Electromagnetic approach to laser resonator analysis

doi:10.1364/OPEX.13.005994

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Electromagnetic approach to laser resonator analysis

Tuomas Vallius, Jani Tervo, Pasi Vahimaa, and Jari Turunen »View Author Affiliations

Optics Express, Vol. 13, Issue 16, pp. 5994-5999 (2005)
http://dx.doi.org/10.1364/OPEX.13.005994

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– Abstract
1. Introduction
2. Theory
3. Results
4. Conclusions
– References and links

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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.

1. Introduction

Electromagnetic analysis of resonators is a challenging problem and exact algebraic solutions are rare [1

A. Yariv, Optical Electronics , 3rd ed. (College Publishing, Holt, 1985).

]. With optically large structures associated with gas and solid-state lasers we may apply geometrical approaches or beam optics [2

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

], and for lossy resonators the iterative Fox-Li algorithm [3

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

] can take into account various defects and alignment errors [4

M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, Cambridge, 2002).

Modeling of semiconductor laser resonators with wavelength-scale features should in principle be based on exact solution of Maxwell’s equations and the appropriate boundary conditions. The lack of exact algebraic solutions that would take into account the boundary conditions at all waveguide discontinuities forces one to resort to numerical methods. One approach is the finite-difference time-domain (FDTD) method, which is suitable for modeling arbitrary time-dependent problems [5

A. Taflove and S. C. Hagness, Computational Electrodymanics: The Finite-Difference Time-Domain Method , 2nd ed. (Artech House, Boston, 2000).

]. However, since each resonator mode in the stable lasing state has an explicit stationary time-dependency, one can apply also frequency domain approaches.

The rigorous Fourier modal method (FMM) is a versatile non-iterative frequency domain approach to various problems involving periodic microstructures [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]

, 8

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications , H. P. Herzig, ed., chap. 2 (Taylor & Francis, London, 1997).

]. It does not contain approximations but is based on solving the allowed modes in the grating region and presenting the transversal structure and grating eigenmodes in the form of Fourier series.

In this paper we apply FMM and the S-matrix (also known as scattering matrix) formalism to laser resonator, considering not the polychromatic field but one harmonic field component of frequency ω = kc, where k = 2π/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

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications , H. P. Herzig, ed., chap. 2 (Taylor & Francis, London, 1997).

, 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]

]. For simplicity, we restrict the discussion to 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

The configuration to be analyzed is presented in Fig. 1. In TE polarization the electric field points in the 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

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]

]. This model is, of course, rigorous only for infinitely periodic laser arrays. To facilitate the analysis of individual resonators we introduce in FMM a series of absorbing regions to decouple the fields within different periods; in the calculation of far-field distributions on individual resonators we only Fourier-transform the field inside a single period between two adjacent absorbing regions.

We divide the structured region into 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.

Mathematically, the absorbing boundaries are introduced by adding an imaginary Gaussian absorber iexp[-(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

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]

]. While FMM (with absorbing boundary conditions) provides the eigenmodes that can exist in all slices, this is not enough to solve resonator problems. Unlike in grating problems of waveguide scattering problems we do not have an incident field from outside the resonator. The stationary field is generated by stimulated emission in the active region inside the resonator and therefore we must formulate an eigenvalue problem also in the z-direction which we proceed to next.

Fig. 1. A general y-invariant periodic resonator structure with imaginary absorption layers inserted at the boundaries of the periods to decouple the interactions between the adjacent resonators in the array.

We can calculate the eigenmodes of the structure in the 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

E_{j y} (x, z) = \sum_{m} {A_{j m} \exp [i β_{j m} (z - z_{j})] + β_{j m} \exp [- i β_{j m} (z - z_{j + 1})]} X_{j m} (x),

(1)

where A_jm and B_jm the amplitudes of the modes propagating to the positive and negative directions, and they constitute the elements of column vectors A_j and B_j ; the eigenvalues β_jm and the eigenvectors X_jm (x) are given by the FMM. We apply this equation to present the field in every layer through j = 1 to j = J-1. Owing to the periodicity of the configuration, transversal dependency X_jm (x) of the modes is expanded in the form of Fourier series. Now the amplitudes in adjacent layers are related through the scattering matrix at the interface z = z _j+1

[\begin{matrix} A_{j + 1} \\ B_{j} \end{matrix}] = [\begin{matrix} T_{u u}^{j ⇄ j + 1} & R_{u d}^{j ⇄ j + 1} \\ R_{d u}^{j ⇄ j + 1} & T_{d d}^{j ⇄ j + 1} \end{matrix}] [\begin{matrix} A_{j} \\ B_{j + 1} \end{matrix}]

(2)

There are several ways to combine the scattering matrices of different layers, but only a few are numerically stable. We apply Li’s well-known recursion formulae [Eq. (15a) in Ref [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]

]], which avoids numerical instabilities even with thick layers, in addition to being numerically feasible.

