Abstract

We examine the angle-impact Wigner function (AIW) as a computational tool for the propagation of nonparaxial quasi-monochromatic light of any degree of coherence past a planar boundary between two homogeneous media. The AIWs of the reflected and transmitted fields in two dimensions are shown to be given by a simple ray-optical transformation of the incident AIW plus a series of corrections in the form of differential operators. The radiometric and leading six correction terms are studied for Gaussian Schell-model fields of varying transverse width, transverse coherence, and angle of incidence.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A.Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. MS69 of Milestone Series (SPIE, 1993).
  2. G. I. Ovchinnikov and V. I. Tatarskii, "On the problem of the relationship between coherence theory and the radiation-transfer equation," Radiophys. Quantum Electron. 15, 1087-1089 (1972).
    [CrossRef]
  3. H. M. Pedersen, "Exact geometrical description of free space radiative energy transfer for scalar wavefields," in Coherence and Quantum Optics VI, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1990), pp. 883-887.
    [CrossRef]
  4. R. G. Littlejohn and R. Winston, "Corrections to classical radiometry," J. Opt. Soc. Am. A 10, 2024-2037 (1993).
    [CrossRef]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-307.
  6. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  7. M.E.Testorf, J.Ojeda-Castanẽda, and A.W.Lohmann, eds., Selected Papers on Phase-Space Optics, Vol. MS181 of Milestone Series (SPIE, 2006).
  8. L. S. Dolin, "Beam description of weakly-inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).
  9. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [CrossRef]
  10. A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. B 9, 1386-1393 (1992).
    [CrossRef]
  11. K. B. Wolf, M. A. Alonso, and G. W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A 16, 2476-2487 (1999).
    [CrossRef]
  12. M. A. Alonso, "Radiometry and wide-angle fields I. Coherent fields in two dimensions," J. Opt. Soc. Am. A 18, 902-909 (2001).
    [CrossRef]
  13. M. A. Alonso, "Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions," J. Opt. Soc. Am. A 18, 910-918 (2001).
    [CrossRef]
  14. M. A. Alonso, "Radiometry and wide-angle wave fields III: partial coherence," J. Opt. Soc. Am. A 18, 2502-2511 (2001).
    [CrossRef]
  15. M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 23, 3128-3135 (2003).
    [CrossRef]
  16. M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized, electromagnetic wave fields in free space," J. Opt. Soc. Am. A 21, 2233-2243 (2004).
    [CrossRef]
  17. L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields," Opt. Commun. 207, 101-112 (2002).
    [CrossRef]
  18. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 40-43.
  19. F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 436, 333-346 (1947).
    [CrossRef]
  20. See Ref. , pp. 252-256, 276-287.
  21. The limits of l¯ have this dependence on ϵ so that the sampling density varies with the AIWs width in l¯.

2004

2003

M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 23, 3128-3135 (2003).
[CrossRef]

2002

L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields," Opt. Commun. 207, 101-112 (2002).
[CrossRef]

2001

1999

1993

1992

1972

G. I. Ovchinnikov and V. I. Tatarskii, "On the problem of the relationship between coherence theory and the radiation-transfer equation," Radiophys. Quantum Electron. 15, 1087-1089 (1972).
[CrossRef]

1968

1964

L. S. Dolin, "Beam description of weakly-inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).

1947

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 436, 333-346 (1947).
[CrossRef]

1932

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Agarwal, G. S.

Alonso, M. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 40-43.

Dolin, L. S.

L. S. Dolin, "Beam description of weakly-inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).

Foley, J. T.

Forbes, G. W.

Friberg, A. T.

Goos, F.

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 436, 333-346 (1947).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 436, 333-346 (1947).
[CrossRef]

Littlejohn, R. G.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-307.

Ovchinnikov, G. I.

G. I. Ovchinnikov and V. I. Tatarskii, "On the problem of the relationship between coherence theory and the radiation-transfer equation," Radiophys. Quantum Electron. 15, 1087-1089 (1972).
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, "Exact geometrical description of free space radiative energy transfer for scalar wavefields," in Coherence and Quantum Optics VI, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1990), pp. 883-887.
[CrossRef]

Tatarskii, V. I.

G. I. Ovchinnikov and V. I. Tatarskii, "On the problem of the relationship between coherence theory and the radiation-transfer equation," Radiophys. Quantum Electron. 15, 1087-1089 (1972).
[CrossRef]

Vicent, L. E.

