Abstract

A geometrical optics theory is used to investigate transmission of radiation in overmoded hollow circular waveguides, with either metal or dielectric walls, in the case that the incident beam is injected into the waveguide by focusing to a waist centered on the guide axis. Expressions are derived for irradiance distributions and transmission coefficients, both for the total radiation and for a plane polarized component surviving in the guide. The effect of directing an initial Gaussian beam at an angle to the guide axis is discussed. Comparisons are made with the work of other authors, and new experimental observations are reported.

© 1985 Optical Society of America

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References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 9 (with corrections to Eqs. 9.85–88).
  2. N. Marcuvitz, Ed., Waveguide Handbook (McGraw-Hill, New York, 1951).
  3. E. A. J. Marcatili, P. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783 (1964).
  4. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Eq. (9.6.19).
  5. Ref. 4, Eq. (11.4.31).
  6. Ref. 4, Eq. (10.2.13).
  7. J. P. Crenn, “Optical Theory of Gaussian Beam Transmission Through a Hollow Circular Dielectric Waveguide,” Appl. Opt. 21, 4533 (1982).
    [CrossRef] [PubMed]
  8. R. L. Abrams, “Coupling Lasers in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8838 (1972).
    [CrossRef]
  9. F. P. Roullard, M. Bass, “Transverse Mode Control in High Gain, Millimeter Bore, Waveguide Lasers,” IEEE J. Quantum Electron. QE-13, 813 (1977).
    [CrossRef]
  10. K. Vogel. “Radiation Characteristics of Light Beams Transmitted Through Straight Dielectric Tubes,” J. Opt. Soc. Am. 56, 1222 (1966).
    [CrossRef]
  11. R. C. Ohlmann, P. L. Richards, M. Tinkham, “Far Infrared Transmission Through Metal Light Pipes,” J. Opt. Soc. Am., 48, 531 (1958).
    [CrossRef]
  12. S. T. Shanahan, N. R. Heckenberg, “A Simple Method for Frequency Controlling CO2 Pump Lasers for Submillimeter Lasers,” J. Phys. 17, 640 (1984).
  13. Ref. 4, Eq. (17.3.35).
  14. J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
    [CrossRef]

1984 (1)

S. T. Shanahan, N. R. Heckenberg, “A Simple Method for Frequency Controlling CO2 Pump Lasers for Submillimeter Lasers,” J. Phys. 17, 640 (1984).

1982 (1)

1977 (1)

F. P. Roullard, M. Bass, “Transverse Mode Control in High Gain, Millimeter Bore, Waveguide Lasers,” IEEE J. Quantum Electron. QE-13, 813 (1977).
[CrossRef]

1975 (1)

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

1972 (1)

R. L. Abrams, “Coupling Lasers in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8838 (1972).
[CrossRef]

1966 (1)

1964 (1)

E. A. J. Marcatili, P. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783 (1964).

1958 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Eq. (9.6.19).

Abrams, R. L.

R. L. Abrams, “Coupling Lasers in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8838 (1972).
[CrossRef]

Bass, M.

F. P. Roullard, M. Bass, “Transverse Mode Control in High Gain, Millimeter Bore, Waveguide Lasers,” IEEE J. Quantum Electron. QE-13, 813 (1977).
[CrossRef]

Birch, J. R.

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

Cook, R. J.

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

Crenn, J. P.

Harding, A. F.

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

Heckenberg, N. R.

S. T. Shanahan, N. R. Heckenberg, “A Simple Method for Frequency Controlling CO2 Pump Lasers for Submillimeter Lasers,” J. Phys. 17, 640 (1984).

Jones, R. G.

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, P. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783 (1964).

Ohlmann, R. C.

Price, G. D.

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

Richards, P. L.

Roullard, F. P.

F. P. Roullard, M. Bass, “Transverse Mode Control in High Gain, Millimeter Bore, Waveguide Lasers,” IEEE J. Quantum Electron. QE-13, 813 (1977).
[CrossRef]

Schmeltzer, P. A.

