Abstract

Coupling losses of rectangular waveguide resonators are discussed in this paper in terms of fourier analysis theorem. Compared to the traditional time-consuming method, the scheme presented in this paper will decrease the simulation time considerably. Under the conditions given in the paper, the EH 11-mode coupling coefficient is calculated numerically. The conclusions can be applied to higher-order mode.

© 2001 Optical Society of America

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References

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  1. J.W. Goodman, Introduction to Fourier Optics, (Second Edition, McGraw-Hill, New York, 1996)
  2. D.R. Hall and H.J. Baker, Laser Focus World. 10, 77 (1989)
  3. C.A. Hill and D.R. Hall, “Coupling Loss theory of single-mode waveguide resonators,” Appl. Opt. 24, 1283–1290 (1985).
    [Crossref] [PubMed]
  4. J.J. Degnan and D.R. Hall, “Finite-aperture waveguide-lasers resonators,” IEEE J.Quant.Electron. QE-9, 901–910 (1973).
    [Crossref]
  5. W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
    [Crossref]

1996 (1)

W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
[Crossref]

1989 (1)

D.R. Hall and H.J. Baker, Laser Focus World. 10, 77 (1989)

1985 (1)

1973 (1)

J.J. Degnan and D.R. Hall, “Finite-aperture waveguide-lasers resonators,” IEEE J.Quant.Electron. QE-9, 901–910 (1973).
[Crossref]

Baker, H.J.

D.R. Hall and H.J. Baker, Laser Focus World. 10, 77 (1989)

Degnan, J.J.

J.J. Degnan and D.R. Hall, “Finite-aperture waveguide-lasers resonators,” IEEE J.Quant.Electron. QE-9, 901–910 (1973).
[Crossref]

Goodman, J.W.

J.W. Goodman, Introduction to Fourier Optics, (Second Edition, McGraw-Hill, New York, 1996)

Hall, D.R.

D.R. Hall and H.J. Baker, Laser Focus World. 10, 77 (1989)

C.A. Hill and D.R. Hall, “Coupling Loss theory of single-mode waveguide resonators,” Appl. Opt. 24, 1283–1290 (1985).
[Crossref] [PubMed]

J.J. Degnan and D.R. Hall, “Finite-aperture waveguide-lasers resonators,” IEEE J.Quant.Electron. QE-9, 901–910 (1973).
[Crossref]

Hill, C.A.

Minjie, X.

W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
[Crossref]

Qiyang, X.

W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
[Crossref]

Xinbing, W.

W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
[Crossref]

Zaiguang, L.

W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
[Crossref]

Appl. Opt. (1)

IEEE J.Quant.Electron. (1)

J.J. Degnan and D.R. Hall, “Finite-aperture waveguide-lasers resonators,” IEEE J.Quant.Electron. QE-9, 901–910 (1973).
[Crossref]

Laser Focus World. (1)

D.R. Hall and H.J. Baker, Laser Focus World. 10, 77 (1989)

Opt.Commun. (1)

W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. 131, 41–46 (1996).
[Crossref]

Other (1)

J.W. Goodman, Introduction to Fourier Optics, (Second Edition, McGraw-Hill, New York, 1996)

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Figures (11)

Fig. 1.
Fig. 1.

Schematic of the rectangular waveguide

Fig.2.
Fig.2.

Equavalent schematic of the waveguide and mirror

Fig.3.
Fig.3.

EH 11 mode-coupling coefficient as a function of the Fresnel number N at q =1 for the square waveguide

Fig.4.
Fig.4.

EH 11 mode-coupling coefficient as a function of the Fresnel number N at q=2 for the square waveguide

Fig. 5.
Fig. 5.

EH 11 mode-coupling coefficient as a function of the Fresnel number N at q = 0 for the square waveguide

Fig.6.
Fig.6.

EH 11 mode-coupling coefficient as a function of the Fresnel number N at several values of q for the square waveguide

Fig.7.
Fig.7.

EH 11 mode-coupling coefficient as a function of the Fresnel number N at several values of q for the square waveguide

Fig.8.
Fig.8.

EH 11 mode-coupling coefficient as a function of the Fresnel number N at q =0.001 for the square waveguide

Fig.9.
Fig.9.

EH 11 mode-coupling coefficient as a function of the Fresnel number N 1 and N 2 at q = 0

Fig.10.
Fig.10.

EH 11 mode-coupling coefficient as a function of the Fresnel number N 1 and N 2 at q =1

Fig. 11.
Fig. 11.

