Derivation of explicit formulas for the fundamental ... - Springer Link

ISSN 1064-2269, Journal of Communications Technology and Electronics, 2007, Vol. 52, No. 12, pp. 1316–1323. © Pleiades Publishing, Inc., 2007. Original Russian Text © A.B. Plachenov, A.M. Radin, V.N. Kudashov, 2007, published in Radiotekhnika i Elektronika, 2007, Vol. 52, No. 12 pp. 1422–1429.

ELECTRODYNAMICS AND WAVE PROPAGATION

Derivation of Explicit Formulas for the Fundamental Resonator Mode in the Form of a Gaussian Beam with Complex Astigmatism A. B. Plachenov, A. M. Radin, and V. N. Kudashov Received September 26, 2006

Abstract—Explicit formulas are derived for a resonator whose fundamental mode is a Gaussian beam with complex astigmatism. The derived formulas can be used to express the beam characteristics directly in terms of the beam matrix and to avoid the procedure of finding the eigenvectors of this matrix. An example is presented. PACS numbers: 42.60.Da, 41.20.Jb DOI: 10.1134/S1064226907120029

1. Propagation of the light field in multimirror ring resonators forming a Gaussian beam with complex astigmatism has been repeatedly analyzed in the literature (see, e.g., [1, 2]). In this case, the function describing the transverse distribution of the fundamental-mode field has the form u(r) = c exp(ikr tHr/2), ⎛ 1/q x 1/q xy ⎞ ⎛ ⎞ where r = ⎜ x ⎟ , r t = (x, y), and H = ⎜ ⎟ is ⎝ 1/q xy 1/q y ⎠ ⎝ y⎠ the matrix of a quadratic form. Matrix H is symmetric. If the beam is concentrated near the resonator axis, this matrix has a positive definite imaginary part. Matrix H satisfies the matrix equation [2] H = (C + DH)(A + BH)–1,

(1)

where A, B, C, and D are the 2 × 2 real (for a lossless passive resonator) matrices. In aggregate, these matrices form the 4 × 4 ray matrix of the complete passage along the resonator (the monodromy matrix [1]): ⎛ T = ⎜ A B ⎝ CD

⎞ ⎟. ⎠

–1

⎛ t t = ⎜ D –B ⎜ t t ⎝ –C A

⎞ ⎟, ⎟ ⎠

Note that the proposed scheme is applicable to both ring and linear double-mirror resonators with elliptic (hyperbolic) mirrors as well as, in the case of a 3D manifold, to the problem of a Gaussian beam concentrated near a closed geodesic line [1]. 2. Let matrix H be a symmetric solution to Eq. (1) with a positive definite imaginary part, and let this matrix be related to some matrix H' via the relationship ˜ +D ˜ + B˜ H' ) –1 , ˜ H' ) ( A H = (C

Matrix T is symmetric [1, 2]. This property is equivalent to satisfaction of the condition T

bers [1]. In this case, Eq. (1) has a symmetric solution with a positive imaginary part. Traditionally, this solution is constructed with the use of the components of the eigenvectors of the monodromy matrix. In this paper, we propose an alternative method for solving Eq. (1). In this method, matrix H is expressed directly in terms of matrices A, B, C, and D. In this case, there is no need to search for the eigenvectors of matrix T. This approach is especially important when matrix T has multiple eigennumbers, because, in this case, eigenvectors cannot be uniquely determined. This circumstance results in formation of the sets of solutions to Eq. (1) that is difficult to describe via a traditional method.

(2)

˜ , and D ˜ , B˜ , C ˜ are the blocks of some real where A symplectic matrix T˜ . Then, H' is likewise a symmetric matrix with a positive definite imaginary part and this matrix satisfies the following equation, similar to Eq. (1): H' = (C' + D'H')(A' + B'H')–1,

where t is the transposition sign. The resonator is stable if the absolute values of all eigennumbers of matrix T are equal to unity and there are no adjoint vectors associated with these eigennum-

(3)

(4)

where A', B', C', and D' are the blocks of symplectic matrix T' related to matrix T via the similarity transformation

1316

–1 T' = T˜ TT˜ .

