## Coupling losses of rectangular waveguide resonators―Fourier analysis

Optics Express, Vol. 8, Issue 1, pp. 11-16 (2001)

http://dx.doi.org/10.1364/OE.8.000011

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### 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.

© Optical Society of America

## 1. Introduction

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]

## 2. Theoretical analysis

*b*

^{′}×

*b*

^{′}and radius of curvature

*R*is put at a distance

*d*from a rectangular waveguide of dimension 4 2

*a*×2

*b*, which coincides with the

*z*-axis.

*X*-polarized(the results are qualitatively the same for

*Y*-polarized case) and there is only

*EH*

_{11}mode in the rectangular waveguide, then we get

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]

*α*and

*β*are determined by cos

*α*=

*f*

_{x}·

*λ*,cos

*β*=

*f*

_{y}·

*λ*, and the amplitude is determined by

*A*

_{1}(

*f*

_{x},

*f*

_{y}). In face, when cos

^{2}α+cos

^{2}

*β*>1, the

*z*component of plane wave attenuates exponentially, so the corresponding component is ignored in our discussion. Equivalently, we consider the spheric mirror as a lens of focal length equals to

*R*/2, as shown in Fig.2, when the constant phase factor is ignored, the complex amplitude permeation ratio is given by

*k*=2

*π*/

*λ*. According to the fourier analysis theorem, we denote the angular spectrum of

*U*

_{1},

*U*

_{2}by term

*A*

_{1},

*A*

_{2}respectively. When the Fresnel condition is satisfied,

*A*

_{2}can be written in the form

*U*

_{3}at the waveguide entrance, viz.,

*F*{….},then

*x*,

*y*components of

*U*

_{3}are decomposible, we rewrite

*U*

_{3}in the form of

*U*

_{3}(

*x*,

*y*)=

*U*

_{3}(

*x*)·

*U*

_{3}(

*y*),and similarly

*P*(

*x*,

*y*)=

*P*(

*x*)·

*P*(

*y*);

*A*

_{1}(

*f*

_{X},

*f*

_{Y})=

*A*

_{1}(

*f*

_{X})·

*A*

_{l}(

*f*

_{Y});

*A*

_{2}(

*f*

_{X},

*f*

_{Y})=

*A*

_{2}(

*f*

_{X})·

*A*

_{2}(

*f*

_{Y});

*EH*

_{11}(

*x*,

*y*,

*z*=0)=

*EH*

_{11}(

*x*,

*z*=0)·

*EH*

_{11}(

*Y*,

*z*=0)

*X*direction can be written as

*Y*direction is

*U*

_{3}(

*x*) in the following discussion, and for the sake of clarity, the constant phase factor is ignored as it has no effect on our conclusion, we can derive from Eq. (7)

*b*

^{′}>>

*a*, viz.,

*l*>>1,then we have,

*N*

_{1}.

*R*≠2

*d*, with different range of Fresnel number

*N*

_{1}

*N*

_{1}, satisfying

^{-6}. For square waveguide,

*a*=

*b*, viz.,

*N*=

*N*

_{1}=

*N*

_{2}. It is desirable to discuss our results in the form of different values of

*q*.

**A***=*

**q***,Numerical simulation result for Eqs. (13),(16) is shown in Fig.(3).1t is evident from the plot that the Fresnel number*

**1***N*should be small enough to achieve large coupling coefficient |C

_{11}(

*N*)|

^{2}.

**B***=*

**q***,as*

**2***N*

_{1}>0 satisfies condition

*N*should be small enough to achieve large coupling coefficient in co-focal geometry case.

**C***=*

**q***, as*

**0***N*<∞ satisfies condition

*N*is large enough then large coupling coefficient |

*C*

_{11}(

*N*)|

^{2}will be obtained, and the distance

*d*between the mirror and the waveguide could be smaller accordingly, which is desirable in the design of compact waveguide laser.

*As similar analysis in b, c cases, we obtain for different values of*

**D***q*a number of curves in which the coupling coefficient |

*C*

_{11}(

*N*)|

^{2}is plotted as a function of the Fresnel number

*N*as shown in Figs.(6),(7),(8),respectively. It turns out that the coupling coefficient |

*C*

_{11}(

*N*)|

^{2}increases as the value of

*q*decrease for large Fresnel number

*N*.

## 3. Conclusion

*q*, one can use the fomula (13) or (14) to calculate |

*C*

_{11}(

*N*

_{1})|

^{2}and |

*C*

_{11}(

*N*

_{2})|

^{2}under different conditions

*N*

_{1,2}in

*X*or

*Y*direction, respectively. As have been seen in the previous section, a compact design of the rectangular waveguide laser requires that we should use the plate-parallel geometry to achieve low coupling losses. The discussion above is for the

*EH*

_{11}mode, higher-order mode cases will be handle in the same manner. As a comparison with the diffraction integral method in space-domain, the most benefit using fourier analysis in frequency-domain is that it avoid double integral calculation which in turn not only simplifies the numerical simulation and also improves the precision.

## References and links

1. | J.W. Goodman, |

2. | D.R. Hall and H.J. Baker, Laser Focus World. |

3. | C.A. Hill and D.R. Hall, “Coupling Loss theory of single-mode waveguide resonators,” Appl. Opt. |

4. | J.J. Degnan and D.R. Hall, “Finite-aperture waveguide-lasers resonators,” IEEE J.Quant.Electron. |

5. | W. Xinbing, X. Qiyang, X. Minjie, and L. Zaiguang, “Coupling Losses and mode properties in planar waveguide resonators,” Opt.Commun. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 29, 2000

Published: January 1, 2001

**Citation**

Liu Erwu, Cao Mingcui, and Wang Xinbing, "Coupling losses of rectangular waveguide
resonators-- Fourier analysis," Opt. Express **8**, 11-16 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-1-11

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### References

- 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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