Abstract

A quantized version of a continuous spiral phase filter with unitary topological charge, here denominated multi-step spiral phase filter (MSSPF), is proposed to extract phase from rotated spiral interferograms. Spiral interferograms are usually obtained from phase objects by registering the interference of its vortex filtered complex amplitude with a reference complex amplitude. The structure found in this kind of interferograms, depend on the number of steps used in the MSSPF that usually are assumed with an infinite number of steps for the continuous spiral phase filter. Reducing the number of steps of the MSSPF affects the residual phase error obtained after the phase extraction method. This error is therefore analysed here using a numerical simulation of a Mach-Zender interferometer with a MSSPF and a reduced number of steps. It is shown that, for our proposed method of rotated spiral interferograms, a residual error persists as the number of steps is increased approaching the residual error reported for the phase extraction method of single-shot spiral interferograms. Furthermore, it is shown that this novel technique can be applied without further modifications for phase contrast measurement. Experimental results show similar performance of this phase extraction technique, when compared to the results obtained with a commercial interferometer and with the numerical simulations.

© 2014 Optical Society of America

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  1. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
    [CrossRef]
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    [CrossRef]
  3. K. Crabtree, J. A. Davis, I. Moreno, “Optical processing with vortex-producing lenses” Appl. Opt.. 43, 1360–1367 (2004).
    [CrossRef] [PubMed]
  4. S. Fürhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, “Spiral interferometry” Opt. Lett. 30, 1953–1955 (2005).
    [CrossRef] [PubMed]
  5. A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy” Phys. Rev. Lett. 94, 233902 (2005).
    [CrossRef] [PubMed]
  6. P. Bouchal, Z. Bouchal, “Selective edge enhancement in three-dimensional vortex imaging with incoherent light” Opt. Lett. 37, 2949–2951 (2012).
    [CrossRef] [PubMed]
  7. M. K. Sharma, J. Joseph, P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter” J. Opt. 42, 1–7 (2012).
    [CrossRef]
  8. M. K. Sharma, J. Joseph, P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter” Opt. Laser Technol.16 (2013).
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    [CrossRef] [PubMed]
  10. C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy” Laser Photonics Rev. 5, 81101 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
    [CrossRef]

2014

A. Aguilar, A. Dávila, J. E. A. Landgrave, “Displacement measurement with multi-level spiral phase filtering in speckle interferometry” Opt. Lasers Eng. 52, 19–26 (2014).
[CrossRef]

2013

M. Baranek, Z. Bouchal, “Rotating vortex imaging implemented by a quantized spiral phase modulation” J. Europ. Opt. Soc. Rap. Public. 8, 130171 (2013).
[CrossRef]

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

E. Rueda, D. Muñetón, J. A. Gómez, A. Lencina, “High-quality optical vortex-beam generation by using a multilevel vortex-producing lens” Opt. Lett. 38, 3941–3944 (2013).
[CrossRef] [PubMed]

2012

P. Bouchal, Z. Bouchal, “Selective edge enhancement in three-dimensional vortex imaging with incoherent light” Opt. Lett. 37, 2949–2951 (2012).
[CrossRef] [PubMed]

J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
[CrossRef]

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory” Opt. Lasers Eng. 501009–1014 (2012).
[CrossRef]

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter” J. Opt. 42, 1–7 (2012).
[CrossRef]

2011

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy” Laser Photonics Rev. 5, 81101 (2011).
[CrossRef]

2006

2005

S. Fürhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, “Spiral interferometry” Opt. Lett. 30, 1953–1955 (2005).
[CrossRef] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

2004

K. Crabtree, J. A. Davis, I. Moreno, “Optical processing with vortex-producing lenses” Appl. Opt.. 43, 1360–1367 (2004).
[CrossRef] [PubMed]

2002

Z. Wang, A. C. Bovik, “A universal image quality index” IEEE Signal Process. Lett. XX, 1–4 (2002).

2001

2000

1996

1992

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Aguilar, A.

A. Aguilar, A. Dávila, J. E. A. Landgrave, “Displacement measurement with multi-level spiral phase filtering in speckle interferometry” Opt. Lasers Eng. 52, 19–26 (2014).
[CrossRef]

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Alayli, Y.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Alcalá-Ochoa, N.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Baranek, M.