(5)

DERIVATION OF EXPLICIT FORMULAS FOR THE FUNDAMENTAL RESONATOR MODE

˜ = D ˜ = E, In particular, matrix T˜ with blocks A ˜ t , where O and E are the 2 × 2 zero ˜ =C B˜ = O, and C and identity matrices, generates the transformation ˜, H = H' + C ˜ ⎛ ⎞ A + BC B T˜ = ⎜ ⎟, ˜ –C ˜A–C ˜ BC ˜ D–C ˜B⎠ ⎝ C + DC

(6)

which consists in isolation of a symmetric real term. A ˜ = O, where ˜ =D ˜ = U and B˜ = C matrix with A ϕ ⎛ U ϕ = ⎜ cos ϕ – sin ϕ ⎝ sin ϕ cos ϕ

⎞ ⎟, ⎠

corresponds to rotation through angle ϕ: H = UϕH'U–ϕ,

⎛ U AU ϕ U –ϕ BU ϕ ⎞ T' = ⎜ –ϕ ⎟. ⎝ U –ϕ CU ϕ U –ϕ DU ϕ ⎠

(7)

˜ = E, we obtain ˜ =D ˜ = O and B˜ = – C Finally, if A H = –(H')–1,

⎛ ⎞ T' = ⎜ D – C ⎟ . ⎝ –B A ⎠

(8)

(For a Gaussian beam, this change corresponds to application of the Fourier transform along the transverse coordinates.) In the following, the strategy of solution of Eq. (1) consists in selection of a sequence of transformations (5) (or, more exactly, (6), (7), and (8)) that reduce the case of a general position to matrices T' of one of the basis types for which Eqs. (4) can be solved directly. 3. The characteristic polynomial of a symplectic matrix is a reciprocal polynomial, i.e., λ and λ–1 are simultaneously or nonsimultaneously the eigenvalues of matrix T. In this case, ν = (λ + λ–1)/2 is an eigenvalue of the matrix ⎛ t t 1 1 –1 X = --- ( T + T ) = --- ⎜ A + D B – B 2 2⎜ t t ⎝ C–C D+A

⎞ ⎟. ⎟ ⎠

(9)

This eigenvalue satisfies the equation ν2 – νSpV + detV + (b12 – b21)(c12 – c21)/4 = 0, (10)

condition for stability) if both roots of Eq. (10) are real and the absolute values of these roots are no more than unity. In this case, parameters θ1, 2 are real. Note that, if matrix B or matrix C is symmetric, the roots of Eq. (10) coincide with the eigennumbers of matrix (11). Evidently, in this case, the eigenvectors of matrix (11) may be chosen as real. In the case when matrix T has multiple eigennumbers, the stability of a resonator requires fulfillment of an additional condition: the absence of adjoint vectors. If ν1 = ν2 = ν = ±1, the multiplicity of the eigennumber coinciding with ν is four. If ν1 = 1 and ν2 = –1, there are two double eigennumbers coinciding with ν1, 2. If ν1 = ν2 = ν ≠ ±1, matrix T has two double eigennumbers that, in this case, are complex conjugate. Finally, the case of |ν1| < 1 and ν2 = ±1 corresponds to one double eigennumber coinciding with ν2 and two simple complex conjugate eigennumbers. Each of the described situations requires individual analysis. 4. Let us begin with the simplest case, when the multiplicity of an eigennumber is four and the eigennumber itself is equal to ±1. In this case, if matrix T does not have adjoint eigenvectors, it evidently coincides with matrix ±E4 (where E4 is the 4 × 4 identity matrix), A = D = ±E, and B = C = O. Hence, arbitrary symmetric matrix H with a positive definite imaginary part satisfies Eq. (1). Exactly this situation is realized in a confocal resonator (see, e.g., [3]). 5. Let us now consider the case when matrix T has two double eigennumbers equal to +1 and –1, respectively. If adjoint eigenvectors are absent, T–1 = T; i.e., blocks B and C are antisymmetric: B = bw and C = cw, where b and c are real numbers, ⎛ ⎞ w = ⎜ 0 –1 ⎟ , ⎝ 1 0 ⎠

