Numerical Simulation of Laser Resonators

Transcription

1 Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004, pp Numerical Simulation of Laser Resonators Jaegwon Yoo, Y. U. Jeong, B. C. Lee and Y. J. Rhee Laboratory for Quantum Optics, Korea Atomic Energy Research Institute, Daejeon S. O. Cho Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Daejeon (Received 30 October 2003) We developed numerical simulation packages for laser resonators on the bases of a pair of integral equations. Two numerical schemes, a matrix formalism and an iterative method, were programmed for finding numeric solutions to the pair of integral equations. The iterative method was tried by Fox and Li, but it was not applicable for high Fresnel numbers since the numerical errors involved propagate and accumulate uncontrollably. In this paper, we implement the matrix method to extend the computational limit further. A great number of case studies are carried out with various configurations of stable and unstable resonators to compute diffractional losses, phase shifts, intensity distributions and phases of the radiation fields on mirrors. Our results presented in this paper show not only a good agreement with the results previously obtained by Fox and Li, but also the legitimacy of our numerical procedures for high Fresnel numbers. PACS numbers: Da, Cb, Pn Keywords: Laser resonator, Numerical computation, Integral equation I. INTRODUCTION Operation of a laser requires a resonant cavity to increase the stimulated emissions induced by the light passing through the gain medium many times. Since the conventional closed resonators are not suitable for operating laser generators in the range of optical frequencies, most resonators have an open cavity structure that is composed of mirrors at the ends of the gain medium. Types of resonant cavities can be categorized into stable, marginally stable, and unstable resonators depending on their mirror configurations. Stable resonators have the lowest diffraction loss, while marginally stable resonators have higher losses, and unstable resonators have the highest losses. The optical resonant cavity determines the frequency and the spatial distribution of the laser beam. The frequencies of the longitudinal modes are separated by the inverse of the round trip time for the light beam in the resonator. Since the gain of a laser is peaked at a transition frequency determined by the energy levels of the gain medium, operation of the laser tends to occur at the longitudinal mode frequency closest to the gain peak. In an open resonator, the transverse mode structure is determined by axial mirrors. The resonator maintains the characteristic configuration of the radiation field with a low loss which is compensated for jgyoo@kaeri.re.kr by the gain from the stimulated emission as the light passes through the gain medium. Thus, it is a very good design strategy to make the mirrors fill all the volume of the active medium contained between the mirrors. The optical resonators must be designed to meet the requirements of low angular divergence and high efficiency by taking into consideration the many aspects of laser physics, such as the geometry of the gain medium, the desired cavity length, the diffractive properties of the radiation, and the single-pass gain. The operational efficiency of a laser generator depends on the quality of the mirror systems, demanding the highest possible reflectance for one mirror and a lossfree output mirror. For high-power lasers, the resonator can be severely damaged if the mirror has some absorption coefficient. For metal-coated mirrors, the reflectance depends on the index of refraction, on the absorption coefficient, and on the angle of incidence. Very high reflectance can be obtained with silver and gold films for the infrared region while the loss in one reflection in the visible range can cause absorption damage to the mirror; especially this is the case for a pulsed high-power laser. The mirrors used in the resonator are coated with multiple layers of dielectric materials that are highly reflective and resistive to damage compared with metal-coated mirrors. Thus, it is necessary to find adequate film materials and film deposition techniques for fabricating mirror structures with extremely low losses suitable for specific