E. A. J. Marcatili, P. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783 (1964).

Shanahan, S. T.

S. T. Shanahan, N. R. Heckenberg, “A Simple Method for Frequency Controlling CO2 Pump Lasers for Submillimeter Lasers,” J. Phys. 17, 640 (1984).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Eq. (9.6.19).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 9 (with corrections to Eqs. 9.85–88).

Tinkham, M.

Vogel, K.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, P. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783 (1964).

IEEE J. Quantum Electron. (2)

R. L. Abrams, “Coupling Lasers in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8838 (1972).
[CrossRef]

F. P. Roullard, M. Bass, “Transverse Mode Control in High Gain, Millimeter Bore, Waveguide Lasers,” IEEE J. Quantum Electron. QE-13, 813 (1977).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. (1)

S. T. Shanahan, N. R. Heckenberg, “A Simple Method for Frequency Controlling CO2 Pump Lasers for Submillimeter Lasers,” J. Phys. 17, 640 (1984).

J. Phys. D (1)

J. R. Birch, R. J. Cook, A. F. Harding, R. G. Jones, G. D. Price, “The Optical Constants of Ordinary Glass from 0.29 to 4000 cm−1,” J. Phys. D 8, 1353 (1975).
[CrossRef]

Other (6)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 9 (with corrections to Eqs. 9.85–88).

N. Marcuvitz, Ed., Waveguide Handbook (McGraw-Hill, New York, 1951).

Ref. 4, Eq. (17.3.35).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Eq. (9.6.19).

Ref. 4, Eq. (11.4.31).

Ref. 4, Eq. (10.2.13).

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Figures (12)

Fig. 1
Fig. 1

Azimuthal plane section (θ = constant) of a cylindrical waveguide showing a pencil of rays leaving point source O and, after reflections off the guide walls enveloping point Q.

Fig. 2
Fig. 2

Transmittance of an axially directed Gaussian initial beam as a function of the ratio z/a of guidelength to guide radius, Eqs. (54), (56), (57), and (58). The solid curves are for a metal waveguide with q = 0.01. The dotted curves are for a dielectric guide, refractive index ν = 2.5. The divergence parameter t0 of the incident beam is defined by Eq. (15).

Fig. 3
Fig. 3

Examples of the effect on the transmittance of a variation of the direction ϕ1 of an initial plane polarized Gaussian beam via Eq. (64). θ1 identifies the azimuthal plane relative to the initial plane of polarization in which the variation of ϕ1 occurs, z/a = 104, and t0 = 0.01 in all cases, (a) metal guide, q = 0.01; (b) dielectric guide, ν = 2.5.

Fig. 4
Fig. 4

Comparisons among mode analysis [curve a, Eq. (89)], the present theory [curves b.1, Eq. (91), and b.2, Eq. (93)], and Crenn’s theory [curves c.1, Eq. (92), and c.2, Eq. (94)], in the case that a Gaussian beam excites only one mode in a dielectric waveguide. The length of the guide is measured by g, Eq. (90).

Fig. 5
Fig. 5

Comparisons between mode analysis [curve a, Eq. (95)], the present theory [curves b.1, Eq. (96), and b.2, Eq. (98)], and Crenn’s theory [curves c.1, Eq. (97), and c.2, Eq. (99)], in the case that a Gaussian beam excites only two modes in a dielectric waveguide. The length of the guide is measured by g, Eq. (90).

Fig. 6
Fig. 6

Comparison of the present theory [curve (a) Eq. (100)], Crenn’s theory (curve (b) Eq. (101)], and Vogel’s experimental data for the case that light from a ruby laser is injected into glass tubes.

Fig. 7
Fig. 7

Tangential focal line images (concentric circles) of the effective point source produced by injecting a He–Ne laser beam into a copper waveguide: z = 730 mm; a = 9.5 mm; t0 = 0.18.

Fig. 8
Fig. 8

Diametral photodiode scans across copper waveguides of various length with incident light from a He–Ne laser: a = 9.5 mm, t0 = 0.18.

