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.

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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.Commim. 131, 41-46 (1996)
    [CrossRef]

Other (5)

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

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]

W. Xinbing, X. Qiyang, X. Minjie and L. Zaiguang, "Coupling Losses and mode properties in planar waveguide resonators," Opt.Commim. 131, 41-46 (1996)
[CrossRef]

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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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