2 -294- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 = K(x, y, z; x, y, z )E(x, y, z )ds,(2) S Fig. 1. Source fields on the observation surface S. applications of laser beams. We begin Section II with an introduction to the diffraction of light in resonant cavities. Section III describes numerical procedures for calculating diffraction integrals. In Section IV, we briefly describe the structure of the code system developed for numerical computations. In Section V, we present typical case studies with stable resonators for understanding the effect of the finite size of the resonator s mirrors on its loss and on the spatial distribution of the modes. Finally, we make our concluding remarks on optical resonant cavities in Section VI. II. DIFFRACTION OF LIGHT IN RESONANT CAVITIES As electromagnetic waves propagate around obstacles, diffraction is an exclusive phenomenon belonging to the realm of wave optics and is inexplicable from the viewpoint of geometrical rays of light. Thus, Huygens principle plays a key role in describing diffraction. For simplicity, we confine ourselves to a scalar-wave theory of electromagnetic waves. Huygens principle says that one can imagine every point on a wave front to be a point source for a spherical wave. Based on this picture, we can calculate the fields on the observation plane S at a distance r away from the known fields in the source plane S as shown in Fig. 1. Setting up coordinate systems (x, y, z) and (x, y, z ) on each plane, we write the contribution of the source field on a field at the observation plane: E(x, y, z) = i e ikr λ r E(x, y, 0) s (1) where λ is the wavelength, k = 2π/λ, r = (x x ) 2 + (y y ) 2 + z 2, and s = x y is an infinitesimal surface element surrounding the point (x, y, z ) on the source plane. Since the complete field at the point (x, y, z) can be obtained by summing up the contribution from all Huygens spherical waves, E(x, y, z) = i λ S e ikr r E(x, y, z )ds where K(x, y, z; x, y, z ) represents a propagator of the scalar wave equation. When a resonator consists of two mirrors having a mirror spacing L and linear dimensions of a 1 and a 2, the Fresnel number of the resonator, F N = a 1 a 2 /λl, plays an important role in the integral kernel. Note that the kernel depends on the shape of the mirror and on the approximations employed in the evaluation of the fields. Depending on the approximations adopted, such as z k(x 2 + y 2 ) max or x 3 k[(x x ) 2 + (y y ) 2 ] 2 max, the diffraction scheme is named the Fraunhofer diffraction or Fresnel diffraction, respectively. Since the z dependent part of the kernel K(x, y, z; x, y, z ) is in a form of on exponential function under one of the above approximations, it satisfies the paraxial wave equation, slightly modified from the Helmholtz equation, ( 2 T + 2ik z ) K(x, y, z; x, y, x ) = 0, (3) where 2 T denotes a transverse Laplacian operator, and the kernel of the integral equation is identical to the Green s function of the Eq. (3). In the case of the light in a resonator, we can apply Eq. (2) repetitively to accomodate waves bouncing between two mirrors in a single integral equation such as E(x, y, 0) = K(x, y ; x, y)e(x, y, z)ds S = K(x, y ; x, y)e(x, y, 0)ds, (4) S where the kernel of the Fredholm integral equation of the first kind is a product of the individual kernels such as K(x, y ; x, y) = K(x, y ; x, y )K(x, y ; x, y)ds S (5) The eigenfunctions of Eq. (5) represent the eigenmodes of the resonator, and the eigenvalues are related to the diffractional power loss and the phase shift per transit. Analytic solutions for the integral equation have been obtained only in the case of the confocal resonator. III. FORMULATION OF NUMERICAL COMPUTATION Our programming target is to find the eigenmodes and the eigenvalues of the integral kernel. Two different approaches are available, depending on the computation procedures: (i) firstly calculate the eigenmode iteratively, and then find the eigenvalue from a simple integration, (ii) find the eigenvalue first, and then the eigenmode by solving the integral equation. Note that the former approach is memory intensive while the later one