Fig. 9
Fig. 9

Diametral pryoelectric detector scan across a glass waveguide with incident radiation from a CO2 laser: z = 668 mm; a = 12.9 mm; t0 = 0.063.

Fig. 10
Fig. 10

Irradiance passing through a polarizer placed over the output end of a copper waveguide (z = 730 mm, a = 9.5 mm) for incident plane polarized light from a He–Ne laser, t0 = 0.18: (a) χ = 0; (b) χ = 90°, where χ is the angle between polarizer axis and direction of polarization of the incident light.

Fig. 11
Fig. 11

Irregular lines: pyroelectric detector scans across a chord of the circular exit end of a glass waveguide for incident plane polarized radiation from a CO2 laser and with a polarizer masking the photodiode. Smooth lines: theoretical predictions in this case via Eqs. (103) and (104): z = 668 mm, t0 = 0.063, ν = 2.5.

Fig. 12
Fig. 12

Irregular lines: the pyroelectric detector scans across a chord of the circular exit end of a copper waveguide for incident plane polarized radiation from a CO2 laser and with a polarizer masking the photodiode. Smooth lines: theoretical predictions in this case via Eqs. (103) and (104). z = 740 mm, t0 = 0.063, q = 0.0075.

Equations (127)

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a λ .
r 0 < a .
r 0 > λ
E = F ( θ , ϕ ) 1 R exp [ i ( ω t k R ) ] t x ,
t x = cos θ ϕ ˆ = sin θ cos ϕ θ ˆ ,
= cos ϕ x ˆ cos θ sin ϕ z ˆ .
A x t x + A y t y = ( A x cos θ + A y sin θ ) ϕ ˆ ( A x sin θ A y cos θ ) cos ϕ θ ˆ ,
| A x | 2 + | A y | 2 = 1 .
P ( θ , ϕ ) = P ( θ , ϕ ) + P ( θ , ϕ ) = p P p ( θ , ϕ ) ,
P p ( θ , ϕ ) = P ( θ , ϕ ) f p ( θ ) ,
f ( θ ) = | A x cos θ sin θ ± A y sin θ cos θ | 2
P ( θ , ϕ ) = | F ( θ , ϕ ) | 2 2 Z 0 .
f = cos 2 θ , f = sin 2 θ ( plane polarized source ) ,
f = f = 1 / 2 ( circularly polarized source ) .
W ( 0 ) = 0 2 π d θ 0 π / 2 sin ϕ d ϕ P ( θ , ϕ ) .
P = P 0 exp ( 2 tan 2 ϕ / t 0 2 ) ,
W ( 0 ) = π t o 2 P 0 / 2 ,
λ a < t 0 < 1 .
P = P 0 for ϕ ϕ u , = 0 for ϕ > ϕ u , }
W ( 0 ) = 2 π ( 1 cos ϕ u ) P 0 = π ξ u 2 P 0 ,
tan 2 ϕ = tan 2 ϕ 1 + tan 2 ϕ 2 tan ϕ 1 tan ϕ cos ( θ θ 1 ) .
P ( θ , ϕ ) = P 1 exp ( 2 tan 2 ϕ / t 0 2 )
