Abstract

We design rotating diffusers with deterministic complex-amplitude transmission functions, which give rise to tailored spatial coherence modulation when transilluminated by an axially incident coherent Gaussian beam. Mathematical expressions are derived for the immediate diffuser output as well as for the far-field response. An experimental demonstration is given using a diffuser fabricated by lithographic techniques.

© 2015 Optical Society of America

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References

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  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2010).
  2. Y. Li, H. Lee, and E. Wolf, “The effect of a moving diffuser on a random electromagnetic beam,” J. Mod. Opt. 52, 791–796 (2005).
    [Crossref]
  3. G. Li, Y. Qiu, and H. Li, “Coherence theory of a laser beam passing through a moving diffuser,” Opt. Express 21, 13032–13039 (2013).
    [Crossref] [PubMed]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  5. Y. Ohtsuka, “Modulation of optical coherence by ultrasonic waves,” J. Opt. Soc. Am. A 3, 1247–1257 (1986).
    [Crossref]
  6. J. Turunen, E. Tervonen, and A. T. Friberg, “Acousto-optic control and modulation of optical coherence by synthetic holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
    [Crossref]
  7. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
    [Crossref]
  8. C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).
  9. L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
    [Crossref]
  10. B. Rodenburg, M. Mirhosseini, O. S. Magaña-Loaiza, and R. W. Boyd, “Experimental generation of an optical field with arbitrary spatial coherence properties,” J. Opt. Soc. Am. B 31, A51–A55 (2014).
    [Crossref]
  11. F. Gori, M. Santarsiero, R. Borghi, and C.-F. Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826–2832 (2008).
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  12. M. Santarsiero, V. Ramírez-Sánchez, and R. Borghi, “Partially correlated thin annular sources: the vectorial case,” J. Opt. Soc. Am. A 27, 1450–1456 (2010).
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  13. G. B. Arfken and H. J. Weber, 5th ed. Mathematical Methods for Physicists (Academic, 2001), p. 681.
  14. H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).
    [Crossref] [PubMed]

2014 (2)

2013 (1)

2012 (2)

C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

2010 (1)

2008 (1)

2005 (1)

Y. Li, H. Lee, and E. Wolf, “The effect of a moving diffuser on a random electromagnetic beam,” J. Mod. Opt. 52, 791–796 (2005).
[Crossref]

1992 (1)

1990 (1)

J. Turunen, E. Tervonen, and A. T. Friberg, “Acousto-optic control and modulation of optical coherence by synthetic holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[Crossref]

1986 (1)

Arfken, G. B.

G. B. Arfken and H. J. Weber, 5th ed. Mathematical Methods for Physicists (Academic, 2001), p. 681.

Borghi, R.

Boyd, R. W.

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Flores, E.

C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Friberg, A. T.

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[Crossref]

J. Turunen, E. Tervonen, and A. T. Friberg, “Acousto-optic control and modulation of optical coherence by synthetic holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[Crossref]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2010).

Gori, F.

Lee, H.

Y. Li, H. Lee, and E. Wolf, “The effect of a moving diffuser on a random electromagnetic beam,” J. Mod. Opt. 52, 791–796 (2005).
[Crossref]

Li, C.-F.

Li, G.

Li, H.

Li, Y.

Y. Li, H. Lee, and E. Wolf, “The effect of a moving diffuser on a random electromagnetic beam,” J. Mod. Opt. 52, 791–796 (2005).
[Crossref]

Magaña-Loaiza, O. S.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Mirhosseini, M.

Ohtsuka, Y.

Olvera-Santamaria, M. A.

C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Ostrovsky, A. S.

C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Partanen, H.

Qiu, Y.

Ramírez-Sánchez, V.

Rickenstorff, C.

C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Rodenburg, B.

Santarsiero, M.

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Tervo, J.

Tervonen, E.

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[Crossref]

J. Turunen, E. Tervonen, and A. T. Friberg, “Acousto-optic control and modulation of optical coherence by synthetic holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[Crossref]

Turunen, J.

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, 5th ed. Mathematical Methods for Physicists (Academic, 2001), p. 681.

Wolf, E.