(12)

w2 = –E, and A = Dt is a traceless matrix such that A2 + BC = E. In this case, the solution procedure depends on whether or not coefficient b is zero. 6. Let b = 0 and B = O. Let us perform transforma˜ = CA/2 owing to the condition C' = O. tion (6) with C Then, A' = (D')t = A, A2 = E (because the eigennumbers of matrix A are equal to +1 and –1), and B' = C' = O. Equation (4) has the form H'A = AtH' and possesses the two-parameter family of solutions

where V = (A + Dt)/2. (11) Let us represent the roots of Eq. (10) in the form ν1, 2 = cosθ1, 2, where θ1, 2 are some numbers. In this case, the eigennumbers of matrix T can be represented as exp(±iθj), where j = 1, 2. The absolute values of these eigennumbers are equal to unity (which is the necessary

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H' = ζ1(E + AtA) + ζ2(At + A), where matrix ImH' is positive definite when Imζ1 > |Imζ2|. ˜ = 7. If b ≠ 0, we perform transformation (6) with C chosen from the condition A' = O. Then, A' = D' = O, B' = B = bw, and C' = –b–1w. In this case, it fol–B–1A

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lows from Eq. (4) that H' is the product of factor |b|–1 and an arbitrary matrix that has a positive definite imaginary part and a determinant equal to –1. Note that this case can be reduced to the previous case by means of symmetrization of block B considered below. 8. Let us pass to the case of ν1 = ν2 = ν ≠ ±1. The eigennumbers of matrix T, which are equal to exp(±iθ)(cosθ = ν), are doubly degenerate; however, ν is a quadruple eigennumber of matrix X determined by formula (9). Hence, in this case, the necessary (and sufficient) stability condition is the fulfillment of the relationship X = νE4. This means that blocks B and C are symmetric and matrix (11) coincides with E to within factor ν: V = νE. In this case, the solution procedure depends on whether or not block B is equal to O. 9. Let B = O. It follows from condition (2) that, in particular, ADt = E and AtC = CtA. (It can be shown that either C = O or detC < 0.) In this case, the eigennumbers of matrix T are the eigennumbers of matrices A and D, which, therefore, are the unimodular matrices with a trace of 2ν. The solution to the system of linear equations derived from (1) can be written as AC – CD ˆ, H = ---------------------- + ζwA t ˆ ˆ Sp ( A A )

(13)

ˆ = A – νE is a traceless where w is matrix (12) and A matrix. Formula (13) with all possible values of complex parameter ζ determines the set of solutions to Eq. (1). The imaginary part of matrix H is positive definite when the sign of Imζ coincides with the sign of the difference a12 – a21 (which is here nonzero). 10. Now, let B = Bt ≠ O. Let us perform transforma˜ chosen from the condition tion (6) with C A' = (D')t;

(14)

˜ = D t – A. 2BC

(15)

i.e., If detB ≠ 0, it follows from the last formula and formula (2) that ˜ = ( DB –1 – B –1 A )/2. C

(16)

In the case under study, the assumption of nondegeneracy of matrix B is valid. Below, we show that degeneracy of block B is related to the following property: One of the eigennumbers of matrix T is ±1. 11. As a result of the described transformation, our problem reduces to the case of A = Dt. (The primes are omitted.) Because, in this case, V = νE, then A = D = νE and, in view of formula (2), BC = –(1 – ν2)E. Equation (1) takes the form HBH = C (17)

or, after premultiplication by –B, becomes –(BH)2 = (1 – ν2)E. Let us make the change H = ih 1 – ν , 2

(18)

(19)

where h is a symmetric matrix with a positive definite real part. Then, Eq. (18) transforms into (20) (Bh)2 = E. It follows from Eq. (20) that the eigennumbers of matrix Bh are equal to ±1 and, if the eigennumbers are coincident, adjoint vectors are absent. This means that Bh either coincides with ±E or represents a traceless matrix whose determinant is equal to –1. In realization of one of these possibilities, the sign of detB is of principal importance. 12. Let B be a matrix of fixed sign, namely, detB > 0. Matrix Reh is positive definite (detRe(h) > 0); consequently, detRe(Bh) > 0. Then, the eigennumbers of matrix Re(Bh) have the same sign and SpRe(Bh) ≠ 0; i.e., Bh is not a traceless matrix. Hence, Bh = ±E and h = ±B–1 (21) is a purely real matrix. The sign in formula (21) should be chosen from the condition of positive definiteness of this matrix, and this sign coincides with the sign of SpB. Finally, for matrix H, we have H = i 1 – ν B sgn SpB. 2

