Abstract

Thin annular sources, either coherent or completely incoherent from the spatial standpoint, have played a significant role in the synthesis of diffraction-free and J0-correlated fields, respectively. Here, we consider thin annular sources with partial correlation. A scalar description is developed under the assumption that the correlation function between two points depends on their angular distance only. We show that for any such source the modal expansion can easily be found. Further, we examine how the correlation properties of the radiated fields change on free propagation. We also give a number of examples and present possible synthesis schemes.

© 2008 Optical Society of America

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  1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  2. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150-152 (1989).
    [CrossRef]
  3. F. Gori, G. Guattari, and C. Padovani, “Modal expansion of J0-correlated sources,” Opt. Commun. 64, 311-316 (1987).
    [CrossRef]
  4. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282-289 (1991).
    [CrossRef]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  7. E. Wolf and W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205-209 (1975).
    [CrossRef]
  8. W. H. Carter and E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785-796 (1977).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  10. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 353-355 (2007).
    [CrossRef]
  11. F. Riesz and B. Sz.-Nagy, Functional Analysis (Blackie and Sons, 1956).
  12. A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes in large-aperture approximation,” Opt. Lett. 32, 611-613 (2007).
    [CrossRef] [PubMed]
  13. M. W. Kowarz and G. S. Agarwal, “Bessel-beam representation for partially coherent fields,” J. Opt. Soc. Am. A 12, 1324-1330 (1995).
    [CrossRef]
  14. C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
    [CrossRef]
  15. R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849-856 (1999).
    [CrossRef]
  16. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150-156 (2001).
    [CrossRef]
  17. G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627-1629 (2003).
    [CrossRef] [PubMed]
  18. J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell-Debye equation,” J. Opt. Soc. Am. A 20, 1409-1417 (2003).
    [CrossRef]
  19. L. Wang and B. Lü, “Propagation and focal shift of J0-correlated Schell-model beams,” Optik (Stuttgart) 117, 167-172 (2006).
    [CrossRef]
  20. L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
    [CrossRef]
  21. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508-2510 (2007).
    [CrossRef] [PubMed]
  22. T. van Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575-581 (2008).
    [CrossRef]
  23. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33, 1857-1859 (2008).
    [CrossRef] [PubMed]
  24. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2000).
  25. Following the standard usage, we denote by J both the mutual intensity and the Bessel functions of the first kind. No confusion should arise since Bessel functions have a numerical index.
  26. R. Borghi, M. Santarsiero, and R. Simon, “Shape invariance and a universal form for the Gouy phase,” J. Opt. Soc. Am. A 21, 572-579 (2004).
    [CrossRef]
  27. Actually, a different coherent limit in which the mutual intensity has the form Ja(φ12)=I0exp(imφ12) with integer m could be considered. This corresponds to illuminating the annulus with a coherent field carrying a vortex of order m. In that case, the intensity of the propagated field would have a Jm2 structure.
  28. Quadratic phase factors can be compensated for through the use of a suitable lens.
  29. Stability performances allowing such an approximation are often ensured by commercial lasers.
  30. P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376-1381 (2006).
    [CrossRef] [PubMed]
  31. In principle, each term could be multiplied by an arbitrary phase factor.

2008 (2)

2007 (4)

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes in large-aperture approximation,” Opt. Lett. 32, 611-613 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508-2510 (2007).
[CrossRef] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 353-355 (2007).
[CrossRef]

L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
[CrossRef]

2006 (2)

L. Wang and B. Lü, “Propagation and focal shift of J0-correlated Schell-model beams,” Optik (Stuttgart) 117, 167-172 (2006).
[CrossRef]

P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376-1381 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627-1629 (2003).
[CrossRef] [PubMed]

J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell-Debye equation,” J. Opt. Soc. Am. A 20, 1409-1417 (2003).
[CrossRef]

2001 (1)

1999 (1)

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849-856 (1999).
[CrossRef]

1997 (1)

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

1995 (1)

1991 (1)

1989 (1)

1987 (2)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion of J0-correlated sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

1977 (1)

1975 (1)

E. Wolf and W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205-209 (1975).
[CrossRef]

Agarwal, G. S.

Ayanides, J.-P.

J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell-Debye equation,” J. Opt. Soc. Am. A 20, 1409-1417 (2003).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Burvall, A.

Cada, M.

Carter, W. H.

W. H. Carter and E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785-796 (1977).
[CrossRef]

E. Wolf and W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205-209 (1975).
[CrossRef]

Cincotti, G.

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Friberg, A. T.

Garnier, J.

J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell-Debye equation,” J. Opt. Soc. Am. A 20, 1409-1417 (2003).
[CrossRef]

Gbur, G.

Gori, F.

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33, 1857-1859 (2008).
[CrossRef] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 353-355 (2007).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion of J0-correlated sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2000).