Y. Li, H. Lee, and E. Wolf, “The effect of a moving diffuser on a random electromagnetic beam,” J. Mod. Opt. 52, 791–796 (2005).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

J. Appl. Phys. (1)

J. Turunen, E. Tervonen, and A. T. Friberg, “Acousto-optic control and modulation of optical coherence by synthetic holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[Crossref]

J. Mod. Opt. (1)

Y. Li, H. Lee, and E. Wolf, “The effect of a moving diffuser on a random electromagnetic beam,” J. Mod. Opt. 52, 791–796 (2005).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Rev. Mex. Fis. (1)

C. Rickenstorff, E. Flores, M. A. Olvera-Santamaria, and A. S. Ostrovsky, “Modulation of coherence and polarization using nematic 90°-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2010).

G. B. Arfken and H. J. Weber, 5th ed. Mathematical Methods for Physicists (Academic, 2001), p. 681.

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Figures (6)

Fig. 1
Fig. 1 The geometry of illuminating a rotating diffuser with a coherent beam and the setup for measuring the spatial coherence properties of the resulting secondary source. L: Laser source, E: Rotating diffuser element, A: Aperture, M: Micromirror device, C: Camera.
Fig. 2
Fig. 2 The schematic structure of the diffuser. The dark and light gray areas mean phase delays of zero and π radians.
Fig. 3
Fig. 3 Diffraction pattern generated by the fabricated diffuser while stationary (left) and rotating (right). The vertical line shows the measurement range for radial modulation with the reference point marked with a dot; the circle shows the measurement range for azimuthal modulation.
Fig. 4
Fig. 4 Left: The absolute value of spatial degree of coherence in radial direction as a function of divergence angle difference to the reference point. Right: Light intensity in the measured range of radial modulation. Measured data marked with dots, theoretical values as solid lines.
Fig. 5
Fig. 5 The absolute value of spatial degree of coherence in azimuthal direction as a function of difference in the azimuthal coordinate φ. Measured data marked with black, theoretical value with red.
Fig. 6
Fig. 6 The measured response similar to that of Fig. 5 with the exception that the diffuser was offset by 50 μm from the rotational axis.

Equations (72)