–1

(22)

13. If B is not a matrix of fixed sign (detB < 0), matrix (21) is no longer the desired solution, because it likewise is not a matrix of fixed sign. Therefore, in this case, Bh is a matrix that has eigennumbers equal to +1 and –1, a zero trace, and a determinant equal to –1. The solution to this problem is certainly ambiguous [4]. In order to describe the family of corresponding matrices, we represent B in the form of a linear combination of identity matrix E and traceless matrix s with a determinant equal to –1: B = αE + βs, α = (SpB)/2, β=

α + d , d = |detB|. 2

Then, the desired family of solutions to Eq. (20) has the form ( 1 + ζ ) ( βE – αs ) + 2ζ dws -, h = -------------------------------------------------------------------------2 ( 1 – ζ )d 2

(23)

where ζ is a complex parameter. If |ζ| < 1, matrix (23) has a positive definite real part; accordingly, under this condition, matrix H (19) likewise has a positive definite imaginary part. The simplest example of a resonator implementing such an ambiguity of the solution is the three-mirror resonator considered in [4]. The resonator has two pla-

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nar mirrors and one elliptic mirror whose radii of curvature are chosen so as to ensure the coincidence of the eigennumbers of the monodromy matrix. A similar situation may arise in the linear resonator with elliptic mirrors that will be analyzed below.

S2

14. In the case when the absolute value of only one eigennumber is equal to unity, we have not revealed any additional limitations on the form of matrix T that can simplify the solution of the posed problem. Therefore, description of the solution procedure at two initial stages that ensure symmetrization of block B is equally related to the case of simple eigennumbers. We assume that matrices B and C are initially not symmetric; otherwise, the corresponding transformations should be omitted. ˜ chosen Let us perform transformation (6) with C from the condition of symmetry of matrix C'. This ˜ in choice is certainly ambiguous. Let us seek matrix C the form of a product of numerical parameter z whose value must be determined and some fixed symmetric real matrix F. As a result, the condition C' = (C')t generates a quadratic equation for parameter z. The free term of this equation is (c12 – c21) and the coefficient of term z2 is (–(b12 – b21) detF). Evidently, if the signs of detF and (b12 – b21)(c12 – c21) are coincident, the discriminant of this equation is certainly positive and the roots are real. If detF = 0, the quadratic equation transforms into a linear equation that has a unique solution if the coefficient of term z is nonzero. In fact, the choice of matrix F is the only informal step in this study, a circumstance that substantially determines the degree of complexity and awkwardness of the subsequent calculations, especially when the solution is sought analytically rather than numerically. However, this step does not affect the final result if the equation for z has real roots. 15. The next step is the passage from a matrix with symmetric block C to a matrix with symmetric block B. This step is performed with the use of transformation (8): B' = –C. Here, block C is meant as the result of transformation performed at the preceding step, i.e., as former matrix C'. In the following, we will act in a similar manner. 16. We reduced the problem to a particular case of a matrix with symmetric block B. (Recall that matrix B relates the transverse projection of the unit vector directed along the near-axial ray outgoing from the origin of coordinates and the position vector of the point through which the ray passes after the complete travel along the resonator.) The condition of symmetry of block B means that this block has two real and mutually orthogonal eigenvectors. Note that some problems may possess this property from the very beginning. Among

1319

S1

Double-mirror resonator with elliptic mirrors. The dashed lines depict the rays passing in the mutually orthogonal planes. Each of these planes contains the optical axis and one of the eigenvectors of symmetric matrix B.

these is the problem of a linear resonator with elliptic mirrors: If the monodromy matrix is considered for the section adjoining one of the mirrors, the directions of the eigenvectors of block B coincide with the directions of the principal curvatures of opposite mirror S2 (see figure). Indeed, if the transverse projection of the wave vector is aligned with one of these directions, both the initial ray and the ray formed after single reflection lie in the plane formed by the wave vector and the optical axis. Hence, the position vector of the crosspoint of the reflected ray and the initial plane is collinear to the wave vector. Properties of the other mirror (S1) have no influence on matrix B, because reflection from this mirror determines the direction of further propagation of the ray but does not change the position of the considered point. The subsequent procedure depends on the rank of matrix B. If B = O, the eigennumbers of matrix T, whose absolute values are equal to unity, coincide with the eigennumbers of blocks A and D, which, as follows from the condition ADt = E, must be pairwise mutually inverse. Since matrices A and D are real, their eigennumbers are either real (and equal to either +1 or –1) or complex conjugate (and common for both matrices). This means that matrix T has either two double eigennumbers or one quadruple eigennumber. We already analyzed all these variants. Note that, from the physical