M. Baranek, Z. Bouchal, “Rotating vortex imaging implemented by a quantized spiral phase modulation” J. Europ. Opt. Soc. Rap. Public. 8, 130171 (2013).
[CrossRef]

Bernet, S.

Bone, D. J.

Bouchal, P.

Bouchal, Z.

M. Baranek, Z. Bouchal, “Rotating vortex imaging implemented by a quantized spiral phase modulation” J. Europ. Opt. Soc. Rap. Public. 8, 130171 (2013).
[CrossRef]

P. Bouchal, Z. Bouchal, “Selective edge enhancement in three-dimensional vortex imaging with incoherent light” Opt. Lett. 37, 2949–2951 (2012).
[CrossRef] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, “A universal image quality index” IEEE Signal Process. Lett. XX, 1–4 (2002).

Calderón-Hermosillo, Y.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Campos, J.

Cottrell, D. M.

Crabtree, K.

K. Crabtree, J. A. Davis, I. Moreno, “Optical processing with vortex-producing lenses” Appl. Opt.. 43, 1360–1367 (2004).
[CrossRef] [PubMed]

Dávila, A.

A. Aguilar, A. Dávila, J. E. A. Landgrave, “Displacement measurement with multi-level spiral phase filtering in speckle interferometry” Opt. Lasers Eng. 52, 19–26 (2014).
[CrossRef]

Davis, J. A.

Ding, J.

C. S. Guo, D. M. Xue, Y. J. Han, J. Ding, “Optimal phase steps of multi-level spiral phase plates” Opt. Commun. 268, 235–239 (2006).
[CrossRef]

Espinosa-Luna, R.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Estrada, J. C.

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory” Opt. Lasers Eng. 501009–1014 (2012).
[CrossRef]

Fürhapter, S.

Galizzi, G. E.

Garcia-Marquez, J.

J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
[CrossRef]

García-Márquez, J.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Gómez, J. A.

Gonzalez-Vega, A.

J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
[CrossRef]

Guo, C. S.

C. S. Guo, D. M. Xue, Y. J. Han, J. Ding, “Optimal phase steps of multi-level spiral phase plates” Opt. Commun. 268, 235–239 (2006).
[CrossRef]

Han, Y. J.

C. S. Guo, D. M. Xue, Y. J. Han, J. Ding, “Optimal phase steps of multi-level spiral phase plates” Opt. Commun. 268, 235–239 (2006).
[CrossRef]

Jesacher, A.

Joseph, J.

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter” J. Opt. 42, 1–7 (2012).
[CrossRef]

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter” Opt. Laser Technol.16 (2013).

Kaufmann, G. H.

Kerr, D.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Landgrave, J. E. A.

A. Aguilar, A. Dávila, J. E. A. Landgrave, “Displacement measurement with multi-level spiral phase filtering in speckle interferometry” Opt. Lasers Eng. 52, 19–26 (2014).
[CrossRef]

Larkin, K. G.

Lencina, A.

Lopez, V.

J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
[CrossRef]

López, V.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy” Laser Photonics Rev. 5, 81101 (2011).
[CrossRef]

S. Bernet, A. Jesacher, F. Severin, C. Maurer, M. Ritsch-marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy” Opt. Express 14, 3792–3805 (2006).
[CrossRef] [PubMed]

McNamara, D. E.

Moreno, I.

K. Crabtree, J. A. Davis, I. Moreno, “Optical processing with vortex-producing lenses” Appl. Opt.. 43, 1360–1367 (2004).
[CrossRef] [PubMed]

Muñetón, D.

Noé, E.

J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
[CrossRef]

Noé-Arias, E.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

Oldfield, M. A.

Ritsch-Marte, M.

Rueda, E.

Senthilkumaran, P.

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter” J. Opt. 42, 1–7 (2012).
[CrossRef]

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter” Opt. Laser Technol.16 (2013).

Servin, M.

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory” Opt. Lasers Eng. 501009–1014 (2012).
[CrossRef]

Severin, F.

Sharma, M. K.

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter” J. Opt. 42, 1–7 (2012).
[CrossRef]

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter” Opt. Laser Technol.16 (2013).

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Wang, Z.