Having outlined the principles of electromagnetic analysis in view of scattering matrices, we are now in a position to apply the treatment to resonators. We look for stable modes whose transversal distributions are sustained after a round-trip in the resonator (see, e.g., Chap. 14 in Ref. [2

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

]). Assuming that no light source exist outside the modulated region, we write 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

R_{ud}^{0 ⇆ j}

and

R_{du}^{j ⇆ J}

. Hence one round-trip of amplitude vector A_j through the resonator is implemented by the operator

R_{ud}^{0 ⇆ j}

R_{du}^{j ⇆ J}

= 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

A^{R} Γ = M^{0 ⇄ j ⇄ J} A^{R},

(3)

where Γ is a diagonal matrix with eigenvalues Γ_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

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

]), 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).

Since light is partially transmitted through planes 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 - iαλ/(2π), where α is a positive coefficient [2

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

] and its value is assumed to be known from theoretical models [1

A. Yariv, Optical Electronics , 3rd ed. (College Publishing, Holt, 1985).

]. 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

Our example structure is illustrated in Fig. 2. It is a non-periodic resonator requiring absorbing boundary conditions. The gain region made of GaAs (refractive index 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 TiO₂ (refractive index n _TiO₂ = 1.9) and MgF₂(refractive index n _MgF₂ = 1.36) each; thus J = 10. The thicknesses of the layers are 0.117 μm for TiO₂ and 0.572 μm for MgF₂ resulting in reflectance 68%. The layers facing the cavity are MgF₂. The wavelength used in the analysis is 0.67 μm. The thickness of the gain region is w = 1.959 μm and the period is chosen to be d = 50 μm 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).

Applying the effective index method to propagation of the fundamental mode within the cavity, we find that constructive interference appears when the length of the cavity is h = 30.931 μm. 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/μm 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 μm and, besides, the amplification should be larger, α = 8.01 × 10^-3/μm.

The time average of the electric energy density distribution of the stable mode is presented in Fig. 3 at a distance of 1 μm 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.

Fig. 2. Cavity surrounded by dielectric mirrors.

Fig. 3. Time average of electric energy density distribution 〈w _e(x)〉 of the resonating mode of the structure of Fig. 2 at z = -1 μm (dashed line) and in the middle of the resonator, i.e. z = (z ₅ + z ₆)/2 (solid line).

With numerical approaches a question on the efficiency always arises. In our example 50 μm 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

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

, 13

, 14

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).

]. Unfortunately the situation is much worse in the numerical analysis of three-dimensionally modulated cavities, which we will consider elsewhere.

4. Conclusions

In conclusion, we have applied Fourier modal method with scattering matrices to the analysis of laser microcavities by forming an eigenvalue problem from which the modal wave forms inside and outside the cavity can be solved, as well as the resonance wavelengths and threshold gain coefficients of different longitudinal modes. The method is completely rigorous for infinitely periodic cavities and for individual resonators or finite arrays when absorbing regions are introduced appropriately. It can be extended to three-dimensionally structured cavities at the expense of increased computation time. However, since we need to apply absorbing boundaries also in the three-dimensional cases, the computational costs are enormous.

Acknowledgments

The work of Tuomas Vallius and Jani Tervo was financed by the Academy of Finland, project numbers 106410 and 203967, respectively. A part of the research was performed when Jani Tervo was visiting LightTrans GmbH in Jena, Germany. His stay was supported by the Alexander von Humboldt Foundation. The authors also acknowledge the support of the Network of Excellence in Micro-Optics (NEMO).

References and links

1.	A. Yariv, Optical Electronics , 3rd ed. (College Publishing, Holt, 1985).
2.	A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).
3.	A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).
4.	M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, Cambridge, 2002).
5.	A. Taflove and S. C. Hagness, Computational Electrodymanics: The Finite-Difference Time-Domain Method , 2nd ed. (Artech House, Boston, 2000).
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]
8.	J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications , H. P. Herzig, ed., chap. 2 (Taylor & Francis, London, 1997).
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]
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]
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]
14.	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).

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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