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion of J0-correlated sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

Huang, W.

Indebetouw, G.

Kowarz, M. W.

Lü, B.

L. Wang and B. Lü, “Propagation and focal shift of J0-correlated Schell-model beams,” Optik (Stuttgart) 117, 167-172 (2006).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martinsson, P.

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Morice, O.

J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell-Debye equation,” J. Opt. Soc. Am. A 20, 1409-1417 (2003).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion of J0-correlated sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

Palma, C.

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

Ponomarenko, S. A.

Rao, L.

L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
[CrossRef]

Riesz, F.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Blackie and Sons, 1956).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2000).

Santarsiero, M.

Simon, R.

Sz.-Nagy, B.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Blackie and Sons, 1956).

Turunen, J.

Vahimaa, P.

van Dijk, T.

Vasara, A.

Visser, T. D.

Wang, L.

L. Wang and B. Lü, “Propagation and focal shift of J0-correlated Schell-model beams,” Optik (Stuttgart) 117, 167-172 (2006).
[CrossRef]

Wang, Z.

L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
[CrossRef]

Wolf, E.

W. H. Carter and E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785-796 (1977).
[CrossRef]

E. Wolf and W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205-209 (1975).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yei, P.

L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
[CrossRef]

Zheng, X.

L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849-856 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (3)

L. Rao, X. Zheng, Z. Wang, and P. Yei, “Generation of optical bottle beams through focusing J0-correlated Schell-model vortex beams,” Opt. Commun. 281, 1358-1365 (2007).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion of J0-correlated sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

E. Wolf and W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205-209 (1975).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Optik (Stuttgart) (1)

L. Wang and B. Lü, “Propagation and focal shift of J0-correlated Schell-model beams,” Optik (Stuttgart) 117, 167-172 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Other (10)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

F. Riesz and B. Sz.-Nagy, Functional Analysis (Blackie and Sons, 1956).

Actually, a different coherent limit in which the mutual intensity has the form Ja(φ12)=I0exp(imφ12) with integer m could be considered. This corresponds to illuminating the annulus with a coherent field carrying a vortex of order m. In that case, the intensity of the propagated field would have a Jm2 structure.

Quadratic phase factors can be compensated for through the use of a suitable lens.

Stability performances allowing such an approximation are often ensured by commercial lasers.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2000).

Following the standard usage, we denote by J both the mutual intensity and the Bessel functions of the first kind. No confusion should arise since Bessel functions have a numerical index.

In principle, each term could be multiplied by an arbitrary phase factor.

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Figures (3)

Fig. 1
Fig. 1

Plot of Eq. (14) for different values of ϵ.

Fig. 2
Fig. 2

Coefficients obtained from Eq. (40) (squares), Eq. (41) (open circles), and Eq. (42) (solid circles) as functions of the index n for α R = α σ I = 10 .

Fig. 3
Fig. 3

Rotating transparency coherently illuminated by an annular field distribution.

Equations (52)