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viewpoint, transformation of matrix B into matrix O means that all rays outgoing from the origin of coordinates and having passed along the resonator are focused at the same point. In the problem of a double-mirror resonator, this case corresponds to the situation when the opposite mirror is spherical and the radius of this mirror coincides with the distance between the mirrors. Let us analyze the case when B = Bt is a nonzero degenerate matrix. It will be shown that, in this case, only one eigennumber of matrix T is a double eigennumber whose value is ±1. From the physical viewpoint, degeneracy of matrix B indicates that the considered section is a focusing section of geometric-optics rays passing along one of the directions. In particular, in the aforementioned problem of a double-mirror resonator, matrix B is degenerate if one of the radii of curvature of the opposite mirror coincides with the distance between the mirrors: In this case, the rays for which the transverse projection of the ray vector is parallel to the direction of the principal curvature return to the origin after reflection. 17. Let B = Bt ≠ O and detB = 0. This matrix can be represented as ⎛ ⎞ 2 B = b ⎜ cos ϕ sin ϕ cos ϕ ⎟ , (24) ⎜ ⎟ 2 ⎝ sin ϕ cos ϕ sin ϕ ⎠ where b is the nonzero eigennumber of matrix B and angle ϕ characterizes the direction of the corresponding eigenvector. Let us perform transformation (7); as a result, block B becomes diagonal: B' = diag[b, 0]. 18. Let us now consider the case when matrix B contains only one nonzero element (b11). It follows from (2) that, in particular, (25) a21 = d12 = 0, a22d22 = 1. Analysis of the structure of matrix T satisfying relationships (25) shows that elements a22 and d22 are the eigennumbers of this matrix and, since the product of these elements is unity, a stable resonator should have a22 = d22 = ν2 = ±1. ˜ Let us perform transformation (6) with matrix C chosen from condition (14). This operation may be performed, because Eq. (15) is solvable; i.e., d 11 – a 11 -, c˜ 11 = ------------------2b 11

d 21 – a 12 -. c˜ 12 = ------------------2b 11

Element c˜ 22 may be chosen arbitrary. 19. The problem with a degenerate nonzero matrix B = Bt is reduced to the case when B = diag[b11, 0] and A = Dt = V is an upper triangular matrix. Hence, the roots of Eq. (7) coincide with the diagonal elements of

this matrix: a11 = ν1 and a22 = ν2 = ±1. In particular, it follows from (2) that ν 1 = 1 + b11c11 and C = Ct. Equation (1), which, in this case, takes the form 2

(26) HBH + HA – AtH – C = O, generates Eq. (17) for the symmetric part and the equation (27) HA – AtH = O for the antisymmetric part. In this case, Eq. (17) is solvable under the condition detC = 0, which ensures the absence of adjoint vectors. This equation generates a system of equations for the elements of matrix H: ⎧ b 11 h 11 = c 11 ⎨ ⎩ b 11 h 11 h 12 = c 12 . 2

In order to ensure positive definiteness of matrix ImH, elements b11 and c11 must have different signs; otherwise, diagonal element h11 becomes real. This suggests, inter alia, that |ν1| < 1. Equation (17) does not impose any constraints on element h22; hence, in this case, the considered family of solutions has the form c 11 ⎛ 1 c 12 /c 11 ⎞ H = i – ------⎜ ⎟, b 11 ⎝ c 12 /c 11 ζ ⎠