L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields," Opt. Commun. 207, 101-112 (2002).
[CrossRef]

Walther, A.

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Winston, R.

Wolf, E.

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. B 9, 1386-1393 (1992).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 40-43.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-307.

Wolf, K. B.

Ann. Phys.

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 436, 333-346 (1947).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz.

L. S. Dolin, "Beam description of weakly-inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields," Opt. Commun. 207, 101-112 (2002).
[CrossRef]

Opt. Express

M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 23, 3128-3135 (2003).
[CrossRef]

Phys. Rev.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Radiophys. Quantum Electron.

G. I. Ovchinnikov and V. I. Tatarskii, "On the problem of the relationship between coherence theory and the radiation-transfer equation," Radiophys. Quantum Electron. 15, 1087-1089 (1972).
[CrossRef]

Other

H. M. Pedersen, "Exact geometrical description of free space radiative energy transfer for scalar wavefields," in Coherence and Quantum Optics VI, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1990), pp. 883-887.
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-307.

M.E.Testorf, J.Ojeda-Castanẽda, and A.W.Lohmann, eds., Selected Papers on Phase-Space Optics, Vol. MS181 of Milestone Series (SPIE, 2006).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 40-43.

See Ref. , pp. 252-256, 276-287.

The limits of l¯ have this dependence on ϵ so that the sampling density varies with the AIWs width in l¯.

A.Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. MS69 of Milestone Series (SPIE, 1993).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Geometrical meaning of the impact parameter, l: It is the distance of closest approach to the origin for a ray whose direction is given by u ( θ ) . For the purposes of this paper, the sign of the impact parameter is chosen to coincide with that of the x intercept of the ray.

Fig. 2
Fig. 2

Reflection and transmission of a plane wave at a planar interface.

Fig. 3
Fig. 3

Ray-optical interpretation of L ( l ¯ , θ ¯ ) [Eq. (21)] when ϕ ¯ = 0 for the case of transmission. This function corresponds to the impact parameter of an incident ray which, after transmission, has impact parameter l ¯ and direction θ ¯ .

Fig. 4
Fig. 4

Relative importance O i j of the correction terms for n 1 = 1 , n 2 = 1.4 for (a) a narrow beam ( θ 0 = 0 , σ θ = 0.1 ), (b) a wide beam ( θ 0 = 0 , σ θ = 0.5 ), and (c) an angled-incidence beam ( θ 0 = π 4 , σ θ = 0.1 ). Solid curves indicate transmission, and dashed curves indicate reflection. The dotted vertical line in (c) indicates the case that is examined in depth (Figs. 6, 7, 8, 9).

Fig. 5
Fig. 5

Relative importance O i j of the correction terms for n 1 = 1.4 , n 2 = 1 for (a) a narrow beam ( θ 0 = 0 , σ θ = 0.1 ), (b) a wide beam ( θ 0 = 0 , σ θ = 0.5 ), and (c) an angled-incidence beam ( θ 0 = π 4 , σ θ = 0.1 ). Solid curves indicate transmission, and dashed curves indicate reflection.

Fig. 6
Fig. 6

AIWs for (a) the incident beam [Eq. (8)], (b) the exact transmitted beam, and (c) the exact reflected beam [Eq. (14)] for n 1 = 1 , n 2 = 1.4 , θ 0 = π 4 , σ θ = 0.1 . Note that in (c), the viewpoint is shifted to make the dip visible.

Fig. 7
Fig. 7

Radiometric AIW estimate and corrections for the same beam as in Fig. 6 for transmission [Eqs. (18, 19, 20)]: (a) M R , (b) M 20 , (c) M 21 , (d) M 30 , (e) M 40 , (f) M 41 , and (g) M 42 . The gray rectangle in (a) denotes the boundary of the domain plotted in (b)–(g).

Fig. 8
Fig. 8

Radiometric AIW estimate and corrections for the same beam as in Fig. 6 for reflection [Eqs. (18, 19, 20)]: (a) M R , (b) M 20 , and (c) M 40 . The gray rectangle in (a) denotes the boundary of the domain plotted in (b) and (c).

Fig. 9
Fig. 9

Reflected (left) and transmitted (right) irradiance calculated from (a) exact, (b) radiometric ( M R ) , and (c) first-correction ( M 20 ) AIWs. The incident beam is narrow ( σ θ = 0.1 ) with central direction θ 0 = π 4 indicated by the arrow. Transmitted irradiance is normalized to the maximal value of the radiometric transmitted irradiance in plots (a) and (b) and is multiplied by 50 in order to make it visible in plot (c). The reflected irradiance is normalized to the maximal value of the radiometric reflected irradiance in all plots. In plots (a) and (b) white indicates normalized irradiance of I = 1 , and black indicates I = 0 . In plot (c) white indicates a normalized irradiance correction of I 20 = 1 , and black indicates I 20 = 1 .