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U ( ρ , ϕ ; τ ) = t ( ρ , ϕ 2 π τ / D ) U 0 ( ρ , ϕ ) .
J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = 1 D 0 D U * ( ρ 1 , ϕ 1 ; τ ) U ( ρ 2 ; ϕ 2 ; τ ) d τ .
J 0 ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = U 0 * ( ρ 1 , ϕ 1 ) U 0 ( ρ 2 , ϕ 2 )
J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = J 0 ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) 1 D 0 D t * ( ρ 1 , ϕ 1 , 2 π τ / D ) t ( ρ 2 , ϕ 2 2 π τ / D ) d τ .
t ( ρ , ϕ ) = m = G m ( ρ ) exp ( i m ϕ ) ,
G m ( ρ ) = 1 2 π 0 2 π t ( ρ , ϕ ) exp ( i m ϕ ) d ϕ
J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = J 0 ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) m = G m * ( ρ 1 ) G m ( ρ 2 ) exp ( i m Δ ϕ )
j ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) J ( ρ 1 , ϕ 1 , ρ 1 , ϕ 1 ) J ( ρ 2 , ϕ 2 , ρ 2 , ϕ 2 ) = exp [ i Φ ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) ] m G m * ( ρ 1 ) G m ( ρ 2 ) exp ( i m Δ ϕ ) m | G m ( ρ 1 ) | 2 m | G m ( ρ 2 ) | 2 ,
T ( f 1 , φ 1 , f 2 , φ 2 ) = 1 ( 2 π ) 4 0 0 2 π J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) exp [ i f 1 ρ 1 cos ( ϕ 1 φ 1 ) ] × exp [ i f 2 ρ 2 cos ( ϕ 2 φ 2 ) ] ρ 1 ρ 2 d ϕ 2 d ϕ 1 d ρ 2 d ρ 1 .
J ( θ , φ ) = ( 2 π k ) 2 cos θ T ( k sin θ , φ , k sin θ , φ ) .
j ( θ 1 , φ 1 , θ 2 , φ 2 ) = T ( k sin θ 1 , φ 1 , k sin θ 2 , φ 2 ) T ( k sin θ 1 , φ 1 , k sin θ 1 , φ 1 ) T ( k sin θ 2 , φ 2 , k sin θ 2 , φ 2 ) .
G m ( ρ ) = G m = 1 2 π 0 2 π t ( ϕ ) exp ( i m ϕ ) d ϕ
j ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = exp [ i Φ ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) ] m | G m | 2 exp ( i m Δ ϕ ) m | G m | 2 .
j ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = exp [ i Φ ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) ] ,
Φ ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = Φ ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) + arg [ G 0 ( ρ 2 ) ] arg [ G 0 ( ρ 1 ) ] .
t ( ρ , ϕ ) = cos ( α ρ + q ϕ ) ,
G m ( ρ ) = 1 2 exp ( i α ρ m / q )
j ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = j ( Δ ρ , Δ ϕ ) = cos ( α Δ ρ + q Δ ϕ ) ,
U 0 ( ρ , ϕ ) = U 0 exp ( ρ 2 w 2 ) ,
J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = 1 2 | U 0 | 2 exp ( ρ 1 2 + ρ 2 2 w 2 ) cos ( α Δ ρ + q Δ ϕ ) .
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 8 π 2 0 ρ 1 ρ 2 exp ( ρ 1 2 + ρ 2 2 w 2 ) J q ( f 1 ρ 1 ) J q ( f 2 ρ 2 ) × cos ( α Δ ρ + q Δ φ ) d ρ 1 d ρ 2 ,
