Angular spectrum theory to calculate coupling efficiency in rectangular waveguide resonators

Optics & Laser Technology 32 (2000) 177±181

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Angular spectrum theory to calculate coupling eciency in rectangular waveguide resonators Xinbing Wang*, Qiyang Xu, Erwu Liu State Key Laboratory of Laser Technology, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People's Republic of China Received 2 December 1999; received in revised form 9 March 2000; accepted 23 March 2000

Abstract Coupling eciency of the rectangular waveguide resonators are discussed in terms of the method of angular spectrum theory. Under the condition given in the paper, the coupling coecients for the EH11 mode in the rectangular waveguide resonator are presented as a function of mirror curvature and position. It is shown that there exist two special geometries to provide low coupling eciency. The method can be applied to the other modes. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Rectangular waveguide; Resonator; Coupling eciency; Angular spectrum

1. Introduction Rectangular waveguide has been increasingly used for high power di€usion-cooled gas laser operating with transverse excitation [1]. Much work has been done on the coupling eciency of square and circular waveguide resonators [2,3]. To calculate coupling eciency, the traditional way is to use scalar di€raction integrals in the spatial domain, but the numerical calculation of the two-fold integral is time-consuming [4]. In this paper a method of angular spectrum theory is used to calculate the coupling eciency in the rectangular waveguide resonator.

2b. Assume that the electric ®eld is x polarized, we only consider the EH11 waveguide modes with the ®eld distribution [5]

U1 …x, y† ˆ U1 …x†U1 … y† ˆ EH11 …x, y, z ˆ 0† ˆ …a  b†

ÿ1=2



px cos 2a





py cos 2b



…1†

According to the angular spectrum theory [6], we obtain

2. Theory Consider a typical waveguide resonator shown in Fig. 1, in which a spherical mirror of radius R is positioned at a distance d from the waveguide end, which coincides with the plane z = 0, the dimension of the mirror is c  c. The cross section of waveguide is 2a  * Corresponding author. Fax: +86-27-8754-3755. E-mail address: [email protected] (X. Wang).

U1 …x, y† ˆ

…1 …1 ÿ1 ÿ1

A 1 … fX , fY †

 exp‰2jp… fX
x ‡ fY
y†Š dfX dfY

where

0030-3992/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 3 7 - 2

…2†

178

X. Wang et al. / Optics & Laser Technology 32 (2000) 177±181

A1 … fX , fY † ˆ A1 … fX †A1 … fY † ˆ

…a …b ÿa ÿb

U1 …x, y†

 exp‰ÿ2jp… fX
x ‡ fY
y†Š dx dy ˆ …a  b† ÿ1=2 ÿp
cos…2pfX
a† ÿp
cos…2pfY
b†  "  2# "  2# p p 2 2 a …2pfX † ÿ b …2pfY † ÿ 2a 2b

…3†

j C11 …N1 , N2 † j 2 ˆj C11 …N1 † j 2 j C11 …N2 † j 2

A2 … fX , fY † ˆ A2 … fX †A2 … fY † ˆ A1 … fX , fY † 

 exp… jk
d † exp ÿ jk
d

l2
f

2 X

‡ l2
f 2

2 Y



…4†

From the Fourier transform and imaging properties of the lens [6], in the waveguide resonator it is true that a 2 =…dl†  …d=a† 2 and b 2 =…dl†  …d=b† 2 , so the Fresnel approximation can be applied, then we can obtain the returned ®eld distribution U3(x, y ) at the waveguide exit from the mirror, U3 …x, y† ˆ U3 …x†U3 … y†  
1 kÿ 2 2 exp… jk
d† exp j x ‡y ˆ jl
d 2d n
 o   ÿ   A2 F p…x, y† F exp jC x 2 ‡ y 2

…5†

x fX ˆ l
d  fY ˆ y l
d where F{ . . . } indicates the Fourier transform, indicates convolution integral, and Cˆ

k …1 ÿ q†, 2d

qˆ

2d , R

…6†

Then the coupling coecient is

The spherical mirror in Fig. 1 can be equivalent to an R/2 lens, then the angular spectrum A2( fX, fY ) of the ®eld distribution U2(x, y ) at the lens is: 

is the pupil function:  1 inside the lens aperture p…x, y† ˆ 0 otherwise

2 … a 2 … b ˆ U3 …x†U1 …x† dx U3 … y†U1 … y† dy ÿa

…7†

ÿb

where N1, N2 is the Fresnel number in the x, y direction, respectively. If we assume c>>a or b, then we have   k 4
exp j x 2 2d r U3 …x† ˆ p
d p a 8   > k
d
l2 2 < f X cos…2pa
fX † exp ÿ j 2 > : …4a
fX † 2 ÿ 1 9 > =  x …8†

F exp… jCx 2 † > ; fX ˆ l
d For the case of R ˆ 2d and R6ˆ2d, we can obtain 1. R ˆ 2d, C = 0, from Eq. (8)   N1
x cos 2p 4 a U3 …x† ˆ r  2 l
d N1
x p ÿ1 4 a a

…9†

p…x, y†

Fig. 1. Schematic of the retangular waveguide resonator.