Equations (77)

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

u B ( r , u ) = 0 .
I ( r ) = B ( r , u ) d Ω .
U ( r ) = k n 2 π 0 2 π A ( θ ) exp [ i k n u ( θ ) r ] d θ ,
W ( r 1 , r 2 ) = U * ( r 1 ) U ( r 2 ) = k n 2 π 0 2 π 0 2 π A * ( θ 1 ) A ( θ 2 ) exp { i k n [ u ( θ 2 ) r 2 u ( θ 1 ) r 1 ] } d θ 1 d θ 2 .
M ( j ) ( l , θ ) = k n 2 π π π A * ( θ α 2 ) A ( θ + α 2 ) exp [ 2 i k n l sin ( α 2 ) ] cos j ( α 2 ) d α ,
B ( j ) [ r , u ( θ ) ] = M ( j ) [ r u ( θ ) , u ( θ ) ] ,
I ( r ) = W ( r , r ) = 0 2 π M [ l ( r , θ ) , θ ] d θ .
M ( l , θ ) = k n 2 π α M α M A * ( θ α 2 ) A ( θ + α 2 ) exp [ 2 i k n l sin ( α 2 ) ] d α ,
n = n 1 ,
n ¯ = { n 2 , in transmission n 1 , in reflection } ,
θ = θ i ,
θ ¯ = { θ t , in transmission θ r , in reflection } .
θ ( θ ¯ ) = arcsin ( n ¯ sin θ ¯ n ) .
A ¯ ( θ ¯ ) = A ( θ ) τ ( θ ) ,
M ¯ ( l ¯ , θ ¯ ) = k n ¯ 2 π α ¯ M α ¯ M A ¯ * ( θ ¯ α ¯ 2 ) A ¯ ( θ ¯ + α ¯ 2 ) exp [ 2 i k n ¯ l ¯ sin ( α ¯ 2 ) ] d α ¯ ,
M ¯ ( l ¯ , θ ¯ ) = k n ¯ 2 π α ¯ M α ¯ M A * [ θ ( θ ¯ α ¯ 2 ) ] A [ θ ( θ ¯ + α ¯ 2 ) ] ρ ¯ ( θ ¯ α ¯ 2 ) ρ ¯ ( θ ¯ + α ¯ 2 ) × exp { i [ ϕ ¯ ( θ ¯ + α ¯ 2 ) ϕ ¯ ( θ ¯ α ¯ 2 ) ] } exp [ 2 i k n ¯ l ¯ sin ( α ¯ 2 ) ] d α ¯ .
α ¯ M = 2 ( θ ¯ M θ ¯ ) ,
θ ¯ M = arcsin ( n n ¯ sin θ M ) .
α ¯ M = { π 2 θ ¯ , if n 2 < n 1 , transmission 2 ( θ C θ ¯ ) , if n 2 < n 1 , reflection 2 ( θ ¯ C θ ¯ ) , if n 2 > n 1 , transmission π 2 θ ¯ , if n 2 > n 1 , reflection } ,
θ C = arcsin ( n 2 n 1 ) , if n 2 < n 1 ,
θ ¯ C = arcsin ( n 1 n 2 ) , if n 2 > n 1 .
M ¯ ( l ¯ , θ ¯ ) [ 1 + i = 2 j = 0 i 2 C i j ( l ¯ , θ ¯ ) i + j l ¯ i θ ¯ j ] M R ( l ¯ , θ ¯ ) ,
M R ( l ¯ , θ ¯ ) = ρ ¯ 2 ( θ ¯ ) n ¯ θ ( θ ¯ ) n M [ L ( l ¯ , θ ¯ ) , θ ( θ ¯ ) ] ,
