## Electromagnetic approach to laser resonator analysis

Optics Express, Vol. 13, Issue 16, pp. 5994-5999 (2005)

http://dx.doi.org/10.1364/OPEX.13.005994

Acrobat PDF (85 KB)

### Abstract

An electromagnetic method based on rigorous diffraction theory of gratings is introduced to analyze the modal structure of semiconductor laser cavities. The approach is based on the use of the Fourier Modal Method, the S-matrix algorithm, and the formulation of an eigenvalue problem from which the wave forms and eigenvalues of the modes can be determined numerically. The method is completely rigorous for infinitely periodic laser arrays and is applicable to individual laser resonators with the introduction of imaginary absorbing regions.

© 2005 Optical Society of America

## 1. Introduction

6. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978). [CrossRef]

7. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

*ω*=

*kc*, where

*k*= 2

*π/λ*,

*c*is the speed of light in vacuum, and

*λ*is the vacuum wavelength. The reader is assumed to be familiar with both approaches since they are well documented [8, 9

9. L. Li, “Formulation and comparison of two recursive matrix algorithms for modelling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996). [CrossRef]

10. L. Li, “Mathematical Reflections on the Fourier modal method in Grating Theory,” in *Mathematical Modeling in Optical Science*,
G. Bao, L. Cowsar, and W. Masters, eds., pp. 111–139 (SIAM, Philadelphia, 2001). [CrossRef]

*y*-invariant geometries and TE-polarization, i.e., the electric field has only a single Cartesian component. Extensions to TM-polarization and three-dimensional geometries are straightforward.

## 2. Theory

*y*-direction, in which the structure is assumed to be invariant. Two semi-infinite homogeneous regions,

*z*<

*z*

_{1}and

*z*>

*z*

_{J}, are separated by an inhomogeneous region that occupies the volume

*z*

_{1}≤

*z*≤

*z*

_{J}and constitutes the actual laser resonator. We assume the structure to be periodic in

*x*-direction, with period

*d*, which allows us to apply grating theory to calculate the field inside it [11

11. P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. **14**, 247–253 (1998). [CrossRef]

*z*-invariant slices and the

*j*:th layer is located between the planes

*z*=

*z*

_{j}and

*z*=

*z*

_{j+1}; the refractive index in the

*j*:th layer is denoted by

*n*

_{j}(

*x*) and it may be complex-valued to allow amplification by stimulated emission as well as absorption. These internal regions may represent, for example, the slice containing the amplifying area and stacks of homogeneous slices made of dielectric materials acting as cavity end mirrors.

*x*/

*W*)

^{2}] (we use

*W*= 1 μm) to the refractive index around the end of the period as illustrated in Fig. 1. This method of isolating the fields in adjacent periods has already been used in the study of various waveguides with abrupt boundaries in the

*z*-direction by FMM [12

12. P. Lalanne and E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

13. J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. **198**, 265–272 (2001). [CrossRef]

*z*-direction which we proceed to next.

*x*-direction by using the standard grating FMM using periodic boundary conditions. The field distribution in each layer

*j*can be expressed as a modal superposition in the form

9. L. Li, “Formulation and comparison of two recursive matrix algorithms for modelling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996). [CrossRef]

*A*

_{0}=

*B*

_{J}= 0. In view of Eq. (2), the propagation of amplitudes

*A*

_{j}and

*B*

_{j}, to the positive and negative directions can be carried out by multiplying them with matrices

*A*

_{j}through the resonator is implemented by the operator

**M**

^{0⇆j⇆J}. If the amplitude distribution is not altered after the round-trip, the multiplication with

**M**

^{0⇆j⇆j}corresponds to a multiplication with a constant complex number. Thus we have an eigenvalue equation

_{l}on the diagonal, and the columns of matrix

**A**

^{R}are the amplitude vectors

*A*

_{j}of the resonance modes that propagate through the resonator without changing their lateral amplitude distribution. We emphasize that Eq. (3) is an exact electromagnetic generalization of classical scalar optical round-trip considerations of the resonators (see, e.g., Chap. 14 in Ref [2]), and correspondingly, the eigenvalue Γ

_{l}must be real for constructive interference to occur. A similar eigenvalue equation can be derived also for amplitudes

*B*

_{j}, and the equation is fulfilled simultaneously with Eq. (3).

*z*=

*z*

_{1}and

*z*=

*z*

_{J}, energy is dissipated outside the resonator. Therefore the condition Γ

_{l}= 1 is not fulfilled without amplification of light. Assume that in layer

*j*we have a region where stimulated emission is possible and coherent light amplification takes place. The saturated value of amplification can be treated by simply replacing the real refractive index

*n*with a complex refractive index

*n*- i

*αλ*/(2

*π*), where

*α*is a positive coefficient [2] and its value is assumed to be known from theoretical models [1]. In practice, the amplification depends on the frequency, but we consider one harmonic eigen frequency of the resonator and thus the dependence can be neglected. Also inhomogeneous gain media can be treated by dividing the structure into

*z*invariant slices where the amplification can be assumed constant enabling analysis of arbitrary gain distributions.