J ( θ , φ ) = 1 2 k 2 | U 0 | 2 cos θ | 0 ρ exp ( ρ 2 w 2 ) J q ( k ρ sin θ ) exp ( i α ρ ) d ρ | 2 .
t ( ρ , ϕ ) = exp [ i ( α ρ + q ϕ ) ] ,
j ( Δ ρ , Δ ϕ ) = exp [ i ( α Δ ρ + q Δ ϕ ) ] ,
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 ( 2 π ) 2 exp ( i q Δ φ ) 0 ρ 1 ρ 2 exp ( ρ 1 2 + ρ 2 2 w 2 ) J q ( f 1 ρ 1 ) J q ( f 2 ρ 2 ) × exp ( i α Δ ρ ) d ρ 1 d ρ 2 .
J ( θ , ϕ ) = k 2 | U 0 | 2 cos θ | 0 ρ exp ( ρ 2 w 2 ) J q ( k ρ sin θ ) exp ( i α ρ ) d ρ | 2 .
t ( ρ , ϕ ) = 1 2 [ 1 + cos ( α ρ + q ϕ ) ] ,
j ( Δ ρ , Δ ϕ ) = 1 3 [ 2 + cos ( α Δ ρ + q Δ ϕ ) ]
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 16 π 2 0 ρ 1 ρ 2 exp ( ρ 1 2 + ρ 2 2 w 2 ) [ J 0 ( f 1 ρ 1 ) J 0 ( f 2 ρ 2 ) + 1 2 J q ( f 1 ρ 1 ) J q ( f 2 ρ 2 ) cos ( α Δ ρ + q Δ φ ) ] d ρ 1 d ρ 2 = | U 0 | 2 w 4 64 π 2 exp [ w 2 4 ( f 1 2 + f 2 2 ) ] + | U 0 | 2 32 π 2 0 ρ 1 ρ 2 exp ( ρ 1 2 + ρ 2 2 w 2 ) × J q ( f 1 ρ 1 ) J q ( f 2 ρ 2 ) cos ( α Δ ρ + q Δ φ ) d ρ 1 d ρ 2
J ( θ , φ ) = k 2 cos θ | U 0 | 2 w 4 16 exp [ w 2 2 k 2 sin 2 θ ] + k 2 cos θ | U 0 | 2 8 | 0 ρ exp ( ρ 2 w 2 ) J q ( k ρ sin θ ) exp ( i α ρ ) d ρ | 2 .
t ( ρ , ϕ ) = exp [ i ϑ ( ρ , ϕ ) ] ,
ϑ ( ρ , ϕ ) = arg [ sin ( α ρ + q ϕ ) ] Δ ϑ π
G m ( ρ ) = { 1 2 [ 1 + exp ( i Δ ϑ ) ] if m = 0 , i q m π exp ( i m α ρ / q ) [ 1 exp ( i Δ ϑ ) ] if m / q is an odd integer , 0 otherwise .
j ( Δ ρ , Δ ϕ ) = 1 2 { 1 + cos ( Δ ϑ ) + [ 1 cos ( Δ ϑ ) ] y ( α Δ ρ + q Δ ϕ ) } ,
y ( ε ) = 8 π 2 p = 0 cos [ ( 2 p + 1 ) ε ] ( 2 p + 1 ) 2
t ( ρ , ϕ ) = 1 2 { 1 + sgn [ sin ( α ρ + q ϕ ) ] }
G 0 ( ρ ) = 1 2
G Nq ( ρ ) = i N π exp ( i N α ρ ) ,
j ( Δ ρ , Δ ϕ ) = 1 2 [ 1 + y ( α Δ ρ + q Δ ϕ ) ] ,
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 2 { [ 1 + cos ( Δ ϑ ) ] I 1 ( 0 ) I 2 ( 0 ) + [ 1 cos ( Δ ϑ ) ] Y ( q Δ φ , I 1 ( N ) I 2 ( N ) ) } ,
Y ( ε , R ( N ) ) = 8 π 2 p = 0 cos [ ( 2 ρ + 1 ) ε ] ( 2 p + 1 ) 2 { R ( 2 p + 1 ) }
I 1 ( N ) = 0 ρ exp ( ρ 2 w 2 i N α ρ ) J Nq ( f 1 ρ ) d ρ
I 2 ( N ) = 0 ρ exp ( ρ 2 w 2 + i N α ρ ) J Nq ( f 2 ρ ) d ρ
T ( f , Δ φ ) = | U 0 | 2 2 { [ 1 + cos ( Δ ϑ ) ] | I ( 0 ) | 2 + [ 1 cos ( Δ ϑ ) ] Y ( q Δ ϕ , | I ( N ) | 2 ) } .
j ( Δ φ ) cos ( q Δ φ )
t ( ρ , ϕ ) = exp [ i ϑ ( ρ , ϕ ) ] ,
ϑ ( ρ , ϕ ) = arg [ sin ( α ρ + q ϕ ) ] Δ ϑ π .
G 0 ( ρ ) = 1 2 π 0 2 π t ( ρ , ϕ ) d ϕ = 1 2 π [ π + π exp ( i Δ ϑ ) ] = 1 2 [ 1 + exp ( i Δ ϑ ) ] .
ϕ ϕ α ρ q
G m ( ρ ) = C m ( ρ ) 0 2 π exp [ i arg ( sin ( q ϕ ) ) Δ ϑ π i m ϕ ] d ϕ ,
C m ( ρ ) = 1 2 π exp ( i m α ρ q ) .