C 20 ( l ¯ , θ ¯ ) = 2 ( ρ ¯ 2 ρ ¯ ρ ¯ ) θ 2 3 ρ ¯ 2 θ 2 + ρ ¯ θ [ 2 ρ ¯ θ + θ ( 3 ) ] 8 ρ ¯ 2 θ 2 k 2 n ¯ 2 ,
C 21 ( l ¯ , θ ¯ ) = θ 8 θ k 2 n ¯ 2 ,
C 30 ( l ¯ , θ ¯ ) = k n ¯ l ¯ [ θ 2 θ 4 3 θ 2 + θ ( 3 ) θ ] + ϕ ¯ [ θ θ ( 3 ) 3 θ 2 θ 4 ] + 3 ϕ ¯ θ θ ϕ ¯ ( 3 ) θ 2 24 ρ ¯ 2 θ 2 k 3 n ¯ 3 ,
C 31 ( l ¯ , θ ¯ ) = 0 ,
C 40 ( l ¯ , θ ¯ ) = 1 384 ρ ¯ 3 θ 4 k 4 n ¯ 4 ( 4 ρ ¯ 3 θ 3 θ 2 ρ ¯ θ 2 { 4 ρ ¯ 2 θ 4 + 21 ρ ¯ 2 θ 2 + 12 ρ ¯ ρ ¯ θ θ + θ 2 [ 4 ρ ¯ ρ ¯ ( 3 ) 3 ρ ¯ 2 ] } + ρ ¯ 3 [ 20 θ 4 θ 2 + 33 θ 4 4 θ 5 θ ( 3 ) + 15 θ θ 2 θ ( 3 ) + 5 θ 2 θ θ ( 4 ) ] + 2 ρ ¯ 2 θ [ 4 ρ ¯ θ 5 + 27 ρ ¯ θ θ 2 + ρ ¯ ( 4 ) θ 3 4 ρ ¯ θ 4 θ 15 ρ ¯ θ 3 ρ ¯ θ 2 θ ( 4 ) ] ) ,
C 41 ( l ¯ , θ ¯ ) = ρ ¯ 2 θ 2 θ ( 4 ) + 2 θ [ 2 ρ ¯ 2 ( θ 4 + 3 θ 2 ) + 3 ρ ¯ θ ( 2 ρ ¯ θ ρ ¯ θ ) 6 ρ ¯ 2 θ 2 ] 384 ρ ¯ 2 θ 3 k 4 n ¯ 4 ,
C 42 ( l ¯ , θ ¯ ) = θ 2 128 θ 2 k 4 n ¯ 4 .
L ( l ¯ , θ ¯ ) 1 k n θ ( k n ¯ l ¯ + ϕ ¯ ) .
W ( x 1 , 0 , x 2 , 0 ) = I 0 exp [ x 1 2 2 σ s 2 ] exp [ x 2 2 2 σ s 2 ] exp [ ( x 2 x 1 ) 2 2 δ 2 ] exp [ i k ( x 2 x 1 ) p 0 ] ,
W ̃ ( p 1 , 0 , p 2 , 0 ) = I 0 σ s k σ θ exp [ ( p 2 p 1 ) 2 2 ϵ 2 ] exp [ ( p 1 p 0 ) 2 2 σ θ 2 ] exp [ ( p 2 p 0 ) 2 2 σ θ 2 ] ,
σ θ 2 = δ 2 + 2 σ s 2 k 2 σ s 2 δ 2 ,
ϵ 2 = δ 2 + 2 σ s 2 k 2 σ s 4 .
A * ( θ 1 ) A ( θ 2 ) = W ̃ [ n sin ( θ 1 ) , 0 , n sin ( θ 2 ) , 0 ] cos ( θ 1 ) cos ( θ 2 ) .
O i j = M i j ( l ¯ , θ ¯ ) d θ ¯ d l ¯ M R ( l ¯ , θ ¯ ) d θ ¯ d l ¯ ,
M i j ( l ¯ , θ ¯ ) = C i j ( l ¯ , θ ¯ ) i + j l ¯ i θ ¯ j M R ( l ¯ , θ ¯ ) .
θ ( θ ¯ ± α ¯ 2 ) = θ ± θ α ¯ 2 + θ 2 ( α ¯ 2 ) 2 ± θ ( 3 ) 6 ( α ¯ 2 ) 3 + θ ( 4 ) 24 ( α ¯ 2 ) 4 + O ( α ¯ 5 ) ,
A [ θ ( θ ¯ ± α ¯ 2 ) ] = A ( θ ± θ α ¯ 2 ) + A ( θ ± θ α ¯ 2 ) [ θ 2 ( α ¯ 2 ) 2 ± θ ( 3 ) 6 ( α ¯ 2 ) 3 + θ ( 4 ) 24 ( α ¯ 2 ) 4 + O ( α ¯ 5 ) ] + 1 2 A ( θ ± θ α ¯ 2 ) [ θ 2 4 ( α ¯ 2 ) 4 + O ( α ¯ 5 ) ] + O ( α ¯ 6 ) .