EH 11 mode-coupling coefficient as a function of the Fresnel number N 1 and N 2 at q =2

Equations (19)

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EH 11 ( x , y , z = 0 ) = ( a × b ) 1 2 · cos ( π × x 2 a ) · cos ( π × y 2 b )
EH 11 ( x , y , z = 0 ) = A 1 ( f X , f Y ) · exp [ 2 ( f X · x + f Y · y ) ] df X df Y
A 1 ( f X , f Y ) = a a b b EH 11 ( x , y , z = 0 ) · exp [ 2 ( f X · x + f Y · y ) ] dx dy
= ( a × b ) 1 2 · π · cos ( 2 πf X · a ) a × [ ( 2 πf X ) 2 ( π 2 a ) 2 ] · π · cos ( 2 πf Y · b ) b × [ ( 2 πf Y ) 2 ( π 2 b ) 2 ]
t ( x , y ) = p ( x , y ) · exp [ j k 2 f 0 ( x 2 + y 2 ) ] = p ( x , y ) · exp [ j k R ( x 2 + y 2 ) ]
A 2 ( f X , f Y ) = A 1 ( f X , f Y ) · exp ( jk · d ) · exp [ jk · d · ( λ 2 · f X 2 + λ 2 · f Y 2 2 ) ]
U 3 = 1 · d · exp ( jk · d ) · exp [ j k 2 d · ( x 2 + y 2 ) ] · F { U 2 · t · exp [ j k 2 d · ( x 2 + y 2 ) ] } f Y = y λ · d f X = x λ · d
U 3 = 1 · d · exp ( jk · d ) · exp [ j k 2 d · ( x 2 + y 2 ) ] · { A 2 * F { p ( x , y ) } * F { exp [ jC · ( x 2 + y 2 ) ] } } f Y · = y λ · d f X = x λ · d
C 11 ( N 1 , N 2 ) 2 = C 11 ( N 1 ) 2 · C 11 ( N 2 ) 2
| C 11 ( N 1 ) | 2 = | a a U 3 ( x ) E H 11 ( x , z = 0 ) d x | 2 ; | C 11 ( N 2 ) | 2 = | a a U 3 ( y ) E H 11 ( y , z = 0 ) d y | 2
U 3 ( x ) = 1 λ · d · exp ( j k 2 d x 2 ) · { A 2 ( f X ) * F { p ( x ) } * F { exp ( jCx 2 ) } } f X = x λ · d
A 2 ( f X ) * F { p ( x ) } * F { exp ( j Cx 2 ) } A 2 ( f X ) * F { exp ( j Cx 2 ) }
A 2 ( f X ) * F { p ( x ) } * F { exp ( j Cx 2 ) } = 4 a π · cos ( 2 πa · f X ) ( 4 a · f X ) 2 1 · exp ( j k · d · λ 2 2 · f X 2 ) * F { exp ( j Cx 2 ) }
U 3 ( x ) = 4 · exp ( j k 2 d x 2 ) π λ · d a · { cos ( 2 π a · f X ) · exp ( j k · d · λ 2 2 · f X 2 ) ( 4 a · f X ) 2 1 * F { exp ( j C x 2 ) } } f X = x λ · d
U 3 ( x ) = 4 π λ · d a · cos ( 2 π N 1 · x a ) ( 4 N 1 · x a ) 2 1
C 11 ( N 1 ) 2 = a a U 3 ( x ) · EH 11 ( x , z = 0 ) dx 2 == 0 1 8 N 1 π · cos ( 2 π N 1 · x ) · cos ( πx 2 ) ( 4 N 1 · x ) 2 1 dx 2
C 11 ( N 1 ) = 0 1 1 1 q · ( sign ( 1 x 1 q ) + sign ( 1 + x 1 q ) ) · cos ( π ( 1 q ) 2 · x ) · exp ( j π N 1 · q · x 2 q 1 ) · cos ( πx 2 ) dx
C 11 ( N 1 ) = 0 5 32 π 2 · ( 1 q ) · 1 q · exp ( j π · x 2 N 1 · 2 q 1 q ) · cos ( 2 π · x 1 q ) · cos ( 2 π · x ) [ 16 · x 2 ( 1 q ) 2 ] · ( 16 · x 2 1 ) dx
C 11 ( N 1 , N 2 ) 2 = C 11 ( N 1 ) 2 · C 11 ( N 2 ) 2 = C 11 ( N ) 2

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