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

rect ( x ) = { 1 x 1 2 0 x > 1 2 } ,
tri ( x ) = { 1 x x 1 0 x > 1 } ,
sinc ( x ) = sin ( π x ) π x ,
dir N ( x ) = 1 2 N + 1 sin [ ( 2 N + 1 ) x 2 ] sin ( x 2 ) .
J ( ρ 1 , ρ 2 , 0 ) = V ( ρ 1 , 0 , t ) V * ( ρ 2 , 0 , t ) ,
Q = J ( ρ 1 , ρ 2 , 0 ) g * ( ρ 1 ) g ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
Q 0
J ( ρ 1 , ρ 2 , 0 ) = n Λ n Φ n ( ρ 1 ) Φ n * ( ρ 2 ) ,
J ( ρ 1 , ρ 2 , 0 ) Φ ( ρ 2 ) d 2 ρ 2 = Λ Φ ( ρ 1 ) ,
J ( ρ 1 , ρ 2 , 0 ) = K δ ( ρ 1 a ) δ ( ρ 2 a ) J a ( φ 1 φ 2 ) ,
J ( ρ 2 , ρ 1 , 0 ) = J * ( ρ 1 , ρ 2 , 0 ) ,
J a ( φ 1 φ 2 ) = n = γ n exp [ i n ( φ 1 φ 2 ) ] .
γ n = 1 2 π π π J a ( φ 12 ) exp ( i n φ 12 ) d φ 12 = 1 2 π π π R { J a ( φ 12 ) exp ( i n φ 12 ) } d φ 12 ,
J ̃ R ( n 2 π ) = π π J a ( φ 12 ) exp ( i n φ 12 ) d φ 12 = 2 π γ n ,
J R ( φ 12 ) = I 0 tri ( φ 12 ε ) ,
J R ( φ 12 ) = I 0 rect ( φ 12 2 ε )
J R ( φ 12 ) = I 0 rect ( φ 12 2 π ) m = tri [ ( φ 12 2 π m ) ε ] ,
2 π γ 0 = I 0 ε ,
2 π γ n = 4 I 0 n 2 ε sin 2 ( n ε 2 ) ( n 0 ) .
J a ( φ 12 ) = I 0 ( 1 q ) 2 1 + q 2 2 q cos φ 12 ,
1 q 2 1 + q 2 2 q cos φ 12 = n = q n exp ( i n φ 12 )
2 π γ n = I 0 1 q 1 + q q n .
J a ( φ 12 ) = I 0 dir N ( φ 12 ) .
dir N ( φ 12 ) = 1 2 N + 1 n = N N exp ( i n φ 12 ) ,
J a ( φ 12 ) = I 0 dir N 2 ( φ 12 )
dir N 2 ( φ 12 ) = 1 ( 2 N + 1 ) 2 n = 2 N 2 N ( 2 N + 1 n ) exp ( i n φ 12 ) ,
n = f ( x + n X ) = n = f ̃ ( n X ) exp ( 2 π i n x X ) .
μ π n = exp [ μ ( φ 12 + 2 π n ) 2 ] = n = exp ( n 2 4 μ + i n φ 12 ) ,
J a ( φ 12 ) = I 0 ϑ 3 [ φ 12 2 , exp ( 1 4 μ ) ] ,
ϑ 3 ( x , q ) = n = q n 2 exp ( 2 i n x ) ,
J a ( φ 12 ) = I 0 J 0 ( 2 q sin φ 12 2 ) ,
γ n = J n 2 ( q ) .
J ( r 1 , r 2 , z ) = 1 λ 2 z 2 J ( ρ 1 , ρ 2 , 0 ) × exp { i k 2 z [ ( r 1 ρ 1 ) 2 ( r 2 ρ 2 ) 2 ] } d 2 ρ 1 d 2 ρ 2 ,
J ( r 1 , r 2 , z ) = K a 2 λ 2 z 2 exp [ i k 2 z ( r 1 2 r 2 2 ) ] J a ( φ 1 φ 2 ) × exp { i k a z [ r 1 cos ( φ 1 ϑ 1 ) r 2 cos ( φ 2 ϑ 2 ) ] } d φ 1 d φ 2 ,
J ( r 1 , r 2 , z ) = K a 2 λ 2 z 2 exp [ i k 2 z ( r 1 2 r 2 2 ) ] n = γ n × exp [ i n φ 1 i k a r 1 z cos ( φ 1 ϑ 1 ) ] d φ 1 × exp [ i n φ 2 + i k a r 2 z cos ( φ 2 ϑ 2 ) ] d φ 2 .
J n ( u ) = 1 2 π 0 2 π exp [ i ( n ϵ u sin ϵ ) ] d ϵ .
J ( r 1 , r 2 , z ) = K k 2 a 2 z 2 exp [ i k 2 z ( r 1 2 r 2 2 ) ] × n = γ n J n ( k a r 1 z ) J n ( k a r 2 z ) exp [ i n ( ϑ 1 ϑ 2 ) ] .
Φ n ( r , z ) = exp ( i k 2 z r 2 ) J n ( k a r z ) exp ( i n ϑ ) ,
I ( r , z ) J ( r , r , z ) = K k 2 a 2 z 2 n = γ n J n 2 ( k a r z ) .
n = J n 2 ( x ) = 1 ,
J a ( φ 12 ) 0 I S ( ρ ) J 0 ( 2 α ρ sin φ 12 2 ) ρ d ρ ,
γ n 0 I S ( ρ ) J n 2 ( α ρ ) ρ d ρ .
γ n = J n 2 ( α R ) .
γ n = J n 2 ( α R ) J n + 1 ( α R ) J n 1 ( α R ) .
γ n = exp ( α 2 σ I 2 2 ) I n ( α 2 σ I 2 2 ) ,
V ( φ , t ) = A i ( t ) τ ( φ ω t ) ,
J ( φ 1 , φ 2 ) = 1 T 0 T V ( φ 1 , t ) V * ( φ 2 , t ) d t = 1 T 0 T A i ( t ) 2 τ ( φ 1 ω t ) τ * ( φ 2 ω t ) d t = I i T 0 T τ ( φ 1 ω t ) τ * ( φ 2 ω t ) d t ,
J ( φ 1 , φ 2 ) = J a ( φ 12 ) = I i C r ( φ 12 ) ,
C r ( φ ) = 1 2 π 0 2 π τ * ( ξ ) τ ( ξ + φ ) d ξ .
τ ( φ ) = rect ( φ ϵ ) .
τ ( φ ) = m = τ m exp ( i m φ ) .
C τ ( φ ) = m = τ m 2 exp ( i m φ ) .

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