A * [ θ ( θ ¯ α ¯ 2 ) ] A [ θ ( θ ¯ + α ¯ 2 ) ] = A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) + [ θ 2 ( α ¯ 2 ) 2 + θ ( 4 ) 24 ( α ¯ 2 ) 4 ] [ A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) + A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) ] + θ ( 3 ) 6 ( α ¯ 2 ) 3 [ A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) ] + θ 2 8 ( α ¯ 2 ) 4 [ A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) + A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) + 2 A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) ] + O ( α ¯ 5 ) ,
A * [ θ ( θ ¯ α ¯ 2 ) ] A [ θ ( θ ¯ + α ¯ 2 ) ] = { 1 + [ θ 2 ( α ¯ 2 ) 2 + θ ( 4 ) 24 ( α ¯ 2 ) 4 ] θ + θ ( 3 ) 3 θ ( α ¯ 2 ) 3 α ¯ + θ 2 8 ( α ¯ 2 ) 4 2 θ 2 + } A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) .
ρ ¯ ( θ ¯ α ¯ 2 ) ρ ¯ ( θ ¯ + α ¯ 2 ) = ρ ¯ 2 + ( α ¯ 2 ) 2 ( ρ ¯ ρ ¯ ρ ¯ 2 ) + ( α ¯ 2 ) 4 [ 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) 12 ] + O ( α ¯ 6 ) ,
{ ρ ¯ 2 + ( α ¯ 2 ) 2 ( ρ ¯ 2 θ 2 θ + ρ ¯ ρ ¯ ρ ¯ 2 ) + ( α ¯ 2 ) 3 ρ ¯ 2 θ ( 3 ) 3 θ α ¯ + ( α ¯ 2 ) 4 [ 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) 12 + 12 θ ( ρ ¯ ρ ¯ ρ ¯ 2 ) + ρ ¯ 2 θ ( 4 ) 24 θ + ρ ¯ 2 θ 2 8 2 θ 2 ] + } × A * ( θ θ α ¯ 2 ) A ( θ + θ α ¯ 2 ) .
θ ± α 2 θ ± θ α ¯ 2 ,
α ̃ θ α ¯ .
θ = 1 θ θ ¯ ,
2 θ 2 = 1 θ 2 2 θ ¯ 2 θ θ 3 θ ¯ .
[ ρ ¯ 2 + ( α ̃ 2 ) 2 ( ρ ¯ 2 θ 2 θ 3 θ ¯ + ρ ¯ ρ ¯ ρ ¯ 2 θ 2 ) + ( α ̃ 2 ) 3 ρ ¯ 2 θ ( 3 ) 3 θ 3 α ̃ + ( α ¯ 2 ) 4 { 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) 12 θ 4 + 12 ( ρ ¯ ρ ¯ ρ ¯ 2 ) θ 2 θ + ρ ¯ 2 [ θ 2 θ ( 4 ) 3 θ 3 ] 24 θ 7 θ ¯ + ρ ¯ 2 θ 2 8 θ 6 2 θ ¯ 2 } + ] A * ( θ α ̃ 2 ) A ( θ + α ̃ 2 ) .