3 Numerical Simulation of Laser Resonators Jaegwon Yoo et al is CPU intensive, especially for the cases of high Fresnel numbers. Thus, assessments of each approach should be taken seriously to use the limited computer resources and CPU times efficiently. For the specific case of a resonator consisting of two infinite plane mirrors separated by a distance L, the problem can be reduced to one dimension because of the symmetry in the y direction. To exploit the iterative method, we write the Fresnel integral equation, Eq. (2), in a recursive form expressing the field at each mirror in terms of the reflected field at the other: u q+o (x 2 ) = u q+e (x 1 ) = i Lλ S 1 K(x 2 ; x 1 )u q+e (x 1 )ds 1, i Lλ S 2 K(x 1 ; x 2 )u q+o (x 2 )ds 2, (6) where S i s represent the mirror surfaces with coordinates x i on each surface S 1 and S 2, respectively. To solve a set of the recursive integral equations, we begin the numerical integration with a properly chosen arbitrary trial function for u 0 (x 1 ), where arbitrary means that any function can be the trial function; on the other hand, the properly limits the number of nodal points of the trial function to be the same as that of the eigenmode that we seek. Since the trial function can be represented by a linear superposition of eigenmodes, the resultant function will converge to a single eigenmode after many iterative integrations. Then, the corresponding eigenvalue can be calculated by comparing numerical results between each iterative step; i.e., γ (1) = u q+3 /u q+1 and γ (2) = u q+1 /u q+2 at arbitrary chosen positions on each mirror. The average diffractional loss and phase shift per each transit are given by 1 γ (1) γ (2) and (phase of γ (1) + phase of γ (2) )/2, respectively. To employ a method based on the eigenvalue problem, we write the Fresnel integral equation, Eq. (2), in a matrix equation xk ij u j = γu i, (7) where γ denotes the eigenvalue of the kernel and where discrete representations of the eigenfunction and the kernel function with a stepsize of x are used; i.e., u i u(x i ) K ij K(x i, x j ) x x i x i+1 (8) Note that the problem domain deals with a complex representation of the kernel operator and the eigenvector in Eq. (7). The real representation of the problem domain can be done by increasing the ranks of the kernel operator and the dimensions of the eigenvector as ( ) Re(Kij ) Im(K K ij = ij ) Im(K ij Re(K ij ) ( ) Re(uj ) u j = (9) Im(u j ) Fig. 2. Sample computation shows mode formation as pass number increases. In most cases, the resonator system requires the eigenvalues to be computed for each transit, since the real representation of the kernel operator, K ij, is not a transposed operator of K ji. For a known eigenvalue γ, the corresponding eigenfunction can be easily computed from Eq. (9). In general, the kernel function has an oscillatory part whose period depends on the Fresnel number. The number of periods can be estimated from the argument of the kernel; πf N [(x + x) 2 x 2 ] = 2π, (10) x 1 F N x max. (11) The kernel operator must include at least O(4FN 2 ) matrix elements, and the total memory allocation for the kernel operator is 8 times the number of matrix elements since each element demands 8 bytes in heap memory. Accordingly, demands on the computing resources grow exponentially as the Fresnel number increases since the numerical procedure is supposed to cover all of the highly oscillating regions. Thus, the matrix method has its own merit in computing eigenvalues while it is not competent in calculating eigenfunctions. For the cases with high Fresnel numbers, the iterative method is advantageous over the matrix approach in the use of physical memories. If the numerical errors occurring in each computational step are small enough not to cause computational instabilities, the output of the sequence of Eq. (6) gets closer to the eigenfunction as the iteration goes on. The typical example illustrated in Fig. 2 shows that the iterative method can be employed in finding the eigenmodes and eigenvalues [1, 2]. However, the iterative method turns out to be very unfavorable from the viewpoints of CPU time management since the number of iterations required increases exponentially in the cases of higher Fresnel numbers. Furthermore, since the numerical error involved in each iteration keeps propagating uncontrollably, repeating iterations many times makes