(28)

where ζ is a complex parameter and matrix ImH is positive definite at Reζ > (c12/c11)2. Equation (27) is automatically satisfied for matrices (28). We have completed the analysis of the cases corresponding to multiple eigennumbers of matrix T. 20. The last case to consider is the case when B = Bt is a nondegenerate matrix and the eigennumbers of matrix V (11) are different. Let us again perform trans˜ chosen according to formula (16); formation (6) with C then, condition (14) is satisfied. 21. Now, let B = Bt be a nondegenerate matrix and A = Dt = V be a matrix with eigennumbers ν1 ≠ ν2. Matrix C is symmetric and Eq. (1), which again takes the form of Eq. (26), splits into Eqs. (17) and (27). After premultiplication by (–B) with consideration for formula (2), these equations take the form –(BH)2 = –BC = E – A2, (29) ABH – BHA = O. (30) Equation (29) shows that, if one of both eigennumbers of matrix A becomes equal to ±1, matrix (BH)2 and, consequently, matrix H become degenerate. Therefore, unlike the cases analyzed above, stability of the resonator requires that the magnitudes of both eigennumbers ν1, 2 are strictly less than unity. This

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means that matrix T has four different single eigennumbers. It follows from Eq. (30) that matrices A and BH commute. This means that matrix BH has the same eigenvectors as matrix A and can be represented in the form of a linear combination of matrices A and E (which are linearly independent because ν1 ≠ ν2) and, accordingly, matrix H can be represented in the form of a linear combination of matrices B–1A and B–1. Hence, our goal is to find the coefficients of this linear combination. Note that it is possible to seek a representation of matrix BH in the form of a linear combination of identity matrix E and some matrix function of matrix A, for example, BC. In this case, matrix H is represented as a linear combination of matrices B–1 and C. However, this representation may be impossible if these matrices are linearly dependent, i.e., when matrix A is traceless. This case requires individual analysis. 22. If SpA = 0, then ν1 = –ν2 = ν = – detA . Hence, 2 A = ν2E and Eq. (29) takes the form of Eq. (18). Repeating the reasoning used for the case of ν1 = ν2 = ν, we again come to formula (22) if detB > 0 and, if detB < 0, we find that, as before, Bh is a traceless matrix whose determinant is –1. However, this matrix is no longer arbitrary but is coincident (to within a factor) with matrix A. Then, the final expression for matrix H has the form 1 – ν –1 –1 H = i ------------------ B A sgn Sp ( B A ). ν 2

23. Now, let SpA ≠ 0 and |ν1| ≠ |ν2|. Then, it follows from Eq. (29) that –(BH)2 is a matrix with different 2 positive eigennumbers equal to 1 – ν 1, 2 . Since the eigenvectors of this matrix are real, we find that BH is a purely imaginary matrix; hence, H = i|H|, where |H| is a positive definite real matrix. Equation (29) takes the form (31) (B|H|)2 = –BC. Let us transform the left-hand side of Eq. (31) with the use of the characteristic relation B|H|Sp(B|H|) – Edet(B|H|) = –BC. (32) As a result, we obtain B det ( B H ) – C H = i ------------------------------------------- , Sp ( B H ) –1

where det ( B H ) = Sp ( B H ) =

det ( BC ) sgn detB, 2det ( B H ) – Sp ( BC )

× sgn Sp ( det ( B H )B – C ) . –1

(33)

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The values of the squared determinant and the squared trace of matrix B|H| can be found via calculation of the determinant and the trace from formulas (31) and (32), respectively. The signs should be chosen from the condition of positive definiteness of matrix |H|. Trace Sp(B|H|) appearing in the denominator of formula (33) is nonzero since, if not, matrix BC would be proportional to an identity matrix, a property that contradicts the condition |ν1| ≠ |ν2|. Note that the alternative form of solution, when matrix H is expressed in terms of blocks A and B [5], is slightly more cumbersome. 24. We have completed the analysis of the variants arising during the solution of Eq. (1). We have revealed a relation between the spectrum of the monodromy matrix, the symmetry properties of the blocks of this matrix, and the structure of the set of symmetric solutions to Eq. (1) that have positive definite imaginary parts. Let us list the spectral types of stable resonators. (i) Assume that we have a quadruple eigennumber equal to either +1 or –1 and T = ±E4. In this case, we have a three-parameter family of solutions (with complex parameters) and the solution to the posed problem is an arbitrary symmetric matrix with a positive definite imaginary part. (ii) Assume that we have two double eigennumbers equal to +1 and –1, respectively; A = Dt, SpA = 0; and B is an antisymmetric matrix that transforms into O after symmetrization. In this case, we have a twoparameter family of solutions. (iii) Assume that we have two double complex conjugate eigennumbers, V = νE, and B is a symmetric matrix. In this case, we have an alternative: (a) detB > 0 and the solution is unique or (b) B = O or detB < 0 and there exists a one-parameter family of solutions. (iv) Assume that we have one double eigennumber equal to either +1 or –1 and two single complex conjugate eigennumbers; after symmetrization, matrix B becomes degenerate but does not transform into O. In this case, we have a one-parameter family of solutions. (v) Assume that we have two pairs of single complex conjugate eigennumbers; after symmetrization, matrix B remains nondegenerate and matrices V and E are linearly independent. In this case, the solution is unique. Since transformation (8) does not change the spectral properties of the monodromy matrix, matrix C has the same properties as matrix B in all listed cases. The symmetrization procedure is meant as similarity transformation (5) after which the corresponding blocks of matrix T' become symmetric. For each spectral type, we derived a sequence of transformations (5) that transform the case of a general position into one of the basis variants and that present explicit analytic formulas for each of these variants.