exp { i [ ϕ ¯ ( θ ¯ + α ¯ 2 ) ϕ ¯ ( θ ¯ α ¯ 2 ) + 2 k n ¯ l ¯ sin ( α ¯ 2 ) ] } = { 1 + i 3 θ 3 ( α ̃ 2 ) 3 [ k n ¯ l ¯ ( θ 2 1 ) + ϕ ¯ θ 2 + ϕ ¯ ( 3 ) ] + O ( α ̃ 6 ) } exp [ i 2 k n L ( l ¯ , θ ¯ ) sin ( α ̃ 2 ) ] ,
L ( l ¯ , θ ¯ ) 1 k n θ ( k n ¯ l ¯ + ϕ ¯ ) .
exp [ i 2 k n L ( l ¯ , θ ¯ ) sin ( α ̃ 2 ) ] ( ρ ¯ 2 + ( α ̃ 2 ) 2 ( ρ ¯ 2 θ 2 θ 3 θ ¯ + ρ ¯ ρ ¯ ρ ¯ 2 θ 2 ) + ( α ̃ 2 ) 3 { ρ ¯ 2 θ ( 3 ) 3 θ 3 α ̃ + i ρ ¯ 2 3 θ 3 [ k n ¯ l ¯ ( θ 2 1 ) + ϕ ¯ θ 2 + ϕ ¯ ( 3 ) ] } + ( α ̃ 2 ) 4 { 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) 12 θ 4 + 12 ( ρ ¯ ρ ¯ ρ ¯ 2 ) θ 2 θ + ρ ¯ 2 [ θ 2 θ ( 4 ) 3 θ 3 ] 24 θ 7 θ ¯ + ρ ¯ 2 θ 2 8 θ 6 2 θ ¯ 2 } + ) A * ( θ α ̃ 2 ) A ( θ + α ̃ 2 ) .
exp [ i 2 k n L sin ( α ̃ 2 ) ] θ ¯ = [ θ ¯ n θ L n ¯ l ¯ ] exp [ i 2 k n L sin ( α ̃ 2 ) ] ,
exp [ i 2 k n L sin ( α ̃ 2 ) ] 2 θ ¯ 2 = [ 2 θ ¯ 2 + O ( α ̃ ) ] exp [ i 2 k n L sin ( α ̃ 2 ) ] ,
exp [ i 2 k n L sin ( α ̃ 2 ) ] α ̃ = [ α ̃ i k n L cos ( α ̃ 2 ) ] exp [ i 2 k n L sin ( α ̃ 2 ) ] ,
( α ̃ 2 ) 3 α ̃ = α ̃ ( α ̃ 2 ) 3 3 2 ( α ̃ 2 ) 2 ,
( ρ ¯ 2 + [ ρ ¯ 2 θ 2 θ 3 θ ¯ + ρ ¯ 2 θ ( ϕ ¯ θ + k n ¯ l ¯ θ ϕ ¯ θ ) 2 k n ¯ θ 4 l ¯ + 2 ρ ¯ ρ ¯ θ 2 ρ ¯ 2 θ ρ ¯ 2 θ ( 3 ) 2 θ 3 ] ( α ̃ 2 ) 2 + { ρ ¯ 2 θ ( 3 ) 3 θ 3 α ̃ + i ρ ¯ 2 3 θ 4 [ k n ¯ l ¯ ( θ 3 θ θ ( 3 ) ) + ϕ ¯ [ θ 3 θ ( 3 ) ] + ϕ ¯ ( 3 ) θ ] } ( α ̃ 2 ) 3 + { 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) 12 θ 4 + 12 ( ρ ¯ ρ ¯ ρ ¯ 2 ) θ 2 θ + ρ ¯ 2 [ θ 2 θ ( 4 ) 3 θ 3 ] 24 θ 7 θ ¯ + ρ ¯ 2 θ 2 8 θ 6 2 θ ¯ 2 } ( α ̃ 2 ) 4 + ) A * ( θ α ̃ 2 ) A ( θ + α ̃ 2 ) exp [ i 2 k n L sin ( α ̃ 2 ) ] .
( ρ ¯ 2 + ρ ¯ 2 θ θ ( 3 ) + 2 ρ ¯ 2 θ 2 2 ρ ¯ ρ ¯ θ 2 2 ρ ¯ 2 θ 2 8 θ 2 k 2 n ¯ 2 2 l ¯ 2 ρ ¯ 2 θ 8 θ k 2 n ¯ 2 3 θ ¯ l ¯ 2 + i ρ ¯ 2 θ ( 3 ) 24 k 3 n ¯ 3 4 l ¯ 3 α ̃ + ρ ¯ 2 { k n ¯ l ¯ [ θ 2 θ 4 3 θ 2 + θ ( 3 ) θ ] + ϕ ¯ [ θ θ ( 3 ) 3 θ 2 θ 4 ] + 3 ϕ ¯ θ θ ϕ ¯ ( 3 ) θ 2 } 