( α ) exp [ j ( k z ω t + ϕ x + k Δ S 1 ( x , y ) ) ] + A y ± 1 sin ( α ) exp [ j ( k z ω t + ϕ y + k Δ S 1 ( x , y ) ) ] ,
E 2 ± 1 ( x , y ) = A x ± 1 sin ( α ) exp [ j ( k z ω t + ϕ x + k Δ S 2 ( x , y ) ) ] + A y ± 1 cos ( α ) exp [ j ( k z ω t + ϕ y + k Δ S 2 ( x , y ) ) ] ,
I image = E 1 image E 1 image * t + E 2 image E 2 image * t = I 1 image + I 2 image ,
I 1 image = 2 ( A x ± 1 ) 2 cos 2 ( α ) + 2 ( A y ± 1 ) 2 sin 2 ( α ) + 4 A x ± 1 A y ± 1 cos ( α ) sin ( α ) cos ( φ x φ y ) + { 2 ( A x ± 1 ) 2 cos 2 ( α ) + 2 ( A y ± 1 ) 2 sin 2 ( α ) } cos [ k Δ S 1 ( x , y ) k Δ S 1 ( x + d shear , y ) ] + { 2 A x ± 1 A y ± 1 cos ( α ) sin ( α ) { cos [ ϕ x ϕ y + k Δ S 1 ( x , y ) k Δ S 1 ( x + d shear , y ) ] + cos [ ϕ y ϕ x + k Δ S 1 ( x , y ) k Δ S 1 ( x + d shear , y ) ] } ,
I 2 image = 2 ( A x ± 1 ) 2 sin 2 ( α ) + 2 ( A y ± 1 ) 2 cos 2 ( α ) 4 A x ± 1 A y ± 1 cos ( α ) sin ( α ) cos ( φ x φ y ) + { 2 ( A x ± 1 ) 2 sin 2 ( α ) + 2 ( A y ± 1 ) 2 cos 2 ( α ) } cos [ k ( Δ S 2 ( x , y ) k ( Δ S 2 ( x + d shear , y ) ] + 2 A x ± 1 A y ± 1 cos ( α ) sin ( α ) { cos [ ϕ x ϕ y + k Δ S 2 ( x , y ) k Δ S 2 ( x + d shear , y ) ] + cos [ ϕ y ϕ x + k Δ S 2 ( x , y ) k Δ S 2 ( x + d shear , y ) ] } .
I image = 2 ( A x ± 1 ) 2 + 2 ( A y ± 1 ) 2 + { 2 ( A x ± 1 ) 2 cos 2 ( α ) + 2 ( A y ± 1 ) 2 sin 2 ( α ) + 2 A x ± 1 A y ± 1     cos ( α ) sin ( α ) cos ( ϕ x ϕ y ) } × cos [ k ( Δ S 1 ( x , y ) Δ S 1 ( x + d shear , y ) ) ] + 2 ( A x ± 1 ) 2 sin 2 ( α ) + 2 ( A y ± 1 ) 2 cos 2 ( α ) 2 A x ± 1 A y ± 1     cos ( α ) sin ( α ) cos ( ϕ x ϕ y ) } × cos [ k ( Δ S 2 ( x , y ) Δ S 2 ( x + d shear , y ) ) ] .
φ 1 , 2 = k ( Δ S 1 , 2 ( x , y ) Δ S 1 , 2 ( x + Δ x , y ) ) k d shear Δ S 1 , 2 ( x , y ) x .
φ 1 , 2 = k d shear C h [ ( σ 1 + σ 2 ) x ± g ( σ 1 σ 2 ) x ] .
I image = I o + I 1 o cos [ φ sum + φ diff ] + I 2 o cos [ φ sum φ diff ] ,
I o = 2 ( A x ± 1 ) 2 + 2 ( A y ± 1 ) 2 ,
I 1 o = 2 ( A x ± 1 ) 2 cos 2 ( α ) + 2 ( A y ± 1 ) 2 sin 2 ( α ) + 2 A x ± 1 A y ± 1 cos ( α ) sin ( α ) cos ( ϕ x ϕ y ) ,
I 2 o = 2 ( A x ± 1 ) 2 sin 2 ( α ) + 2 ( A y ± 1 ) 2 cos 2 ( α ) 2 A x ± 1 A y ± 1 cos ( α ) sin ( α ) cos ( ϕ x ϕ y ) ,
φ sum = k d shear C h ( σ 1 + σ 2 ) x ,
φ diff = k d shear C h g ( σ 1 σ 2 ) x .
I image = I o + I c cos [ φ sum + φ c ] ,
I c = I 1 o 2 + I 2 o 2 + 2 I 1 o I 2 o cos ( 2 φ diff ) ,
φ c = arctan [ ( I 1 o I 2 o ) sin ( φ diff ) ( I 1 o + I 2 o ) cos ( φ diff ) ] .
φ sum = 2 π Δ C h p ( σ 1 + σ 2 ) x ,
φ diff = 2 π Δ C h g p ( σ 1 σ 2 ) x .
φ = arctan [ I 4 I 2 I 1 I 3 ] = arctan [ sin ( φ ) cos ( φ ) ] .
I 1 = I o + I c cos [ φ sum + φ c ] ,
I 2 = I o + I c cos [ φ sum + φ c + π 2 ] ,
I 3 = I o + I c cos [ φ sum + φ c + π ] ,
I 4 = I o + I c cos [ φ sum + φ c + 3 π 2 ] .
φ sum + φ c = arctan [ I 4 I 2 I 1 I 3 ] = arctan [ I c sin ( φ sum + φ c ) I c cos ( φ sum + φ c ) ] ,
I E x = I o E x + I c E x cos [ φ E x ] ,
φ E x = φ sum + φ α d ,
I o E x = 2 ( A o ± 1 ) 2 ,
I c E x = I o E x 1 sin 2 ( 2 α ) sin 2 ( ϕ diff ) ,
φ α d = arctan [ cos ( 2 α ) tan ( ϕ diff ) ] ,
I E y = I o E y + I c E y cos [ φ E y ] ,
φ E y = φ sum φ α d ,
I o E y = 2 ( A o ± 1 ) 2 ,
I c E y = I o E y 1 sin 2 ( 2 α ) sin 2 ( ϕ diff ) .
I circ = I o circ + I c circ cos [ φ sum ] ,
I o circ = 2 ( A o ± 1 ) 2 ,
I c circ = I o circ cos [ φ diff ] .
φ sum = arctan [ I 4 I 2 I 1 I 3 ] = arctan [ sin ( φ sum ) cos ( φ diff ) cos ( φ sum ) cos ( φ diff ) ] .
C h ( σ 1 + σ 2 ) x = ± p 2 π Δ φ sum .
W i , ĵ = k = 1 8 1 2 { cos ( 2 π Δ ψ k h ) + 1 } .

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