Z. Wang, A. C. Bovik, “A universal image quality index” IEEE Signal Process. Lett. XX, 1–4 (2002).

Xue, D. M.

C. S. Guo, D. M. Xue, Y. J. Han, J. Ding, “Optimal phase steps of multi-level spiral phase plates” Opt. Commun. 268, 235–239 (2006).
[CrossRef]

Appl. Opt.

Appl. Opt..

K. Crabtree, J. A. Davis, I. Moreno, “Optical processing with vortex-producing lenses” Appl. Opt.. 43, 1360–1367 (2004).
[CrossRef] [PubMed]

IEEE Signal Process. Lett.

Z. Wang, A. C. Bovik, “A universal image quality index” IEEE Signal Process. Lett. XX, 1–4 (2002).

J. Europ. Opt. Soc. Rap. Public.

M. Baranek, Z. Bouchal, “Rotating vortex imaging implemented by a quantized spiral phase modulation” J. Europ. Opt. Soc. Rap. Public. 8, 130171 (2013).
[CrossRef]

J. Mod. Opt.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

J. Opt.

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter” J. Opt. 42, 1–7 (2012).
[CrossRef]

J. Opt. Soc. Am. A

Laser Photonics Rev.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy” Laser Photonics Rev. 5, 81101 (2011).
[CrossRef]

Opt. Commun.

C. S. Guo, D. M. Xue, Y. J. Han, J. Ding, “Optimal phase steps of multi-level spiral phase plates” Opt. Commun. 268, 235–239 (2006).
[CrossRef]

Opt. Express

S. Bernet, A. Jesacher, F. Severin, C. Maurer, M. Ritsch-marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy” Opt. Express 14, 3792–3805 (2006).
[CrossRef] [PubMed]

J. Garcia-Marquez, V. Lopez, A. Gonzalez-Vega, E. Noé, “Flicker minimization in an LCoS spatial light modulator” Opt. Express, 8, 8431–8441 (2012).
[CrossRef]

Opt. Lasers Eng.

Y. Calderón-Hermosillo, J. García-Márquez, R. Espinosa-Luna, N. Alcalá-Ochoa, V. López, A. Aguilar, E. Noé-Arias, Y. Alayli, “Flicker in a twisted nematic spatial light modulator” Opt. Lasers Eng. 51, 741–748 (2013).
[CrossRef]

A. Aguilar, A. Dávila, J. E. A. Landgrave, “Displacement measurement with multi-level spiral phase filtering in speckle interferometry” Opt. Lasers Eng. 52, 19–26 (2014).
[CrossRef]

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory” Opt. Lasers Eng. 501009–1014 (2012).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Other

M. K. Sharma, J. Joseph, P. Senthilkumaran, “Directional edge enhancement using superposed vortex filter” Opt. Laser Technol.16 (2013).

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Figures (11)

Fig. 1
Fig. 1

Phase of the MSSPF with variation of the n and p parameters

Fig. 2
Fig. 2

Point Spread Functions of the multi-step spiral phase filter. a) |h2(xi, yi)|2, b) |h3(xi, yi)|2, c) | h4(xi, yi)|2, d) |h5(xi, yi)|2, e) |h(xi, yi)|2

Fig. 3
Fig. 3

Schematic diagram of a 4 f optical correlator with a phase vortex filter represented in gray levels at the common focus to L1 and L2

Fig. 4
Fig. 4

Computed Interference pattern for p = 4, a) I 4 ( 1 ) ( x i , y i ), b) I 4 ( 2 ) ( x i , y i ), c) I 4 ( 3 ) ( x i , y i ), d) I 4 ( 4 ) ( x i , y i ) of a double element phase object

Fig. 5
Fig. 5

Extracted wrapped phase using the interferograms of Fig. 4. a) Convolved phase Θ4(xi, yi), b) Deconvolved phase φ o e ( x o , y o )

Fig. 6
Fig. 6

Computed Interference pattern for p = 4, a) I 4 ( 1 ) ( x i , y i ), b) I 4 ( 2 ) ( x i , y i ), c) I 4 ( 3 ) ( x i , y i ), d) I 4 ( 4 ) ( x i , y i ) of an only phase USAF optical test pattern

Fig. 7
Fig. 7

Extracted phase using the interferograms of Fig. 6.

