Abstract

A compact extended Mie solution to electromagnetic scattering from a cluster of spheres is obtained through indirect mode matching. Our interest is focused on interactions among member spheres, as manifested in multiple scattering from the cluster. We therefore define and investigate the interactive-backscattering cross section of the cluster and the interaction length of member spheres. Numerical results are presented to probe the effect of sphere spacing as well as the effect of look direction and incident polarization on interactive backscattering from silicate clusters. Moreover, we present a brief extinction and absorption study of a soot particle near a cloud droplet.

© 1995 Optical Society of America

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  1. G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377 (1908).
    [Crossref]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. W. Trinks, “Zur Vielfachstreuung an kleinen Kugeln,” Ann. Phys. 22, 561–590 (1935).
    [Crossref]
  4. O. A. Germogenova, “The scattering of a plane electromagnetic wave by two spheres,” Izv. Akad. Nauk. USSR Ser. Geofiz. 4, 403–405 (1963).
  5. S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
    [Crossref]
  6. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  7. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [Crossref]
  8. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II. Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
    [Crossref]
  9. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
    [Crossref] [PubMed]
  10. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
    [Crossref]
  11. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I. Linear chains,” Opt. Lett. 13, 90–92 (1988).
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  12. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II. Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [Crossref] [PubMed]
  13. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [Crossref]
  14. J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
    [Crossref] [PubMed]
  15. K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
    [Crossref] [PubMed]
  16. K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
    [Crossref]
  17. S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).
  18. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).
  19. N. C. Skaropoulos, M. P. Ioannidou, D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
    [Crossref]
  20. P. M. Morse, H. Feshbach, Methods of Theoretical Physics. Part II (McGraw-Hill, New York, 1953).
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  22. R. T. Wang, J. M. Greenberg, D. W. Schuerman, “Experimental results of dependent light scattering by two spheres,” Opt. Lett. 6, 543–545 (1981).
    [Crossref] [PubMed]
  23. G. W. Kattawar, C. E. Dean, “Electromagnetic scattering from two dielectric spheres: comparison between theory and experiment,” Opt. Lett. 8, 48–50 (1983).
    [Crossref] [PubMed]
  24. K. A. Fuller, G. W. Kattawar, R. T. Wang, “Electromagnetic scattering from two dielectric spheres: further comparisons between theory and experiment,” Appl. Opt. 25, 2521–2529 (1986).
    [Crossref] [PubMed]
  25. G. L. Stephens, S. C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
    [Crossref]
  26. J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

1994 (1)

1991 (3)

1990 (1)

G. L. Stephens, S. C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[Crossref]

1988 (2)

1986 (1)

1984 (1)

1983 (1)

1981 (1)

1979 (2)

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
[Crossref] [PubMed]

1971 (2)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II. Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[Crossref]

1968 (1)

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

1967 (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

1963 (1)

O. A. Germogenova, “The scattering of a plane electromagnetic wave by two spheres,” Izv. Akad. Nauk. USSR Ser. Geofiz. 4, 403–405 (1963).

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

1935 (1)

W. Trinks, “Zur Vielfachstreuung an kleinen Kugeln,” Ann. Phys. 22, 561–590 (1935).
[Crossref]

1908 (1)

G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377 (1908).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Alexander, D. R.

Barton, J. P.

Borghese, F.

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II. Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[Crossref]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

Chrissoulidis, D. P.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

Dean, C. E.

Denti, P.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics. Part II (McGraw-Hill, New York, 1953).

Fikioris, J. G.

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

Fuller, K. A.

Germogenova, O. A.

O. A. Germogenova, “The scattering of a plane electromagnetic wave by two spheres,” Izv. Akad. Nauk. USSR Ser. Geofiz. 4, 403–405 (1963).

Greenberg, J. M.

Ioannidou, M. P.

Kanellopoulos, J. D.

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

Kattawar, G. W.

Levine, S.

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II. Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[Crossref]

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Ma, W.

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[Crossref]

Mie, G.

G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377 (1908).
[Crossref]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics. Part II (McGraw-Hill, New York, 1953).

Olaofe, G. O.

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

Saija, R.

Schaub, S. A.

Schuerman, D. W.

Sindoni, O. I.

Skaropoulos, N. C.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

Stephens, G. L.

G. L. Stephens, S. C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[Crossref]

Toscano, G.

Trinks, W.

W. Trinks, “Zur Vielfachstreuung an kleinen Kugeln,” Ann. Phys. 22, 561–590 (1935).
[Crossref]

Tsay, S. C.

G. L. Stephens, S. C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wang, R. T.

Ann. Phys. (2)

W. Trinks, “Zur Vielfachstreuung an kleinen Kugeln,” Ann. Phys. 22, 561–590 (1935).
[Crossref]

G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377 (1908).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (2)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part II. Numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[Crossref]

Izv. Akad. Nauk. USSR Ser. Geofiz. (1)

O. A. Germogenova, “The scattering of a plane electromagnetic wave by two spheres,” Izv. Akad. Nauk. USSR Ser. Geofiz. 4, 403–405 (1963).

