Abstract

The field configurations and propagation constants of the normal modes of a hollow rectangular dielectric waveguide have been determined. In addition, the coupling coefficients of a Gaussian free-space mode into the normal modes of a square guide were calculated. The attenuation of each mode is found to be inversely proportional to the cube of the guide aperture 2a and proportional to the square of the free-space wavelength λ. For a hollow dielectric square guide with 2a = 1 mm and λ = 10.6 μm, an attenuation of 0.140 dB/m is predicted for SiO2 and 0.032 dB/m for BeO. All modes are found to be hybrid modes, although they very closely approximate linearly polarized TEM modes.

© 1976 Optical Society of America

Full Article  |  PDF Article

Corrections

K. D. Laakmann, "Hollow rectangular dielectric waveguides: errata," Appl. Opt. 15, 2029-2029 (1976)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-15-9-2029

References

  • View by:
  • |
  • |
  • |

  1. E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
  2. P. W. Smith, P. J. Maloney, O. R. Wood, Appl. Phys. Lett. 23, No. 9, 524 (1973).
    [CrossRef]
  3. R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 000 (1973).
  4. O. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
  5. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2133 (1969).
  6. E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1950), pp. 260–261.

1973 (2)

P. W. Smith, P. J. Maloney, O. R. Wood, Appl. Phys. Lett. 23, No. 9, 524 (1973).
[CrossRef]

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 000 (1973).

1969 (2)

O. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2133 (1969).

1964 (1)

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Abrams, R. L.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 000 (1973).

Bridges, W. B.

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 000 (1973).

Goell, O. E.

O. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).

Jordan, E. C.

E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1950), pp. 260–261.

Maloney, P. J.

P. W. Smith, P. J. Maloney, O. R. Wood, Appl. Phys. Lett. 23, No. 9, 524 (1973).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2133 (1969).

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Smith, P. W.

P. W. Smith, P. J. Maloney, O. R. Wood, Appl. Phys. Lett. 23, No. 9, 524 (1973).
[CrossRef]

Wood, O. R.

P. W. Smith, P. J. Maloney, O. R. Wood, Appl. Phys. Lett. 23, No. 9, 524 (1973).
[CrossRef]

Appl. Phys. Lett. (1)

P. W. Smith, P. J. Maloney, O. R. Wood, Appl. Phys. Lett. 23, No. 9, 524 (1973).
[CrossRef]

Bell Syst. Tech. J. (3)

O. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2133 (1969).

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

IEEE J. Quantum Electron. (1)

R. L. Abrams, W. B. Bridges, IEEE J. Quantum Electron. QE-9, 000 (1973).

Other (1)

E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1950), pp. 260–261.

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 (4)

Fig. 1
Fig. 1

Geometry of the hollow rectangular dielectric waveguide.

Fig. 2
Fig. 2

Intensity of coupling coefficients of a Gaussian beam into several modes of a square guide.

Fig. 3
Fig. 3

Amplitude of coupling coefficient of a Gaussian beam into the HE11 mode as a function of beam waist.

Fig. 4
Fig. 4

Amplitude of coupling coefficient of a Gaussian beam into HE13 (HE31) mode of a square guide.

Tables (1)

Tables Icon

Table I Attenuation

Equations (42)