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25. Let us illustrate these constructions by the example of the aforementioned double-mirror resonator with elliptic (hyperbolic) mirrors (see figure). The distance between mirrors S1, 2 is L, and the shapes of the mirrors are described by real symmetric matrices Yj, where j = 1, 2. The eigenvectors of these matrices are aligned with the directions of principal curvatures of the mirrors, and the absolute values of the matrix eigennum( 1, 2 ) ( 1, 2 ) bers are 1/ R j (where R j are the principal radii of curvature). The eigennumbers are positive for the focusing mirror and negative for the defocusing mirror. For a spherical mirror, the eigennumbers are equal; for a hyperbolic mirror, the eigennumbers have different signs. If axis z is aligned with the resonator’s optical axis and intersects mirror S1 at the origin of coordinates, the surfaces of mirrors S1, 2 are described near this axis by the equations z = r tY1r/2 and z = L – r tY2r/2, respectively. Note that the coincidence of the eigenvectors of matrices Y1, 2 and, accordingly, the possibility of simultaneous diagonalization of these matrices are not assumed. By analogy with a 2D problem (see, e.g., [6]), we represent each mirror Sj in the form of an equivalent combination of a planar mirror and a thin lens with ( 1, 2 ) focal distances R j . Transmission of light from section S1 to section S2 and backward is described with the help of the matrices ⎛ G1 LE ⎞ T 12 = ⎜ ⎟, ⎝ ( G 2 G 1 – E )/L G 2 ⎠

If even one of the eigennumbers of matrix F is equal to unity, the resonator is instable. The resonator is likewise instable if only one of matrices G1, 2 is a zero matrix or only one of matrices G1, 2 is degenerate. If matrix F has different eigennumbers that are not equal to zero or unity and SpF ≠ 1, then i ( 1 – SpF )G 1 + ( detF + d )G 2 -, H = --- ----------------------------------------------------------------------L t –1

where detF ( 1 – SpF + det F ) sgn detG 2 ,

d =

SpF ( 1 – SpF ) + 2 ( detF + d )

t =

× sgn [ ( 1 – SpF )SpG 1 + ( detF + d )SpG 2 ]. –1

If matrix F has different eigennumbers that are not equal to zero or unity and SpF = 1, then i –1 H = --- detFG 2 sgn SpG 2 L for detG1, 2 > 0 and i detF –1 –1 H = --- ----------------------- ( 2G 1 – G 2 ) sgn Sp ( 2G 1 – G 2 ) L 1 – 4detF for detG1, 2 < 0. The geometric condition detG1, 2 < 0 means that, for each mirror S1, 2, one principal radius is larger than L and another principal radius is less than L. If F = fE, where 0 < f < 1, then

⎛ G2 LE ⎞ T 21 = ⎜ ⎟, ⎝ ( G 1 G 2 – E )/L G 1 ⎠ where Gj = E – LYj are the matrix analogues of wellknown g parameters of a 2D resonator. Then, the monodromy matrix corresponding to section S1 has the form T 1 = T 21 T 12

elsewhere. Here, we briefly formulate some results of this study, which will be obtained through repetition of the constructions of this study with consideration for the specific features of the posed problem.