P 0 = P 1 exp ( 2 tan 2 ϕ 1 / t 0 2 ) .
W ( 0 ) = π t 0 2 P 1 / 2 .
| α | = | ν 2 cos 2 ϕ ν 2 sin ϕ ν 2 cos 2 ϕ + ν 2 sin ϕ | , δ = π for ϕ < ϕ B , = 0 for ϕ > ϕ B , | α | = | ν 2 cos 2 ϕ sin ϕ ν 2 cos 2 ϕ + sin ϕ | , δ = π ,
| α | = 2 sin 2 ϕ 2 q sin ϕ + q 2 2 sin 2 ϕ + 2 q sin ϕ + q 2 , δ = tan 1 ( 2 q sin ϕ 2 / sin 2 ϕ q 2 ) , | α | = 1 2 q sin ϕ , δ = tan 1 ( q sin ϕ ) ,
| α p | = exp ( β p ξ ) .
| α | = exp ( β ξ ) for ξ < ξ m = q / 2 = exp ( 2 / β ξ ) for ξ > ξ m . }
ξ = tan ϕ , β p = ( d | α p | d ξ ) ξ ¯ 0
β = 2 ν 2 ν 2 1 , β = 2 ν 2 1 ,
β = 2 / q , β = q .
d A n = r d θ s d ϕ = r s sin ϕ d ω ,
tan ϕ n ± = 2 n a ± r z .
θ = θ for s = s n = even + or s n = odd , θ = θ + π for s = s n = odd + or s n = even . }
| α p | 2 n P p ( θ , ϕ ) d ω d A n = | α p | 2 n P p ( θ , ϕ ) sin ϕ r s .
I p n ± ( z , r , θ ) = | α p ( ϕ n ± ) | 2 n P p ( θ , ϕ n ± ) sin ϕ n ± cos ϕ n ± r s n ± .
W p n ± ( z ) = 0 2 π d θ 0 a r d r I p n ± .
I ( z , r , θ ) = p + n I p n ± .
W ( z ) = p + n W p n ±
= 0 2 π d θ 0 a r d r J ( z , r , θ ) .
z < a 2 / λ .
z a 3 β λ 2 .
ξ n = ( tan ϕ n ) r = 0 = 2 n a z ,
I p n ± = 1 r z exp ( z a β p ξ n 2 ) P ( θ , ξ n ) f p ( θ ) ξ n .
I n ± = 1 r z exp ( 2 β z a ) P ( θ , ξ n ) f p ( θ ) ξ n .
P ¯ ( θ , ξ ) = 1 2 [ P ( θ , ξ ) + P ( θ + π , ξ ) ] .
I ( z , r , θ ) = I + I ,
I p = f p ( θ ) J p ( θ , z ) a r ,
J p = 0 P ¯ ( θ , ξ ) exp ( z a β p ξ 2 ) ξ d ξ
J = 0 ξ m P ¯ ( θ , ξ ) exp ( z a β ξ 2 ) ξ d ξ + exp ( 2 z β a ) ξ m P ¯ ( θ , ξ ) ξ d ξ .
0 2 π f p ( θ ) d θ = π ,
W ( z ) = W + W ,
W p ( z ) = π J p ( z ) .
J p ( z ) = P 0 2 β p ( z / a ) .
I W ( z ) = 1 π a r β f ( θ ) + β f ( θ ) β + β ,
I W ( z ) = 1 2 π a r ( 1 γ cos 2 θ )
γ = β β β + β ,
I W ( z ) = 1 2 π a r .
z a > 2 β t 0 2 .
T ( z ) = W ( z ) W ( 0 ) = a z t 0 2 ( 1 β + 1 β ) .
T ( z ) = a q z t 0 2 .
I W ( z ) = 1 π a r f ( θ ) ,
T ( z ) = T ( z ) + T ( z ) ,
T p ( z ) = π J p ( z ) W ( 0 ) .
I ( z , r , θ ) W ( z ) = 1 π a r f ( θ ) T + f ( θ ) T T .
T p ( z ) = 1 / ( 2 l p ) .
T ( z ) = 1 2 l [ 1 + ( l 1 ) exp ( q 2 l / t 0 2 ) ] .
l p = 1 + 1 2 t 0 2 β p z a .