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

( n λ / 4 b ) 1 , ( m λ / 4 a ) 1.
¯ b - 1 1 / 2 n λ 4 b ¯ a - 1 1 / 2 ¯ a m λ 4 a }             for x polarized modes , ¯ a - 1 1 / 2 m λ 4 a ¯ b - 1 1 / 2 ¯ b n λ 4 b }             for y polarized modes ,
k x i k ( m λ 4 a ) 1 ψ n , m k y i k ( n λ 4 a ) 1 ,
L | ϕ n ( ¯ - 1 ) 1 / 2 | 1 ,
L | ϕ n ¯ ( ¯ - 1 ) 1 / 2 | 1.
E z i = - j ( m λ 4 a ) cos ( m π 2 a x + φ x ) cos ( n π 2 b y + φ y ) exp ( - j k g z ) , H z i = j ( 0 μ ) 1 / 2 ( n λ 4 b ) sin ( m λ 2 a x + φ x ) × sin ( n π 2 b y + φ y ) exp ( - j k g z ) , E x i = { sin ( m π 2 a x + φ x ) + sin [ j ¯ a ( ¯ a - 1 ) 1 / 2 k a ( m π 2 a x ) ] × cos ( m π 2 a x + φ x ) } { cos ( n π 2 b y + φ y ) - sin [ j 1 ( ¯ b - 1 ) 1 / 2 1 k b ( n π 2 b y ) ] sin ( n π 2 b y + φ y ) } , H y i = ( 0 μ ) 1 / 2 E x i , E y i = 0 , H x i = 0. }
Im ( k g ) = - 1 a ( m λ 4 a ) 2 Re [ ¯ a ( ¯ a - 1 ) 1 / 2 ] - 1 b ( n λ 4 b ) 2 Re [ 1 ( ¯ b - 1 ) 1 / 2 ] , Re ( k g ) = 2 π λ [ 1 - 1 2 ( λ m 4 a ) 2 - 1 2 ( λ n 4 b ) 2 ] .
E z i = - j ( n λ 4 b ) cos ( m π 2 a x + φ x ) cos ( n π 2 b y + φ y ) exp ( - j k g z ) , H z i = - j ( 0 μ ) 1 / 2 ( m λ 4 a ) sin ( m π 2 a x + φ x ) × sin ( n π 2 b y + φ y ) exp ( - j k g z ) , E y i = { cos ( m π 2 a x + φ x ) - sin [ j ( ¯ a - 1 ) 1 / 2 1 k a ( m π 2 a x ) ] × sin ( m π 2 a x + φ x ) } · { sin ( n π 2 b y + φ y ) + sin [ j ¯ b ( ¯ b - 1 ) 1 / 2 1 k b ( n π 2 b y ) ] × cos ( n π 2 b y + φ y ) } , H x i = - ( 0 μ ) 1 / 2 E x i , E x i = 0 , H x i = 0. }
Im ( k g ) = - 1 a ( m λ 4 a ) 2 Re [ 1 ( ¯ a - 1 ) 1 / 2 ] - 1 b ( n λ 4 b ) 2 Re [ ¯ b ( ¯ b - 1 ) 1 / 2 ] , Re ( k g ) = 2 π λ [ 1 - 1 2 ( λ m 4 a ) 2 - 1 2 ( λ n 4 b ) 2 ] .
E x i = ( μ 0 ) 1 / 2 × H y i = [ sin ( m π x 2 a ) cos ( m π x 2 a ) ] · [ sin ( n π y 2 b ) cos ( n π y 2 b ) ] m , even ; n , even , m , odd ; n , odd .
E y i = ( μ 0 ) 1 / 2 H x i = [ sin ( m π x 2 a ) cos ( m π x 2 a ) ] [ sin ( n π y 2 b ) cos ( n π y 2 b ) ] n , even ; m , even , n , odd ; m , odd .
1 / ( ¯ - 1 ) 1 / 2 .
¯ / ( ¯ - 1 ) 1 / 2 .
ψ ( r ) = exp [ - ( r / w 0 ) 2 ] ( π w 0 2 / 2 ) 1 / 2 ,
- ψ ψ * d x d y = 1.
A n m ( w 0 ) = C n x ( w 0 ) C m y ( w 0 ) ,
C n x ( w 0 ) = ( 2 π ) 1 / 4 ( w 0 a ) - 1 / 2 - a a exp [ - ( 1 w 0 ) 2 ] cos ( n π x 2 a ) d x ,
C n x ( w 0 ) = ( 2 π ) 1 / 4 ( w 0 a ) 1 / 2 exp [ - ( n π w 0 4 a ) 2 ] × Re [ erf ( a w 0 - j n π 4 a · w 0 ) + c . c . ] ,
C n x ( w 0 ) = 0.
E z i = E 3 i cos ( k y i y + φ y ) cos ( k x i x + φ x ) exp ( - j k g 3 ) , H z i = H 3 i sin ( k y i y + φ y ) sin ( k x i x + φ x ) exp ( - j k g 3 ) , E y i = j w μ k 3 - k g 2 + ( k g k y i w u E 3 i + k x i H 3 i ) × cos ( k x i x + φ x ) sin ( k y i y + φ y ) exp ( - j k g 3 ) , E x i = i w μ k 2 - k g 2 - ( k g w u k x i E 3 i + k y i H 3 i ) sin ( k x i x + φ x ) cos ( k y i y + φ y ) exp ( - j k g 3 ) , H x i = j w 0 k 2 - k g 2 - ( k y i E 3 i - k g k x i w 0 H 3 i ) cos ( k x i x + φ x ) sin ( k y i y + φ y ) exp ( - j k g 3 ) , H y i = - j w 0 k 2 - k g 2 - ( k x i E 3 i + k g k y i w 0 H 3 i ) sin ( k x i x + φ x ) cos ( k y i y + φ y ) exp ( - j k g 3 ) ,