J. Colloid Interface Sci. (1)

S. Levine, G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Lett. (4)

Proc. R. Soc. London Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[Crossref]

Q. J. R. Meteorol. Soc. (1)

G. L. Stephens, S. C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[Crossref]

Quart. Appl. Math. (3)

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

Radio Sci. (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Other (4)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
[Crossref]

P. M. Morse, H. Feshbach, Methods of Theoretical Physics. Part II (McGraw-Hill, New York, 1953).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (7)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Truncation number M versus size parameter k0α. The cluster is composed of two identical acrylic spheres (n1 = n2 = 1.61 + j0.004; α1 = α2 = α; O1, O2 on z axis; θi = 0°; polarization 1).

Fig. 3
Fig. 3

Endfire incidence (θi = 0°) upon a bisphere (n1 = n2 = 1.61 + j0.004, k0α1 = k0α2 = 3). (a) Normalized backscattering cross section σ0(î)/σ10 and normalized non-interactive-backscattering cross section σNI0(î)/σ10 versus center-to-center separation k0d. (b) Δσ0(î)/σ10 and normalized interactive-backscattering cross section σI0(î)/σ10 versus k0d.

Fig. 4
Fig. 4

Endfire incidence (θi = 0°) upon a linear cluster composed of N = 2, 3, 4, 5 spheres (n1 = n2 = n3 = n4 = n5 = 1.61 + j0.004, k0α1 = k0α2 = k0α3 = k0α4 = k0α5 = k0α = 3). (a) Normalized backscattering cross section σ0(î)/σ10 and (b) normalized interactive-backscattering cross section σI0(î)/σ10 versus center-to-center separation k0d.

Fig. 5
Fig. 5

Broadside incidence (θi = 90°) upon a linear cluster composed of N = 2, 3, 4, 5 spheres (n1 = n2 = n3 = n4 = n5 = 1.61 + j0.004, k0α1 = k0α2 = k0α3 = k0α4 = k0α5 = k0α = 3). (a) Normalized backscattering cross section σ0(î)/σ10 and (b) normalized interactive-backscattering cross section σI0(î)/σ10 versus center-to-center separation k0d.

Fig. 6
Fig. 6

Effect of the angle of incidence θi on the normalized interaction length L−10dB/α. The linear cluster is composed of N = 2, 3, 4, 5 spheres (n1 = n2 = n3 = n4 = n5 = 1.61 + j0.004, k0α1 = k0α2 = k0α3 = k0α4 = k0α5 = k0α = 3). Incident polarizations (a) 1 (horizontal) and (b) 2 (vertical).

Fig. 7
Fig. 7

Endfire incidence (θi = 0°) upon a water–carbon bisphere (nW = 1.33, nC = 1.8 + j0.5 at λ = 0.55 μm, αW = 1 μm, αC = 0.1 μm, or αC = 0.05 μm). (a) Normalized extinction cross section σe/(σeW + σeC) and (b) albedo W0 versus center-to-center separation k0d.

Equations (31)