max { 1 q , q t 0 2 } < z a < 2 q t 0 2 ,
T p ( z ) = 1 exp ( l p ) 2 l p
T ( z ) = 1 2 l [ 1 + ( l 1 q z a ) exp ( q z / a ) ] .
l p = ξ u 2 β p z a = 1 2 t 0 2 β p z a ,
I p ( z , r , θ ) = W ( 0 ) exp ( 2 t 1 2 / t 0 2 ) 2 π a r l p f p ( θ ) × [ 1 + π F p exp ( F p 2 ) erf ( F p ) ] ,
F p ( z , θ ) = 2 l p t 1 t 0 cos ( θ θ 1 ) ,
I p = W ( 0 ) exp ( 2 t 1 2 / t 0 2 ) 2 π a r l p f p ( θ ) ( 1 + 2 F p 2 ) .
T p ( z ) = exp ( G p 2 t 1 2 / t 0 2 ) 2 l p { 1 + H p ( θ 1 ) [ 1 1 exp ( G p ) G p ] } ,
G p ( z ) = 2 t 1 2 t 0 2 l p , H ( θ 1 ) = H ( θ 1 ) = f ( θ 1 ) f ( θ 1 ) = ( | A x | 2 | A y | 2 ) cos 2 θ 1 + 2 | A x | | A y | × cos ( arg A x arg A y ) sin 2 θ 1
T p ( z ) a exp ( 2 t 1 2 / t 0 2 ) z t 0 2 β p .
I χ n ± = I n ± cos 2 ( θ χ ) + I n ± sin 2 ( θ χ ) + I n ± I n ± sin 2 ( θ χ ) cos [ n δ ( ξ n ) ] .
I a n ± = 1 r z exp ( z a β ¯ ξ n 2 ) P ( θ , ξ n ) ξ n sin θ cos θ ,
I a n ± = 1 r z exp ( 1 2 q z a ) P ( θ , ξ n ) ξ n sin θ cos θ .
I χ ( z , r , 0 ) = 1 a r [ J ( θ , z ) cos 2 θ cos 2 ( θ χ ) + J ( θ , z ) sin 2 θ sin 2 ( θ χ ) + 1 2 J a ( θ , z ) sin 2 θ sin 2 ( θ χ ) ]
T χ ( z ) = T [ 1 2 + 1 4 ( 1 + 2 T a T ) cos χ ] .
Γ ( z ) = T 0 T 90 T 0 + T 90
Γ ( z ) = 1 2 + T a T .
Γ = 1 2 + 2 ( ν ν 2 + 1 ) 2 .
f cos 2 θ cos 2 ( θ χ ) , f sin 2 θ sin 2 ( θ χ ) , f a 1 2 sin 2 θ sin 2 ( θ χ ) ,
T χ ( z ) = 1 4 exp ( 2 t 1 2 / t 0 2 ) × { l 1 exp ( G ) [ H 1 ( χ ) + H 2 ( θ 1 , χ ) F 2 ( G ) + H 3 ( θ 1 , χ ) F 3 ( G ) ] + l 1 exp ( G ) [ H 1 ( χ ) H 2 ( θ 1 , χ ) F 2 ( G ) + H 3 ( θ 1 , χ ) F 3 ( G ) ] + l a 1 exp ( G a ) [ cos 2 χ 4 H 3 ( θ 1 , χ ) F 3 ( G a ) } ,
H 1 = 1 + 1 2 cos 2 χ , H 2 = cos 2 θ 1 + cos 2 ( θ 1 χ ) , H 3 = 1 4 cos ( 4 θ 1 2 χ ) , F 2 ( G ) = 1 1 exp ( G ) G , F 3 ( G ) = 1 2 [ 2 + exp ( G ) ] G + 6 [ 1 exp ( G ) ] G 2 , G p ( z ) = 2 t 1 2 t 0 2 l p ( p = , , a ) .
δ = π 2 ξ q for ξ < ξ m , = q ξ for ξ > ξ m .
cos [ ( n δ ( ξ n ) ] = cos ( n 2 ξ n q ) for ξ n < ξ m = cos ( n π ) cos ( n q ξ n ) for ξ n > ξ m .
J a = Re { 0 ξ m P ¯ ( θ , ξ ) exp [ ( 1 + i ) z ξ 2 q a ] ξ d ξ + cos ( q z 2 a ) exp ( q z 2 a ) ξ m P ¯ ( θ , ξ ) exp ( i π z ξ 2 a ) ξ d ξ } .