4 -296- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 the case even worse. Besides, the CPU time management problem, for higher Fresnel numbers, the iterative method does not enable one to calculate the eigenvalues based on those inaccurate eigenfunctions. IV. DESCRIPTION OF THE CODE SYSTEM From the viewpoint of mathematics, the calculation of the eigenvalue is closely related with that of the eigenmode; however, in numerical computations, different numerical techniques can be used for calculating them. Our computer codes [3] are written in the C++ programming language on the basis of two methods: (i) The eigenvalues are calculated by making use of the matrix approach, and (ii) the eigenfunctions are approximately computed by using the iterative method. Numerical computations are carried out with a desktop computer. This machine configuration enables one to solve the eigenvalue problem up to the Fresnel number of 360, the maximum Fresnel number the physical memory of the desktop computer can accommodate. Even though the machine limit may be relieved by exploiting virtual memories swapped to a computer s hard disk drive, we have not considered this trick because of the unrealistic demand on CPU time. The code system, OC MODE, is composed of two packages: EIGENV for computing the eigenvalues and EIGENF for calculating the eigenmodes. Note that the eigenvalue computation for an asymmetric resonator takes twice the CPU time as that for a symmetric case since the EIGENV package is composed of two almost identical subroutines to deal with the asymmetric configuration. Each package requires its own input parameters, such as the mirror shape, the mirror size, the gap distance between mirrors, and the wavelength. Notice that the EIGENV must be run before executing the EIGENF package since the latter requires the corresponding eigenvalue as one of the input parameters. The key mathematical routines employed in writing the OC MODE are adopted from Ref. 4-6 after making minor modifications. The numerical procedures implemented in the EIGENV package compute the eigenvalues of the kernel operator. The QR factorization has been known as the most efficient and widely used general method for calculating all of the eigenvalues of a matrix. Since the QR method is CPU time intensive when repeated many times, the matrix, however, should be reduced to a simpler form for which the QR factorization is much less expensive. Notice that the real matrix representation of the kernel operator, Eq. (9), is a nonsymmetric matrix. A nonsymmetric matrix can be reduced to a Hessenberg matrix which is upper triangular, except for a single nonzero subdiagonal. Since the eigenvalues of a nonsymmetric matrix are very defective and sensitive to small changes in the matrix elements, no numerical procedure can overcome these intrinsic difficulties, and the presence of a rounding error can make the situation even worse. To reduce the sensitivity of the eigenvalues to rounding errors, we balance the matrix by using a procedure that requires corresponding rows and columns of the matrix to have comparable norms, reducing the overall norm of the matrix while leaving the eigenvalues unchanged. The procedure for balancing must track down the accumulated similarity transformation of the original matrix to compute the eigenvectors, however, it is extremely complicated to implement and requires excessive computing resources. The numerical procedures implemented in the EIGENF package calculates the eigenmode of the resonator system iteratively. The integrand is composed of two parts: One is the kernel function of the Fredholm integral equation which is a highly oscillating function for the cases of high Fresnel numbers, the other is the field function which is a smooth function. Since the recursive representation of the integral equation takes the previous result of the integration as an approximate eigenmode, it should be properly normalized before being inserted into the right-hand side of the integral equation and the discreteness in the representation of the approximate eigenmode should be relieved by making use of an interpolation procedure or by averaging two adjacent values. For the cases of higher Fresnel numbers, the iterative numerical integration must be performed as precise as possible to make the result converge to the proper eigenmode. Very sophisticated routines developed for numerical integration that have the capability of checking the convergence and adaptively adjusting the step size are very expensive from the viewpoint of CPU time management. On the other hand, the routines based on the Gaussian quadrature are straightforward to implement. Note that the Gaussian quadrature formula works accurately only for cases with a smooth integrand. Even