E z a = E 3 a cos ( k y i y + φ y ) exp ( - k x a x ) exp ( - j k g 3 ) , H z a = H 3 a sin ( k y i y + φ y ) exp ( - k x a x ) exp ( - j k g 3 ) , E y a = j w μ k a 2 - k g 2 ( + k g w u k y i 3 a - k x a H 3 a ) × sin ( k y i y + φ y ) exp ( - k x a x ) exp ( - j k g 3 ) , E x a = j w μ k a 2 - k g 2 ( - k g w u k x a E 3 a - k y i H 3 z ) × cos ( k y i y + φ y ) exp ( - k x a x ) exp ( - j k g 3 ) , H x a = j w a k a 2 - k g 2 ( - k y i E 3 a + k g w a k x a H 3 a ) × sin ( k y i y + φ y ) exp ( - k x a x ) exp ( - j k g 3 ) , H y a = - j w a k a 2 - k g 2 ( - k x a E 3 a + k g w a k y i H 3 a ) × cos ( K y i y + φ y ) exp ( - k x a x ) exp ( - j k g 3 ) ,
E z b = E 3 b cos ( k x i x + φ x ) exp ( - k y b y ) exp ( - j k g 3 ) , H z b = H 3 b sin ( k x i x + φ x ) exp ( - k y b y ) exp ( - j k g 3 ) , E y b = j w μ k b 2 - k g 2 ( + k g w u k y b E 3 b - k x i H 3 b ) × cos ( k x i x + φ y ) exp ( - j k g 3 - k y b y ) , E x b = - j w μ k b 2 - k g 2 ( - k g w u k x i E 3 b - k y b H 3 b ) × sin ( k x i x + φ x ) exp ( - j k g 3 - k y b y ) , H x b = j w b k b 2 - k g 2 ( - k y b E 3 b - k g w b k x i H 3 b ) × cos ( k x i x + φ x ) exp ( - j k g 3 - k x b y ) , H y b = j w b k b 2 - k g 2 ( - k x i E 3 b - k g w b k y b H 3 b ) × sin ( k x i x + φ x ) exp ( - j k g 3 - k y b y ) .
E 3 b = E 3 i cos ( k y i b + φ y ) exp ( + k y b b ) , H 3 b = H 3 i sin ( k y i b + φ y ) exp ( k y b b ) .
O = ( - k y b ¯ b Δ k b 2 cot ( k y i b + φ y ) + k y i Δ k i 2 k g k x i w ( - 1 Δ k b 2 + 1 Δ k i 2 ) k x i ¯ b Δ k b 2 - k x i Δ k i 2             k g w ( k y b δ k b 2 tan ( k y i b + φ y ) + k y i Δ k i 2 ) ) ( E 3 i H 3 i ) ,
¯ b = b / 0 , Δ k b 2 = k b 2 - k g 2 , Δ k i 2 = k 2 - k g 2 .
O = ( - k y i ¯ a Δ k a 2 + k y i Δ k i 2             k g w [ k x a Δ k a 2 tan ( k x i a + φ x ) + k x i Δ k i 2 ] k x a Δ a Δ k a 2 cot ( k x i a + φ x ) - k x i Δ k i 2             k g w ( - k y i k a 2 + k y i k i 2 ) ) ( E 3 i H 3 i )
k y 2 + k x i 2 + k y i 2 = w 2 μ 0 = k 2 , k y 2 + k x a 2 + k y i 2 = w 2 μ a = k a 2 , k g 2 - k y b 2 + k x i 2 = w 2 μ b = k b 2 .
( - k y i Δ k a 2 + k y i Δ k i 2 ) ( - k y i Δ k a 2 ¯ a + k y i Δ k i 2 ) = [ k x a Δ k a 2 tan ( k x i a + φ x ) + k x i Δ k i 2 ] · [ k x a Δ k a 2 ¯ a cot ( k x i a + φ x ) - k x i Δ k i 2 ] .
( - k x i Δ k b 2 + k x i Δ k i 2 ) ( - k x i ¯ a Δ k b 2 + k x i Δ k i 2 ) = [ k y b Δ k b 2 tan ( k y i b + φ y ) + k y i Δ k i 2 ] × [ k y b Δ k b 2 ¯ b cot ( k y b + φ y ) - k y i Δ k i 2 ] .
a b ; x y , b a ; y x .
tan ( k x i a + φ x ) = { - ¯ a k x i k x a k x a k x i } ,
- k x a 2 = k x i 2 + k 2 ( ¯ a - 1 ) .
| ¯ a k x i k x a | 1 or | k x a k x i | 1.
j ¯ a ( ¯ a - 1 ) 1 / 2 k x i k 1
tan ( k x i a + φ x ) = j ( ¯ a - 1 ) 1 / 2 k k x i 1.
k x i a + φ x = l ( ρ / 2 ) + δ a ,
δ a 1 ; l is an integer .
φ x = 0 ; k x i m π a [ 1 + j ¯ a ( ¯ a - 1 ) 1 / 2 1 k a ] , φ x = π 2 ; k x i ( π / 2 ) + m π a [ 1 + j ¯ a ( ¯ a - 1 ) 1 / 2 1 k a ] .
φ x = 0 ; k x i = ( π / 2 ) + m π a [ 1 + j 1 ( ¯ a - 1 ) 1 / 2 1 k a ] , φ x = π 2 ; k x i = m π a [ 1 + j 1 ( ¯ a - 1 ) 1 / 2 1 k a ] ,
E 3 i H 3 i = ( μ 0 ) 1 / 2 ( - k x i k y i + O ( λ / a ) 2 ) .
cot ( k x i a + φ x ) 0 , tan ( k y i b + φ y ) 0 ,
E 3 i H 3 i = ( μ 0 ) 1 / 2 [ k y i k y i + O ( λ / a ) 2 ] ,

Metrics