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E s ( r s ) = m n C m n s W m n T ( k s , r s ) ,
E 0 ( r ) = E inc ι ( r α ) + E sca ι ( r ) = m n s C m n s 0 W m n T ( k 0 , r s ) .
C m n s 0 = [ δ s , α ξ m n λ m n s ι             a m n s ι             - j δ s , α ξ m n μ m n s ι             b m n s ι ] ,
C m n s = [ c m n s ι             0             d m n s ι             0 ]             s = 1 , 2 , , N ,
ξ m n = ( - 1 ) m j n ( 2 n + 1 ) [ n ( n + 1 ) ] 1 / 2 ,
λ m n s ι = λ m n ι exp ( j k 0 i ^ · d s ) = e ^ ι · C - m n ( θ i , ϕ i ) exp ( j k 0 i ^ · d s ) ,
μ m n s ι = μ m n ι exp ( j k 0 i ^ · d s ) = e ^ ι · B - m n ( θ i , ϕ i ) exp ( j k 0 i ^ · d s ) ,
W m n ( k , r ) = [ M m n ( 1 ) ( k , r )             M m n ( 3 ) ( k , r )             N m n ( 1 ) ( k , r )             N m n ( 3 ) ( k , r ) ] ,
S α ( E 0 × × Q - Q × × E 0 ) · r ^ α d s = 0             α = 1 , 2 , , N ,
I = 2 π α α 2 ( - 1 ) k m n s C m n s 0 U m n s ,
U m n s = ( 1 - δ s , α ) U l 11 ( k 0 , k α , α α ) [ A - k l , 1 m n ( k 0 d s α ) A - k l , 3 m n ( k 0 d s α ) B - k l , 1 m n ( k 0 d s α ) B - k l , 3 m n ( k 0 d s α ) ] + δ m , - k δ n , l δ s , α [ U l 11 ( k 0 , k α , α α ) U l 31 ( k 0 , k α , α α 0 0 ] ,
U n i j ( u , v , r ) = 2 n ( n + 1 ) 2 n + 1 [ v z n ( i ) ( u r ) η n ( j ) ( v r ) - u η n ( i ) ( u r ) z n ( j ) ( v r ) ] ,
I = 2 π α α 2 ( - 1 ) k m n s C m n s 0 V m n s ,
V m n s = ( 1 - δ s , α ) V l 11 ( k 0 , k α , α α ) [ B - k l , 1 m n ( k 0 d s α ) B - k l , 3 m n ( k 0 d s α ) A - k l , 1 m n ( k 0 d s α ) A - k l , 3 m n ( k 0 d s α ) ] + δ m , - k δ n , l δ s , α [ 0 0 V l 11 ( k 0 , k α , α α ) V l 31 ( k 0 , k α , α α ) ] , V n i j ( u , v , r ) = 2 n ( n + 1 ) 2 n + 1 [ u z n ( i ) ( u r ) η n ( j ) ( v r ) - v η n ( i ) ( u r ) z n ( j ) ( v r ) ] .
m n s C m n s 0 U m n s = 0 ,
m n s C m n s 0 V m n s = 0.
n s C - k n s 0 U - k n s = 0 ,
n s C - k n s 0 V - k n s = 0.
S α ( E 0 × × Q - Q × × E 0 ) · r ^ α d s = S α ( E α × × Q - Q × × E α ) · r ^ α d s ,
c m n s ι = U n 13 ( k 0 , k 0 , α s ) U n 11 ( k 0 , k s , α s ) a m n s ι ,
d m n s ι = V n 13 ( k 0 , k 0 , α s ) V n 11 ( k 0 , k s , α s ) b m n s ι ,
E sca ι ( r ) = f ι ( i ^ , s ^ ) exp ( j k 0 r ) r ,
f ι ( i ^ , s ^ ) = 1 k 0 m n s j - n exp ( - j k 0 d s · s ^ ) { [ m a m n s ι P n m ( cos θ ) sin θ + b m n s ι d P n m ( cos θ ) d θ ] θ ^ + j [ m b m n s ι P n m ( cos θ ) sin θ + a m n s ι d P n m ( cos θ ) d θ ] ϕ ^ } exp ( j m ϕ ) .
a m n α ι = - ξ m n λ m n α ι U n 11 ( k 0 , k α , α α ) U n 31 ( k 0 , k α , α α ) ,
b m n α ι = j ξ m n μ m n α ι V n 11 ( k 0 , k α , α α ) V n 31 ( k 0 , k α , α α ) .
f α ι ( i ^ , s ^ ) = 1 k 0 exp ( - j k 0 d α · s ^ ) m n j - n { [ m a m n α ι P n m ( cos θ ) sin θ + b m n α ι d P n m ( cos θ ) d θ ] θ ^ + j [ m b m n α ι P n m ( cos θ ) sin θ + a m n α ι d P n m ( cos θ ) d θ ] ϕ ^ } exp ( j m ϕ ) ,
f NI ι ( i ^ , s ^ ) = s = 1 N f s ι ( i ^ , s ^ ) .
σ I 0 ( i ^ ) = 4 π f ι ( i ^ , - i ^ ) - f NI ι ( i ^ , - i ^ ) 2
r d s M m n ( i ) ( k , r s ) = ν = 1 μ = - ν ν [ A μ ν , i m n M μ ν ( 1 ) ( k , r ) + B μ ν , i m n N μ ν ( 1 ) ( k , r ) ] , N m n ( i ) ( k , r s ) = ν = 1 μ = - ν ν [ A μ ν , i m n N μ ν ( 1 ) ( k , r ) + B μ ν , i m n M μ ν ( 1 ) ( k , r ) ] ,
r d s M m n ( i ) ( k , r s ) = ν = 1 μ = - ν ν [ A μ ν , 1 m n M μ ν ( i ) ( k , r ) + B μ ν , 1 m n N μ ν ( i ) ( k , r ) ] , N m n ( i ) ( k , r s ) = ν = 1 μ = - ν ν [ A μ ν , 1 m n N μ ν ( i ) ( k , r ) + B μ ν , 1 m n M μ ν ( i ) ( k , r ) ] ,
A μ ν , i m n = ( - 1 ) μ p a ( m , n - μ , ν p ) a ( n , ν , p ) × z p ( i ) ( k d s ) P p m - μ ( cos θ s ) exp [ j ( m - μ ) ϕ s ] , B μ ν , i m n = ( - 1 ) μ p a ( m , n - μ , ν p + 1 , p ) b ( n , ν , p + 1 ) × z p + 1 ( i ) ( k d s ) P p + 1 m - μ ( cos θ s ) exp [ j ( m - μ ) ϕ s ] ,

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