though the integrands are highly oscillating functions for the cases of high Fresnel numbers, the range of integration can be divided by using the subranges estimated in Eq. (11) in which the integrand becomes smooth. The convergence is checked by comparing the normalized result with the previously normalized one. Note that the eigenvalues can be obtained from the EIGENF package by implementing four sets of storage vectors that can keep unnormalized results for comparison purposes. V. SIMULATION RESULTS 1. Rectangular Plane Mirror System A resonator composed of rectangular plane mirrors can be reduced to a one-dimensional problem of infinite strip mirrors. Figure 3(a) illustrates the resonator configuration composed of two identical plane mirrors parallel to the x y plane. The integral equation for this configuration can be written as

5 Numerical Simulation of Laser Resonators Jaegwon Yoo et al Fig. 4. Diffractional loss per transit in an infinite strip plane mirror system. Fig. 3. Various mirror configurations. γv(x 2 ) = +a a K(x 2 ; x 1 )v(x 1 )dx 1. (12) Because of the symmetric configuration, the kernel is symmetric and continuous K(x 2 ; x 1 ) = eiπ/4 Lλ e ik(x1 x2)2 /2L. (13) The products of the one-dimensional eigenmodes of the infinite strip mirrors in the x and the y directions enable one to find the eigenfunctions and eigenvalues for rectangular plane mirrors; Fig. 5. Normalized intensity distributions for the m=0 mode on an infinite strip plane mirror. v mn (x, y) = v x,m (x)v y,n (y) (14) γ mn = γ x,m γ y,n (15) The loss and the phase shift are given by 1 γ mn 2 and tan 1 (Im[γ mn ]/Re[γ mn ]), respectively. Figure 4 shows that the loss gets smaller as F N increases. For one and the same value of F N, they increase when the mode order increases; that is, higher modes suffer much larger spill-over loss. The eigenfunctions of the integral equation turn out to be complex functions. For the lowest mode, the amplitude at the edges is smaller than it is on the axis. Figure 5 shows that the field intensity at the edge of the mirror is smaller for larger F N. However, the amplitude distributions are almost the same as those for a sufficiently large F N, which means that the propagation of short-wavelength waves is very directional while long-wavelength waves suffer from diffraction. Figure 6 shows the amplitude distribution of the odd symmetric normal modes. For the same values of F N, the amplitude at the edge is higher than for the Fig. 6. Normalized intensity distributions for the m=1 mode on an infinite strip plane mirror. lowest mode. The phase has a waving tendency, but is substantially different from the value on the axis only in the narrow marginal zones of the mirrors. Though the shape of the field distribution depends on F N, we can say that it is a function of x/a rather than of x and a

6 -298- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 separately, so the region of the mirrors where the field is appreciably different from zero (field spot) varies almost proportionally to a. Parallel plane mirrors have a good filling because the light rays can fill the entire volume between the mirrors. However, the parallel plane resonator is not used for practical lasers because it becomes unstable with only slight misalignment of the mirrors. The resonators of most lasers have at least one spherical mirror surface. For large F N, asymptotic expressions for the even- and the odd-mode eigenfunctions are, respectively, given by v m = cos ( π x 2 a ( π x v m = sin 2 a m β(1 + i)/2 2πF N m β(1 + i)/2 2πF N ), (16) ), (17) Fig. 7. Diffractional loss per transit in a cylindrical mirror system. where β is a fitting constant. The power loss per transit, δ, and the resonant frequencies, ν, are asymptotically given by δ = (m + 1)2 β 8 ν = c (q + 2L where q = arg(γ/π). π 2F 3 N, (18) ) (m + 1)2, (19) 16F N 2. Cylindrical Mirror System For sufficiently large mirror sizes, the properties of curved resonators depend on the g parameters which measures the mirror curvatures relative to the mirror spacing. Figure 3(b) illustrates the resonator configuration composed of two identical cylindrical mirrors facing each other. The integral equation has the same form as Eq. (12), but the kernel for the cylindrical resonators can be read Fig. 8. Normalized intensity distributions for the m=0 mode on a cylindrical mirror. K(x 2 ; x 1 ) = eiπ/4 λl e ik(g1x2 1 +g2x2 2 2x1x2)/2L, (20) where the g i is a measure of the flatness of each mirror; g i 1 L/R i. These resonators can be divided into two classes, depending on the signs of the g i, stable and unstable resonators. In comparison with the cases of parallel plane mirrors, the loss decreases much faster as F N increases, and, for one and the same value of F N, it decreases for decreasing g i, as shown in Fig. 7. For the same values of F N and g i, the loss increases for increasing mode number. The eigenfunctions of the integral equation turn out to be complex functions, except for the confocal resonator (g i = 0). For the lowest mode, the amplitude at the edges is smaller than it is on the axis. Figure 8 shows that Fig. 9. Normalized intensity distributions for the m=1 mode on a cylindrical mirror. the distribution of the field intensity gets much sharper as g i decreases, which shows that the curved mirror

7 Numerical Simulation of Laser Resonators Jaegwon Yoo et al system is very convergent. Figure 9 shows the amplitude distribution of the odd symmetric normal modes. For the same values of F N, the amplitude at the edge is higher than it is for the lowest mode. Though the shape of the field distribution depends on F N, the same as for cases of parallel plane mirrors, it depends on x only, not on x/a, so the region of the mirrors where the field is appreciably different from zero (field spot) does not vary much as a varies. Because of the symmetry of the mirror system in rectangular coordinates, for a reversal of the signs of g i, the eigenvalues are complex conjugates of each other; i.e., they have the same losses for g i = g i. For stable configurations of the curved mirrors, the diffractional losses for a large F N are small enough to be neglected. The resonance frequencies are given by ν = c [q + (1 + m + n)a], (21) 2L where a = (1/π) cos 1 (1 + L/R 1 )(1 L/R 2 ). Fig. 10. Diffractional loss per transit in a circular spherical mirror system. 3. Circular Spherical Mirror System Figure 3(c) illustrates the resonator configuration composed of two identical circular plane mirrors parallel to the x y plane. Since the mirror system has an axial symmetry, the azimuthal mode can be easily separated as e ±imφ, and for the n th mode, the radial integral equation can be written in a symmetric form as γ n v n (r 2 ) r 2 = a 0 K n (r 2 ; r 1 )v n (r 1 ) r 1 dr 1 (22) Notice that the eigenfunction of the radial integral equation is v n (r) r for the radial field function v n (r). The radial kernel of the n th mode can be expressed as K n (r 2 ; r 1 ) = i n+1 k L J n ( k r 1r 2 L ) r1 r 2 e ik(r2 1 +r2 2 )/2L, (23) where J n is the n th order Bessel function of the first kind. Figure 3(d) illustrates the resonator configuration composed of two identical circular spherical mirrors facing each other. The kernel function can be read from Eq. (23) by including the curvatures of each mirror: K n (r 2 ; r 1 ) = i n+1 k L J n ( k r 1r 2 L ) r1 r 2 e ik(g1r2 1 +g2r2 2 )/2L. The radial integral equation for the n th mode has the same form as Eq. (22). Because the mirror systems have a finite circular aperture, the loss of the circular mirror system is slightly larger than that of the rectangular system. The losses of circular spherical mirror systems decreases much faster than those of the circular plane mirror systems as F N increases, and for one and the same value of the F N, they (24) Fig. 11. Normalized intensity distributions for the m=0 mode on a circular spherical mirror. Fig. 12. Normalized intensity distributions for the m=1 mode on a circular spherical mirror. decrease for decreasing g i, as shown in Figs. 8(a) and 8(b). For the same values of F N and g i, the loss increases with increasing mode number. Since the systems have