Fig. 8
Fig. 8

a) Unwrapped extracted phase, b) Absolute error map, between the unwrapped extracted phase and the simulated phase

Fig. 9
Fig. 9

Schematic Diagram of the experimental Setup. A detailed description of the elements and their function is given in the text.

Fig. 10
Fig. 10

Experimental Interference pattern I p ( n ) ( x i , y i ) with the MSSPF gray representation in the right corner. a) I 3 ( 1 ) ( x i , y i ), b) I 4 ( 1 ) ( x i , y i ), c) I 5 ( 1 ) ( x i , y i )

Fig. 11
Fig. 11

Experimental extracted wrapped phase. a) Convolved phase Θ4(xi, yi), b) Deconvolved phase φ o e ( x o , y o )

Tables (1)

Tables Icon

Table 1 Dependance of the Quality index and the number of steps of the MSSPF

Equations (19)

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

p ( n ) ( u , v ) = circ ( ρ ρ o ) exp [ i Ψ p ( n ) ( u , v ) ] ,
Ψ p ( u , v ) = [ Ψ p ( 1 ) ( u , v ) , Ψ p ( 2 ) ( u , v ) , , Ψ p ( n ) ( u , v ) ] T = 2 π p R q p ( u , v ) ,
R = [ r i , j ] = [ r 1 , r 2 , , r p ] T = ( 0 1 2 p 1 p 1 0 1 p 2 2 p 1 0 1 1 2 p 1 0 ) ,
q p ( u , v ) = [ q 1 ( u , v ) , q 2 ( u , v ) , , q j ( u , v ) ] T ,
q j ( u , v ) = { 1 2 π ( j 1 ) p < θ 2 π j p , 0 otherwise
p ( n ) ( u , v ) = circ ( ρ ρ o ) exp [ i 2 π p j = 1 p r n , j q j ( u , v ) ] ,
r n , j = r 1 , j + n 1 mod p ,
p ( n ) ( u , v ) = circ ( ρ ρ o ) exp [ i Ψ p ( 1 ) ( u , v ) ] × exp [ i 2 π ( n 1 ) p j = 1 p q j ( u , v ) ] = circ ( ρ ρ o ) exp [ i 2 π ( n 1 ) p ] p ( 1 ) ( u , v ) ,
h p ( x i , y i ) = 1 { p ( u , v ) }
U i p ( n ) ( x i , y i ) = 1 { p ( n ) ( u , v ) { U o ( x o , y o ) P ( x o , y o ) } } ,
U o ( x o , y o ) = E o ( x o , y o ) exp [ i φ o ( x o , y o ) ] ,
I p ( n ) ( x i , y i ) = | U r ( x i , y i ) + U i p ( n ) ( x i , y i ) | 2 = | E r + 1 { p ( n ) ( u , v ) G f ( u , v ) } | 2
I p ( n ) ( x i , y i ) = | E r + exp [ i ( 2 π ( n 1 ) p ) ] × 1 { p ( 1 ) ( u , v ) G f ( u , v ) } | 2 = | E r + exp [ i ( 2 π ( n 1 ) p ) ] U i p ( 1 ) ( x i , y i ) | 2
I p ( n ) ( x i , y i ) = 𝒜 p ( x i , y i ) + p ( x i , y i ) cos [ Θ p ( x i , y i ) + 2 π ( n 1 ) p ]
𝒜 p ( x i , y i ) = E r 2 + | U i p ( 1 ) ( x i , y i ) | 2 , p ( x i , y i ) = 2 E r | U i p ( 1 ) ( x i , y i ) | 2 , Θ p ( x i , y i ) = arg [ U i p ( 1 ) ( x i , y i ) ] .
U i p ( 1 ) ( x i , y i ) = n = 1 p I p ( n ) ( x i , y i ) exp [ i 2 π ( n 1 ) p ]
U o ( x i , y i ) = U i p ( 1 ) ( x i , y i ) h p 1 ( x i , y i ) .
φ o ( e ) ( x o , y o ) arg [ U o ( x i , y i ) ] ,
Q = σ x y σ x σ y 2 x y ¯ x ¯ 2 + y ¯ 2 2 σ x σ y σ x 2 + σ y 2

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