Fig. 2. EH11 mode-coupling coecient as a function of the Fresnel number N for the three special values of q for the square waveguide.

X. Wang et al. / Optics & Laser Technology 32 (2000) 177±181

179

3. Results and discussions

then … a 2 U3 …x†U1 …x† dx j C11 …N1 † j 2 ˆ ÿa

2   … p cos…2pN
x† cos px 1 18 N 2 1 dx ˆ 2 0 p …4N1
x† ÿ 1

…10†

1. q = 1, corresponding to the semiconfocal geometry, Eq. (10) is used to calculate the coupling eciency, and the results are shown as the curve (a) in Fig. 2. It is shown that high coupling eciency can be obtained when the mirror is properly positioned. As can be seen the Fresnel number is small. 2. q = 2, corresponding to the half-concentric geometry, the condition 2 ÿ q N1 > 25 1 ÿ q

2. R6ˆ2d, from Eq. (8), we obtain 2.1. if 2 ÿ q N1 > 25 1 ÿ q … 1 s   1 x sign 1 ÿ C11 …N1 † ˆ 1ÿq 1ÿq 0  ‡ sign 1 ‡

x 1ÿq





p…1 ÿ q† cos x 2 !   jpN1
q
x 2 px dx cos  exp qÿ1 2

 …11†

where sign ( . . .) is the symbol function. 2.2. if N1 <

jqÿ1j 20 j q j

then …5

p 32 …1 ÿ q† 1 ÿ q 2 0 p !   p
x22ÿq 2p
x cos cos…2p
x† exp ÿ j 1ÿq N1 1 ÿ q   dx 16
x 2 ÿ …1 ÿ q† 2 …16
x 2 ÿ 1† …12†

C11 …N1 † ˆ

A Newton Cotes method was used to calculate the integral to obtain high accuracy. For the case of square waveguide, a=b, which means the Fresnel p number N ˆ N1
N2 ˆ N1 ˆ N2 : We can discuss the coupling eciency according to the parameter q and the Fresnel number N.

is true, so Eq. (11) is used to calculate the coupling eciency, the results are shown as the curve (b) in Fig. 2. It is shown that high coupling eciency can be obtained when the mirror is far away from the waveguide. As can be seen the Fresnel number is very small. 3. q = 0, corresponding to the case of parallel plane resonator, the condition N1 <

jqÿ1j 20 j q j

is true, so Eq. (12) is used to calculate the coupling eciency, the results are shown as the curve (c) in Fig. 2. It is shown that high coupling eciency can be obtained when the mirror is very close to the waveguide. The Fresnel number is large. 4. The same detailed analyses can be done for the di€erent valuez of q, and can obtain a series of curves j C11 …N † j 2 as a function of Fresnel number N (Fig. 3). We can see if the Fresnel number N is

Fig. 3. EH11 mode-coupling coecient as a function of the Fresnel number N at several values of q for the square waveguide.

180

X. Wang et al. / Optics & Laser Technology 32 (2000) 177±181

Fig. 4. EH11 mode-coupling coecient as a function of the Fresnel number N1 and N2 for the retangular waveguide.

large. The coupling coecient increases with the decrease of q. The same analyses can be done for the rectangular waveguide. The results are shown in Fig. 4(a±c) for the case of q = 0, q = 1 and q = 2, which shows the three dimensional distribution of the coupling coecient for N1, N2.

4. Conclusion For di€erent q parameter, according to the di€erent Fresnel number N1,2 in the x or y direction, for di€erent condition we can use Eqs. (10), (11) or (12) to calculate coupling coecient. We only treat the EH11 mode, the other mode can be treated in the same manner, compared with the scalar di€ract integral in the

spatial domain. The angular spectrum analysis in the frequency domain can avoid the two-fold integral calculation, not only simplify the calculation, but also improve the calculation accuracy. In this paper, applying a Fourier optical approach, we presented a semi-analytical model to calculate the coupling eciencies in the rectangular waveguide resonator. One referee pointed out that this problem could be solved with the scalar di€raction integral method utilizing a high-speed numerical procedure like the Fast Fourier Transform (FFT) algorithm. The FFT algorithm has used to optical beam and resonator calculations [7], but there are no reports on the waveguide resonator calculations using the FFT algorithm. In the future, we consider using the FFT algorithm to model the waveguide resonator (including the coupling eciencies).

X. Wang et al. / Optics & Laser Technology 32 (2000) 177±181

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[4] Wang Xinbing, Xu Qiyang, Xie Minjie, Li Zaiguang. Optics Commun 1996;131:41. [5] Leakmann KD, Steier WH. Appl Optics 1976;15:1334. [6] Goodman JW. Introduction to Fourier optics. McGraw-Hill, 1988. [7] Siegman E. Lasers. Oxford University Press, 1986.