8 -300- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 axial symmetry, the eigenfunctions of the integral equation are Gaussian-Laguerre type [7], and they turn out to be complex functions except for the confocal resonator (g i = 0). For the lowest mode, the amplitude at the edge is smaller than it is on the axis. Figure 9(a) shows that the distribution of the field intensity gets much sharper for decreasing g i, which shows that the curved mirror system is very convergent. Figure 9(b) shows the amplitude distribution of the odd symmetric normal modes. For the same values of F N, the amplitude at the edge is higher than it is for the lowest mode. Though the shape of the field distribution depends on F N, the same as for cases of parallel plane mirrors, it depends on r only, not on r/a, so the region of the mirrors where the field is appreciably different from zero (field spot) does not vary much when a varies. Notice that, for the symmetric mirror system, the eigenfunctions that satisfy the integral equations are complex conjugates of each other for a reversal of the signs of g i, and that the eigenvalues are also complex conjugates of each other; i.e., the losses are symmetric about g = 0. Since the mode volume occupied by the fundamental mode of the confocal resonator is generally very small, in practice, it is desirable to use quasi-hemispherical resonators to enhance the spot size on the mirrors. This type of resonator has the additional advantage that it does not require a critical adjustment. 4. Asymmetric Mirror Unstable Resonators Optical power loss inside a resonant cavity occurs due to the imperfect mirror reflectivites, to the transmissive output coupling through one of the mirrors, to scattering and absorption. Unstable resonators [8] suffer much larger power losses associated with the escape of radiation past the mirrors than stable resonators. Since the diffraction losses of unstable resonators are sizable, but not intolerable for sustaining laser oscillations, unstable resonators typically require active media with higher gain to compensate for this additional loss factor. Conventional devices, such as commercial He-Ne lasers, employ stable resonators. However, stable resonators are not suitable for high-power laser devices because of their small mode volumes and mirror damages. On the contrary, unstable resonators have several advantages for certain high-power lasers, such as large mode volumes, small thermal distortion and damage of the mirrors, and diffractional output coupling. The idea of diffraction coupled resonators is based on geometrically unstable cavities; in particular, the output mirror incorporates a small reflecting dot, and the output mode has a very characteristic doughnut shape. In addition, the unstable geometry is much less sensitive to misalignment. In general, high-power laser generators employ one of the three types of resonators, i.e., plane-plane, concave-plane asymmetric, and confocal positive and negative branch unstable resonators. Fig. 13. Various unstable resonator configurations. Figures 13 show the mirror configurations of unstable resonators losing large radiation fields. Place a set of points P 1 and P 2 that are virtual objects for the other mirror. Suppose that P 1 is located r 1 L from mirror 1 and (r 1 + 1)L from mirror 2, then the reflected wave appears to come from the object at point P 2 and the image and the object distance satisfy the mirror formulas 1 1 r 2 r = 2(g 2 1), (25) Applying the same arguments to P 2, one can find the image and the object distance similar to Eq. (25) by exchanging subscripts 1 and 2. Introducing the concept of dimensional magnification, one can compute the size of the beam on the other mirror. The magnification factors of mirror 1 and 2 are calculated from the g parameters, M i = r i + 1 r i. (26) The effective reflection [4,5] of the power at mirror 2 is proportional to the ratio of the wave emitted from mirror 1 and intercepted by mirror 2 as well as the reflectivity

9 Numerical Simulation of Laser Resonators Jaegwon Yoo et al Fig. 14. Diffractional loss per transit in an unstable resonator with asymmetric mirrors. Fig. 16. Normalized intensity distributions of the m=0 mode for mirrors in an unstable resonator system. Fig. 15. Diffractional loss per transit in an unstable resonator with symmetric mirrors. Fig. 17. Normalized intensity distributions of the m=1 mode for mirrors in an unstable resonator system. of mirror 2, Γ 2 2: ρ 2 = Γ 2 2 [ 1 r ] 2 ( a2 a 1 ) 2 ( ) 2 a 2 = Γ 2 M 1. (27) a 1 By the same token, the effective reflection of the power at mirror 1 can be obtained by exchanging the subscripts 1 and 2. Then, the reflection of the power for the round trip is given by ( ) 2 R 2 Γ1 Γ 2 R 1 R 2 =. (28) M 1 M 2 Note that the net power gain should be larger than the loss; i.e., GR > 1. The mean reflection for a one-way trip can be expressed as g1 g 2 g 1 g 2 1 ρ = ± g1 g 2 + g 1 g 2 1, (29) where Γ 1 Γ 2 is absorbed in ρ. Since the ρ should be positive and less than unity, there are