⎛ 2G 2 G 1 – E 2LG 2 ⎞ = ⎜ ⎟. ⎝ 2 ( G 1 G 2 G 1 – G 1 )/L 2G 1 G 2 – E ⎠

This is a matrix with symmetric blocks B and C for which the condition A = Dt is fulfilled. The condition |ν1, 2| ≤ 1 is fulfilled when the eigennumbers of the matrix F = G2G1 lie between zero and unity (in the 2D case, the resonator is stable if 0 < g2g1 < 1)) and Eqs. (17) and (27) take the form L2HG2H = G1G2G1 – G1,

(34)

HG2G1 = G1G2H.

(35)

A detailed analysis of a double-mirror resonator and, in particular, system (34), (35) will be presented

i –1 H = --- f ( 1 – f )G 2 sgn SpG 2 L for detG1, 2 > 0 and i H = --- f ( 1 – f ) L ( 1 + ζ )(β 2 E + α 2 s) + 2ζ detG 2 ws -, × -------------------------------------------------------------------------------------------2 ( 1 – ζ ) detG 2 2

|ζ| < 1 for detG1, 2 < 0. Here, α2 = SpG2/2, β2 = α 2 + detG 2 , G2 = α2E + β2s, s2 = –E, and w is the matrix defined by formula (12). If detG1, 2 = 0 and G1, 2 ≠ O, these matrices have form (24). The resonator is stable if 0 < f < 1, where f = g1g2 cos2∆ϕ is the nonzero eigennumber of matrix F, g1, 2 ≠ 0 are the nonzero eigennumbers of matrices G1, 2,

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DERIVATION OF EXPLICIT FORMULAS FOR THE FUNDAMENTAL RESONATOR MODE

and ∆ϕ is the angle between the directions of the corresponding eigenvectors. Then, i –1 H = --- ( f – 1 G 1 sgn g 1 – ζwG 2 w ), L g2Reζ > 0. If G1, 2 = O, the resonator is confocal and H is an arbitrary matrix with a positive definite imaginary part. It seems that the presented formulas clearly demonstrate the advantages of the proposed method: These formulas are difficult (if even possible) to obtain via a traditional method, i.e., via expression of matrix H in terms of the eigenvectors of matrix T1. 26. Let us make some conclusions. In this paper, we proposed a new method for construction of the resonator’s fundamental mode in the form of a Gaussian beam with complex astigmatism. Unlike a traditional method, in which the beam characteristics are expressed in terms of the eigenvectors of a 4 × 4 monodromy matrix, the proposed procedure does not include the search for such vectors. It seems that another advantage of our method is that it allows derivation of explicit analytic formulas for the beam in terms of the elements of the initial matrix and, thus, tracing of the dependence of the beam characteristics of the problem parameters. At the same time, this algorithm can serve as a base for numerical calculations. The proposed procedure consists in successive simplification of the problem with the help of the similarity

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transformations and, finally, reduction of the initial problem to one of the basis variants allowing explicit solution (in some cases, in the form of a family of functions). For each of these variants, the stability condition has been derived. Relations between these variants, spectral characteristics of the initial monodromy matrix, and the symmetry properties of the blocks of this matrix have been found. The potentialities of this method have been illustrated by a model example of a resonator with nonspherical mirrors. REFERENCES 1. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in the Diffraction Problems of Short Waves (Nauka, Moscow, 1972) [in Russian]. 2. I. V. Golovnin, A. I. Kovrigin, A. N. Konovalov, and G. D. Laptev, Kvantovaya Elektron. (Moscow) 22, 461 (1995). 3. A. A. Malyutin, Kvantovaya Elektron. (Moscow) 24, 736 (1997). 4. V. N. Kudashov, A. B. Plachenov, and A. M. Radin, Opt. Spektrosk. 88, 130 (2000) [Opt. Spectrosc. 88, 118 (2000)]. 5. V. N. Kudashov, A. B. Plachenov, and A. M. Radin, Opt. Spektrosk. 88, 127 (2000) [Opt. Spectrosc. 88, 291 (2000)]. 6. Yu. A. Anan’ev, Optical Resonators and Laser Beams (Nauka, Moscow, 1990) [in Russian].

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2007