## 3. Results

*n*

_{GaAs}= 3.578) is surrounded on both sides by layers of GaAlAs (refractive index

*n*

_{GaAlAs}= 3.5) to form the waveguide. At both the ends of the cavity there are dielectric mirrors composed of two layers of TiO

_{2}(refractive index

*n*

_{TiO2}= 1.9) and MgF

_{2}(refractive index

*n*

_{MgF2}= 1.36) each; thus

*J*= 10. The thicknesses of the layers are 0.117

*μ*m for TiO

_{2}and 0.572

*μ*m for MgF

_{2}resulting in reflectance 68%. The layers facing the cavity are MgF

_{2}. The wavelength used in the analysis is 0.67

*μ*m. The thickness of the gain region is

*w*= 1.959

*μ*m and the period is chosen to be

*d*= 50

*μ*m to ensure that the tails of the guided modes do not reach the absorbers. Finally, we use refractive indices

*n*

_{0}=

*n*

_{10}= 1 (the cavity is in air).

*h*= 30.931

*μ*m. This approach neglects the absorbers and replaces the modulated region with a constant refractive index. Even though we use Eq. (3), scattering at the ends of the cavity is not taken into account exactly. However, when calculating the eigenvalue Γ by using the above-presented rigorous approach, we get Γ = 1.0983 – 0.1048i with phase angle –0.5° [phase angle i.e. arg(Γ)] indicating that strong destructive interference will take place after a few round-trips. Further, the effective index approach yields

*α*= 6.24 × 10

^{-3}/

*μ*m to compensate for the transmission losses. Analyzing the configuration with the rigorous method, we find the resonance by using parametric optimization to define the parameters that produce the constructive interference. Now the stable mode is obtained at

*h*= 30.924

*μ*m and, besides, the amplification should be larger,

*α*= 8.01 × 10

^{-3}/

*μ*m.

*μ*m outside the resonator mirror and at in the middle of the active region within the cavity, i.e.

*z*= (

*z*

_{5}+

*z*

_{6})/2. We notice that the energy decays rapidly after the higher index core and is essentially zero outside the central region. In consequence, the absorbing boundaries do not have an effect on the field distribution. In addition, the dark central region indicates that the resonating mode is supported by the second mode of the central waveguide.

*μ*m is more than a sufficient value for the (artificial) period and if diffraction orders over the interval [–120,120] are retained in the analysis, excellent convergence is obtained. With these parameters we can carry out the analysis in a few seconds using a standard PC. With small

*y*-invariant waveguide structures the FMM with absorbing boundary conditions is numerically stable and efficient [12

12. P. Lalanne and E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

13. J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. **198**, 265–272 (2001). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | A. Yariv, |

2. | A. E. Siegman, |

3. | A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. |

4. | M. Mansuripur, |

5. | A. Taflove and S. C. Hagness, |

6. | K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. |

7. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

8. | J. Turunen, “Diffraction theory of microrelief gratings,” in |

9. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modelling layered diffraction gratings,” J. Opt. Soc. Am. A |

10. | L. Li, “Mathematical Reflections on the Fourier modal method in Grating Theory,” in |

11. | P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. |

12. | P. Lalanne and E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. |

13. | J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. |

14. | M. G. Moharam and A. Greenwell, “Integrated Output Grating Coupler in Semiconductor Lasers,” in |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(050.1970) Diffraction and gratings : Diffractive optics

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3410) Lasers and laser optics : Laser resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 23, 2005

Revised Manuscript: July 18, 2005

Published: August 8, 2005

**Citation**

Tuomas Vallius, Jani Tervo, Pasi Vahimaa, and Jari Turunen, "Electromagnetic approach to laser resonator analysis," Opt. Express **13**, 5994-5999 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-5994

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### References

- A. Yariv, Optical Electronics, 3rd ed. (College Publishing, Holt, 1985).
- A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).
- A. G. Fox and T. Li, �??Resonant modes in a maser interferometer,�?? Bell Syst. Tech. J. 40, 453�??458 (1961).
- M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, Cambridge, 2002).
- A. Taflove and S. C. Hagness, Computational Electrodymanics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, 2000).
- K. Knop, �??Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,�?? J. Opt. Soc. Am. 68, 1206�??1210 (1978). [CrossRef]
- L. Li, �??Use of Fourier series in the analysis of discontinuous periodic structures,�?? J. Opt. Soc. Am. A 13, 1870�??1876 (1996). [CrossRef]
- J. Turunen, �??Diffraction theory of microrelief gratings,�?? in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed., chap. 2 (Taylor & Francis, London, 1997).
- L. Li, �??Formulation and comparison of two recursive matrix algorithms for modelling layered diffraction gratings,�?? J. Opt. Soc. Am. A 13, 1024�??1035 (1996). [CrossRef]
- L. Li, �??Mathematical Reflections on the Fourier modal method in Grating Theory,�?? in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., pp. 111�??139 (SIAM, Philadelphia, 2001). [CrossRef]
- P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, �??Guided-mode propagation through an ion-exchanged graded-index boundary,�?? Opt. Commun. 14, 247�??253 (1998). [CrossRef]
- P. Lalanne and E. Silberstein, �??Fourier-modal method applied to waveguide computational problems,�?? Opt. Lett. 25, 1092�??1094 (2000). [CrossRef]
- J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, �??Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer�??s star product,�?? Opt. Commun. 198, 265�??272 (2001). [CrossRef]
- M. G. Moharam and A. Greenwell, �??Integrated Output Grating Coupler in Semiconductor Lasers,�?? in 2004 ICO International Conference Optics & Photonics in Technology Frontier, pp. 543�??544 (ICO, Tokyo, 2004).

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