two branches for unstable resonators, i.e., a positive or a negative branch, depending on g 1 g 2. The equiloss contours satisfy g 1 g 2 = (1 + ρ) 2 /ρ for positive branches and g 1 g 2 = (1 ρ) 2 /2ρ for negative branches. The plots of the equiloss contours are hyperbolas on the stability diagram. Since the integral equation method for optical resonators is general enough to cover any cavity configuration, stable or unstable, it can generate numerical data for the loss per transit and fields for asymmetric cavities shown in Fig. 14 and 15. Numerical computations are carried out with asymmetric resonators having perfect reflectivity at each mirrors. Figures 16 shows that symmetric resonators with g i > 1 have oscillatory losses as F N increases. The cusps in the oscillatory loss curves are due to mode crossings at which two different modes have exactly the same diffraction losses. However, only the magnitudes, and not the phase angles, of the eigenvalues

10 -302- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 are equal at these crossing points. Figures 17 and 18 show that diffraction effects push the peak intensity outward. The higher-order modes have substantially larger diffraction spread and, thus, much increased diffraction losses for the TEM 10 mode. The general features will always be very much the same as those for either a symmetric or an asymmetric mirror. For asymmetric mirror resonators, the intensity profiles of mirror 1 reflect a large spill-over outside the mirror edge while those of mirror 2 are negligibly small as illustrated in Figs 15(a), 15(b), 16(a) and 16(b). The complex field amplitude falling outside the mirror edges in those figures will be coupled from the resonator past the mirror edges. VI. CONCLUDING REMARKS The integral equation method for optical resonators is general enough to cover any cavity configuration, stable or unstable, it can generate numerical data for the fields and the loss per transit for various optical cavities. The integral equation method turns out to be incompetent for dealing with the cases of a high Fresnel number because of limited computing resources and numerical accuracy. Also, the technique of the integral equation method should be extended to cover cases of resonators containing a gain media. The use of an optical cavity sustaining high-q resonances offers a number of features in operating lasers: (i) The Q of the fundamental mode determines the pumppower threshold for laser oscillations, (ii) the number of high-q resonances per unit band is small, (iii) resolving the high-q resonances by simple devices can improve the mode selectivity and the monochromaticity of the laser beam, (iv) optimal output coupling is used, (v) optimal mode volume determines the amount of gain medium necessary for amplifying light. The use of unstable resonators offers a number of advantages in the operation of high power lasers: (i) a greater portion of the gain medium contributing to the laser output power as a result of the availability of a larger modal volume, (ii) higher output powers attained from operation of the lowest-order transverse mode rather than from operation of higher-order transverse modes, and (iii) high output power with minimal optical damage to the resonator mirrors. These are a result of the use of purely reflective optics that permit the laser light to spill out around the mirror edges (this configuration also permits the optics to be water-cooled and thereby to tolerate high optical powers without damage). ACKNOWLEDGMENTS This work was supported by the Nuclear Long-mid Term Research and Development Fund. REFERENCES [1] A. G. Fox and T. Li, Bell Sys. Tech J. 40, 453 (1961). [2] A. G. Fox and T. Li, Proc. IEEE, 51, 80 (1963). [3] Jaegwon Yoo, et. al., KAERI/TR-1661 (Daejon, 2000). [4] W. P. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, Cambridge, 1992). [5] Per Christian Hansen, Rank-Deficient and Discrete Ill- Posed Problems : numerical aspects of linear inversion (SIAM, Philadelphia, 1997). [6] Kendall E. Atkinson, An Introduction to NUMERICAL ANALYSIS, 2nd ed. (John Wiley & Sons, New York, 1989). [7] F. Gori, G. Guattari and C. Padovani, Optics Comm. 64, 491 (1987). [8] Uwe D. Zeitner and Frank Wyrowski, IEEE J. Quantum Electron. 37, 1594 (2001). [9] G. D. Boyd and J. P. Gordon, Bell Sys. Tech J. 40, 489 (1961). [10] G. D. Boyd and H. Kogelnik, Bell Sys. Tech J. 41, 1347 (1962).