G m ( ρ ) = C m ( ρ ) [ 0 π / q exp ( i m ϕ ) d ϕ + exp ( i Δ ϑ ) π / q 2 π / q exp ( i m ϕ ) d ϕ + 2 π / q 3 π / q exp ( i m ϕ ) d ϕ + + exp ( i Δ ϑ ) 2 π π / q 2 π exp ( i m ϕ ) d ϕ ] .
G m ( ρ ) = C m ( ρ ) i m [ 1 exp ( i Δ ϑ ) ] { n = 0 q 1 exp [ ( 2 n + 1 ) i m π / q ] n = 0 q 1 exp [ 2 n i m π / q ] } ,
G m ( ρ ) = C m ( ρ ) i m [ 1 exp ( i Δ ϑ ) ] [ exp ( i m π / q ) 1 ] n = 0 q 1 exp [ 2 n i m π / q ] .
G Nq ( ρ ) = C Nq ( ρ ) i N [ 1 exp ( i Δ ϑ ) ] [ exp ( i N π ) 1 ] .
G Nq ( ρ ) = i N π [ 1 exp ( i Δ ϑ ) ] exp ( i N α ρ ) .
n = 0 q 1 exp ( 2 i m π / q ) n = 1 exp ( 2 i m π / q ) q 1 exp ( 2 i m π / q ) ,
G 0 * ( ρ 1 ) G 0 ( ρ 2 ) = 1 2 [ 1 + cos ( Δ ϑ ) ] .
G Nq * ( ρ 1 ) G Nq ( ρ 2 ) = 2 N 2 π 2 [ 1 cos ( Δ ϑ ) ] exp ( i N α Δ ρ ) .
S N ( ρ 1 , ρ 2 , Δ ϕ ) = G Nq * ( ρ 1 ) G Nq ( ρ 2 ) exp ( i Nq Δ ϕ ) .
j ( Δ ρ , Δ ϕ ) = 1 + cos ( Δ ϑ ) + [ 1 cos ( Δ ϑ ) ] y ( α Δ ρ + q Δ ϕ ) 2 .
J ( ρ 1 , ϕ 1 , ρ 2 , ϕ 2 ) = | U 0 | 2 exp ( ρ 1 2 w 2 ) exp ( ρ 2 2 w 2 ) m = G m * ( ρ 1 ) G m ( ρ 2 ) exp ( i m Δ ϕ ) ,
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 ( 2 π ) 2 0 ρ 1 exp ( ρ 1 2 w 2 ) ρ 2 exp ( ρ 2 2 w 2 ) m = G m * ( ρ 1 ) G m ( ρ 2 ) × F m ( ρ 1 , ρ 2 , f 1 , φ 1 , f 2 , φ 2 ) d ρ 1 d ρ 2 ,
F m ( ρ 1 , ρ 2 , f 1 , φ 1 , f 2 , φ 2 ) = 0 2 π exp ( i m Δ φ ) exp [ i f 1 ρ 1 cos ( ϕ 1 φ 1 ) ] × exp [ i f 2 ρ 2 cos ( ϕ 2 φ 2 ) ] d ϕ 1 d ϕ 2 .
F m ( ρ 1 , ρ 2 , f 1 , φ 1 , f 2 , φ 2 ) = n , r i n ( i ) r J n ( f 1 ρ 1 ) J r ( f 2 ρ 2 ) exp ( i n 1 φ 1 ) exp ( i r φ 2 ) × 0 2 π exp [ i ( n m ) ϕ 1 ] d ϕ 1 0 2 π exp [ i ( m r ) ϕ 2 ] d ϕ 2 .
F m ( ρ 1 , ρ 2 , f 1 , φ 1 , f 2 , φ 2 ) = ( 2 π ) 2 J m ( f 1 ρ 1 ) J m ( f 2 ρ 2 ) exp ( i m Δ φ ) .
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 m = exp ( i m Δ φ ) 0 ρ 1 exp ( ρ 1 2 w 2 ) ρ 2 exp ( ρ 2 2 w 2 )
× G m * ( ρ 1 ) G m ( ρ 2 ) J m ( f 1 ρ 1 ) J m ( f 2 ρ 2 ) d ρ 1 d ρ 2 ,
T ( f 1 , φ 1 , f 2 , φ 2 ) = | U 0 | 2 T c ( f 1 , f 2 ) + | U 0 | 2 p = T p ( f 1 , f 2 , Δ φ )
T c ( f 1 , f 2 ) = 1 2 [ 1 + cos ( Δ ϑ ) ] 0 H ( ρ 1 ) J 0 ( f 1 ρ 1 ) d ρ 1 0 H ( ρ 2 ) J 0 ( f 2 ρ 2 ) d ρ 2 ,
T p ( f 1 , f 2 , Δ φ ) = [ 1 cos ( Δ ϑ ) ] 2 exp [ i ( 2 p + 1 ) q Δ φ ] ( 2 p + 1 ) 2 π 2 × 0 H ( ρ 1 ) J ( 2 p + 1 ) q ( f 1 ρ 1 ) exp [ i ( 2 p + 1 ) α ρ 1 ] d ρ 1 × 0 H ( ρ 2 ) J ( 2 p + 1 ) q ( f 2 ρ 2 ) exp [ i ( 2 p + 1 ) α ρ 2 ] d ρ 2 ,
p = T p ( f 1 , f 2 , Δ φ ) = 2 p = 0 { T p ( f 1 , f 2 , Δ φ ) } ,

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