24 θ 2 k 3 n ¯ 3 3 l ¯ 3 + 1 192 θ 4 k 4 n ¯ 4 { 4 ( ρ ¯ ρ ¯ ρ ¯ 2 ) ( θ 6 + 6 θ 2 θ 2 ) + θ 4 [ 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) ] + 2 ρ ¯ 2 [ 4 θ 4 θ 2 + θ 2 θ θ ( 4 ) + 6 θ 4 θ 5 θ ( 3 ) + 3 θ θ 2 θ ( 3 ) ] } 4 l ¯ 4 + 12 ( ρ ¯ ρ ¯ ρ ¯ 2 ) θ 2 θ + ρ ¯ 2 [ θ 2 θ ( 4 ) + 9 θ 3 + 4 θ 4 θ ] 384 θ 3 k 4 n ¯ 4 5 θ ¯ l ¯ 4 + ρ ¯ 2 θ 2 128 θ 2 k 4 n ¯ 4 6 θ ¯ 2 l ¯ 4 + ) A * ( θ α ̃ 2 ) A ( θ + α ̃ 2 ) exp [ i 2 k n L sin ( α ̃ 2 ) ] .
M ¯ ( l ¯ , θ ¯ ) = ρ ¯ 2 n ¯ θ n ( 1 + ρ ¯ 2 θ θ ( 3 ) + 2 ρ ¯ 2 θ 2 2 ρ ¯ ρ ¯ θ 2 2 ρ ¯ 2 θ 2 8 ρ ¯ 2 θ 2 k 2 n ¯ 2 2 l ¯ 2 θ 8 θ k 2 n ¯ 2 3 θ ¯ l ¯ 2 + [ k n ¯ l ¯ ( θ 2 θ 4 3 θ 2 + θ ( 3 ) θ ) + ϕ ¯ ( θ θ ( 3 ) 3 θ 2 θ 4 ) + 3 ϕ ¯ θ θ ϕ ¯ ( 3 ) θ 2 ] 24 θ 2 k 3 n ¯ 3 3 l ¯ 3 + 1 192 ρ ¯ 2 θ 4 k 4 n ¯ 4 { 4 ( ρ ¯ ρ ¯ ρ ¯ 2 ) ( θ 6 + 6 θ 2 θ 2 ) + θ 4 [ 3 ρ ¯ 2 + ρ ¯ ρ ¯ ( 4 ) 4 ρ ¯ ρ ¯ ( 3 ) ] + 2 ρ ¯ 2 [ 4 θ 4 θ 2 + θ 2 θ θ ( 4 ) + 6 θ 4 θ 5 θ ( 3 ) + 3 θ θ 2 θ ( 3 ) ] } 4 l ¯ 4 + 12 ( ρ ¯ ρ ¯ ρ ¯ 2 ) θ 2 θ + ρ ¯ 2 [ θ 2 θ ( 4 ) + 9 θ 3 + 4 θ 4 θ ] 384 ρ ¯ 2 θ 3 k 4 n ¯ 4 5 θ ¯ l ¯ 4 + θ 2 128 θ 2 k 4 n ¯ 4 6 θ ¯ 2 l ¯ 4 + ) M ̃ ( L , θ ) + i ρ ¯ 2 θ ( 3 ) 24 k 3 θ n ¯ 2 n 3 l ¯ 3 k n 2 π α ̃ M α ̃ M d d α ̃ { A * ( θ α ̃ 2 ) A ( θ + α ̃ 2 ) exp [ i 2 k n L sin ( α ̃ 2 ) ] } d α ̃ ,
M ̃ ( l , θ ) = k n 2 π α ̃ M α ̃ M A * ( θ α ̃ 2 ) A ( θ + α ̃ 2 ) exp [ i 2 k n l sin ( α ̃ 2 ) ] d α ̃ .
M ̃ ( l , θ ) M ( l , θ ) .
M ¯ ( l ¯ , θ ¯ ) [ 1 + 2 ( ρ ¯ 2 ρ ¯ ρ ¯ ) θ 2 3 ρ ¯ 2 θ 2 + ρ ¯ θ [ 2 ρ ¯ θ + θ ( 3 ) ] 8 ρ ¯ 2 θ 2 k 2 n ¯ 2 θ 8 θ k 2 n ¯ 2 3 θ ¯ l ¯ 2 + k n ¯ l ¯ [ θ 2 θ 4 3 θ 2 + θ ( 3 ) θ ] + ϕ ¯ [ θ θ ( 3 ) 3 θ 2 θ 4 ] + 3 ϕ ¯ θ θ ϕ ¯ ( 3 ) θ 2 24 ρ ¯ 2 θ 2 k 3 n ¯ 3 3 l ¯ 3 + 1 384 ρ ¯ 3 θ 4 k 4 n ¯ 4 ( 4 ρ ¯ 3 θ 3 θ 2 ρ ¯ θ 2 { 4 ρ ¯ 2 θ 4 + 21 ρ ¯ 2 θ 2 + 