T a ( z ) = Re ( 1 2 l a [ 1 exp ( q 2 t 0 2 l a ) ] + 1 2 cos L exp ( L ) { exp [ q 2 t 0 2 ( 1 2 i L 1 ) ] + i π q t 0 L 1 exp ( q 2 t 0 2 L 1 2 ) erfc [ q t 0 ( 1 i L 1 ] } ) ,
l a = 1 + ( 1 + i ) t 0 2 z 2 q a , L = q z a , L 1 = π t 0 2 z 4 2 q a . }
T a ( z ) = 1 2 l [ 1 exp ( 1 2 l ) ( cos 1 2 l sin 1 2 l ) ]
T a = 1 2 l [ 1 exp ( L ) ( cos L sin L ) ] + L 1 2 cos L exp ( L ) [ ( L 1 sin L 1 + cos L 1 ) ( L 2 sin L 2 + cos L 2 ) ]
L = q z 2 a , L 1 = π z ξ u 2 a , L 2 = π z ξ m 2 a .
z a > max { 1 q , q t 0 2 } .
P ( ξ ) = P 0 exp ( 2 ξ 2 / t 0 2 ) for ξ < ξ u = a r 0 t 0 , = 0 for ξ > ξ u ,
T ( z ) = W ( z ) W beam .
T ( z ) = T + T ,
T p = 1 exp ( 2 a 2 l p / r 0 2 ) 2 l p
T ( z ) = 2 T a = 1 exp ( 2 a 2 l a / r 0 2 ) l a ,
l a = 1 + 1 2 t 0 2 β ¯ z a , β ¯ = ν 2 + 1 ν 2 1 ,
T = 1 2 l [ 1 + ( l 1 ) exp ( q 2 l / t 0 2 ) ] 1 2 exp ( q z / a ) exp ( 2 a 2 / r 0 2 ) .
t 0 = t 0 1 exp ( a 2 / r 0 2 ) ,
W ( 0 ) W beam = 1 exp ( 2 a 2 / r 0 2 ) .
T ( z ) = [ 1 exp ( 2 a 2 / r 0 2 ) ] ( 1 2 l + 1 2 l ) ,
l p = 1 + 1 2 t 0 β p z a .
T ( z ) = 1 exp ( 2 a 2 / r 0 2 ) l a ,
T = 0.992 exp ( 5.78 g ) .
g = β ¯ λ 2 4 π 2 a 2 z a
T = 1 exp ( 4.82 l ) 2 l + 1 exp ( 4.82 l ) 2 l ,
T = 1 exp ( 4.82 l a ) l a ,
T = 0.496 1 + 10.0 g + 0.496 1 + 1.61 g .
T = 0.992 1 + 5.82 g .
T = 0.90 exp ( 5.78 g ) + 0.10 exp ( 30.5 g ) .
T = 1 exp ( 7.94 l ) 2 l + 1 exp ( 7.94 l ) 2 l ,
T = 1 exp ( 7.94 l a ) l a ,
T = 0.4998 1 + 14.2 g + 0.4998 1 + 2.28 g ,
T = 0.9996 1 + 8.24 g .
T = 0.17 [ exp ( 0.36 / l ) l + exp ( 0.36 / l ) l ] + 0.22 [ exp ( 0.14 / l ) l + exp ( 0.14 / l ) l ] ,
l = 1 + 0.0059 z / a , l = 1 + 0.0027 z / a , l = 1 + 0.0148 z / a , l = 1 + 0.0068 z / a .
T = 0.34 exp ( 0.36 / l a ) l a + 0.44 exp ( 0.14 / l a ) l a ,
f ( r ) = 4 R E ( r 2 / R 2 ) for r < R = π R 2 r n = 0 ( 2 n n ) 2 16 n ( n + 1 ) ( R r ) 2 n for r > R . }
I χ = 0 ° = W ( 0 ) 8 π a r [ ( 3 T + 3 T + 2 T a ) + 4 ( T T ) cos 2 θ + ( T + T 2 T a ) cos 4 θ ] ,
I χ ¯ 90 ° = W ( 0 ) 8 π a r ( T + T 2 T a ) ( 1 cos 4 θ ) .

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