12 ρ ¯ ρ ¯ θ θ + θ 2 [ 4 ρ ¯ ρ ¯ ( 3 ) 3 ρ ¯ 2 ] } + ρ ¯ 3 [ 20 θ 4 θ 2 + 33 θ 4 4 θ 5 θ ( 3 ) + 15 θ θ 2 θ ( 3 ) + 5 θ 2 θ θ ( 4 ) ] + 2 ρ ¯ 2 θ [ 4 ρ ¯ θ 5 + 27 ρ ¯ θ θ 2 + ρ ¯ ( 4 ) θ 3 4 ρ ¯ θ 4 θ 15 ρ ¯ θ ( 3 ) ρ ¯ θ 2 θ ( 4 ) ] ) 4 l ¯ 4 + ρ ¯ 2 θ 2 θ ( 4 ) + 2 θ [ 2 ρ ¯ 2 ( θ 4 + 3 θ 2 ) + 3 ρ ¯ θ ( 2 ρ ¯ θ ρ ¯ θ ) 6 θ 2 ρ ¯ 2 ] 384 ρ ¯ 2 θ 3 k 4 n ¯ 4 5 θ ¯ l ¯ 4 + θ 2 128 θ 2 k 4 n ¯ 4 6 θ ¯ 2 l ¯ 4 + ] M R ( l ¯ , θ ¯ ) ,
m l ¯ m exp [ i 2 k n L sin ( α ̃ 2 ) ] = ( 2 i k n ¯ θ ) m sin m ( α ̃ 2 ) exp [ i 2 k n L sin ( α ̃ 2 ) ] = ( 2 i k n ¯ θ ) m [ ( α ̃ 2 ) m m 6 ( α ̃ 2 ) m + 2 + O ( α ̃ m + 4 ) ] exp [ i 2 k n L sin ( α ̃ 2 ) ] .
( α ̃ 2 ) 4 exp [ i 2 k n L sin ( α ̃ 2 ) ] = θ 4 16 k 4 n ¯ 4 4 l ¯ 4 exp [ i 2 k n L sin ( α ̃ 2 ) ] + O ( α ̃ 6 ) ,
( α ̃ 2 ) 3 exp [ i 2 k n L sin ( α ̃ 2 ) ] = i θ 3 8 k 3 n ¯ 3 3 l ¯ 3 exp [ i 2 k n L sin ( α ̃ 2 ) ] + O ( α ̃ 5 ) ,
[ ( α ̃ 2 ) 2 1 3 ( α ̃ 2 ) 4 ] exp [ i 2 k n L sin ( α ̃ 2 ) ] + O ( α ̃ 6 ) = θ 2 4 k 2 n ¯ 2 2 l ¯ 2 exp [ i 2 k n L sin ( α ̃ 2 ) ] ,
( α ̃ 2 ) 2 exp [ i 2 k n L sin ( α ̃ 2 ) ] = ( θ 2 4 k 2 n ¯ 2 2 l ¯ 2 + θ 4 48 k 4 n ¯ 4 4 l ¯ 4 ) exp [ i 2 k n L sin ( α ̃ 2 ) ] + O ( α ̃ 6 ) .
θ ¯ M = θ ¯ + α ¯ M 2 .
θ M = θ ( θ ¯ ) + α M 2 .
θ ( θ ¯ M ) = θ M .
θ ( θ ¯ + α ¯ M 2 ) = θ ( θ ¯ ) + α M 2 .
θ ( θ ¯ + α ¯ M 2 ) = θ ( θ ¯ ) + θ ( θ ¯ ) α ¯ M 2 + θ ( θ ¯ ) 2 ( α ¯ M 2 ) 2 + = θ ( θ ¯ ) + α ̃ M 2 + θ ( θ ¯ ) 2 θ 2 ( θ ¯ ) ( α ̃ M 2 ) 2 + ,
α ̃ M = α M θ θ 2 ( α ̃ M 2 ) 2 + .
θ ( m ) α ̃ M m θ m m ! 2 m 1 = B m ( θ ¯ ) ( θ ¯ M θ ¯ ) m { [ 1 ( n ¯ sin θ ¯ n ) 2 ] ( 2 m 1 ) 2 + R m ( θ ¯ ) } ,
B m ( θ ¯ ) = 2 ( 2 m 3 ) ! ! m ! ( n ¯ n ) ( 2 m 1 ) cos m θ ¯ sin ( m 1 ) θ ¯ ,
lim θ ¯ θ ¯ M θ ( m ) α ̃ M m θ m m ! 2 m 1 = 0 .
α ̃ M α M .

Metrics