Optics and Spectroscopy, Vol. 88, No. 1, 2000, pp. 121–126. Translated from Optika i Spektroskopiya, Vol. 88, No. 1, 2000, pp. 130–136. Original Russian Text Copyright © 2000 by Kudashov, Plachenov, Radin.
LASERS AND THEIR APPLICATION
Optical Beams with Complex Astigmatism in the Case of Coincident Eigenvalues of the Cavity Ray Matrix V. N. Kudashov, A. B. Plachenov, and A. M. Radin St. Petersburg State Academy of Cold and Food Technologies, St. Petersburg, Russia Received April 5, 1999
Abstract—An analysis is carried out of the optical cavities whose natural modes have the form of Gaussian beams with complex astigmatism. In the case of coincident eigenvalues of the cavity ray matrix, the wave beams were found to be determined not only by the geometrical optics characteristics of a cavity, but also by additional cavity-independent parameters. A detailed analysis is given for the fundamental mode of such cavities. Concrete cavities possessing these properties are noted. © 2000 MAIK “Nauka/Interperiodica”.
INTRODUCTION The propagation of a light field in optical cavities without loss, in which a Gaussian beam with complex astigmatism is formed, has already been analyzed in the literature (see, e.g., [1, 2]). According to [2], a Gaussian beam is referred to as a beam with complex astigmatism when the principal axes of its elliptic cross section (the section perpendicular to the optical axis) are not parallel to the transverse axes of the coordinate system fixed to the beam. A beam is described by three complex parameters qx, qy, and qx, y representing a nondiagonal symmetric matrix H [see formula (1)]. In the general case, the spatial variables in the corresponding three-dimensional stationary wave equation (Helmholtz equation) cannot be separated. If the principal axes of the elliptic cross section are parallel to the transverse axes of the coordinate system, the matrix H is diagonal and such beams are characterized by simple astigmatism. In this case, one can separate the variables in the three-dimensional Helmholtz equation, and its solution is reduced to the solution of two two-dimensional equations. However, a technique that makes it possible to classify cavities with complex astigmatism by properties of their ray matrices has not yet been developed. It was found that the terms of Lagrange planes [3] were naturally suitable to this aim. They make it possible to find new properties of wave beams in three-dimensional optical cavities without complex numerical calculations whose direct performance is not always possible. The problem posed above calls for adequate mathematical means, i.e., a modern mathematical formalism. We apologize to the reader beforehand, because our paper contains more modern mathematics than conventional papers in the areas of physics and engineering. Unfortunately, it is impossible to elucidate deeper properties of oscillations in optical cavities without the use of the modern mathematical formalism.
From the formal point of view, the fulfillment of this work offers the means to purposefully control the beam astigmatism, which is inevitably formed (or used) in actual systems. Note that even the study of ideal cavities (without loss and gain; for example, a model of a laser operating at the oscillation threshold) by this means can give new information on the properties of modes maintained in a cavity. In this paper, the analytical technique developed based on the approach mentioned above is demonstrated, for example, by describing properties of the fundamental mode in a cavity with complex astigmatism (higher modes can be analyzed in a similar way). In this case, the function describing the transverse field distribution for the fundamental mode in a certain section has the form ϕ ( r ) = ce
t
ikr Hr/2
,
x where r = , rt = (x y), and H is the matrix of qua y dratic form 1 ⁄ q x 1 ⁄ q xy H = , 1 ⁄ q xy 1 ⁄ q y 2i 1 1 -, ------- = ------- + ---------q xy R xy kw 2xy
2 1 1 ----- = ----- + ---------2 , qx R x kw x 1 2i 1 ----- = ----- + ---------2 . qy R y kw y
The matrix H is symmetric, and for a beam concentrated near the cavity axis, it has only a positive definite imaginary part. Let us present some facts concerning the propagation of such beams in cavities. For the most part, we shall use the notation of paper [2]. A laser beam is characterized by the four-component column vector r Y = , α where α is the projection of the unit vector onto the (x, y) plane, α = Hr. To describe the transformation of this
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ray passing through an optical system, let us use the ray matrix T of dimension 4 × 4: T = A B C D
,
where A, B, C, and D are real matrices of dimension 2 × 2. The matrix T satisfies the equality [4] TtJT = J,
(1)
where J = i 0 –E E 0
,
E is the two-dimensional unit matrix, 0 is the twodimensional zero matrix, and Tt is the transposed matrix T. The matrices with properties (1) are said to be symplectic [4]. The vector Y is transformed according to the rule Y out = TY in , where Yin and Yout are the initial and transformed rays, or r out A B r in = , H out r out C D H in r in from which follows the relation H out = ( C + DH in ) ( A + BH in ) , –1
describing the corresponding transformation of the matrix H. If T is the matrix for the round trip along a cavity (such matrices are also referred to as the ray matrices or the monodromy matrices [1]), a stable configuration is bound to satisfy the conditions Hout = Hin = H,
t
If Y1, 2 = 0, the vectors Y 1 JY2 are referred to as skew-orthogonal [3]. The symmetry of the matrix H implies that any pair of vectors belonging to L is skeworthogonal. The plane possessing such properties is said to be Lagrangian. The matrix J is Hermitian, and therefore, the quadratic form Y*tJY has a real value for any vector Y. The vector Y is said to be positive if Y*tJY > 0 (and negative in the case where the opposite inequality is fulfilled). The positive definiteness of the imaginary part of H (the condition of cavity stability) implies that all the vectors belonging to L are positive. Thus, the ray matrix of a stable cavity has an invariant positive Lagrangian plane. Conversely, if a plane of this kind is found, the matrix H can be uniquely reconstructed from an arbir trary pair of linearly independent vectors Y1, 2 = 1, 2 α 1, 2 belonging to L by using the condition α1, 2 = Hr1, 2. One can easily show that, from the positiveness of L, it follows that the vectors r1, 2 are linearly independent, and therefore, the matrix (r1 r2) is nondegenerate, which gives H = ( α1 α2 ) ( r1 r2 ) . –1
(4)
CONSTRUCTION OF THE FUNDAMENTAL MODE OF AN OPTICAL CAVITY IN THE CASE OF A SIMPLE SPECTRUM OF ITS RAY MATRIX It is known [1] that a cavity is stable in the case where all eigenvalues of the matrix T are equal in modulus to unity and have no adjoint vectors. In this case, the eigenvalues can be represented in the form ±
λj = e
± iθ j
,
j = 1, 2,
with real θj, and νj = cosθj, which represent eigenvalues of the matrix (T + T–1)/2, satisfying the equation ν + a 1 ν + a 0 = 0, 2
r out = T r in Hr in Hr out
(2) where
and
a 1 = – tr ( A + D ) ⁄ 2, t
H = ( C + DH ) ( A + BH )
–1
(3)
(a beam is reconstructed upon a round trip). A resonator is stable in the case where equation (3) has a symmetric solution with a positive imaginary part. INVARIANT PROPERTIES OF THE RAY MATRIX OF AN OPTICAL CAVITY Condition (2) means that the plane L formed by the r vectors of form (for a fixed H and all possible r) Hr is invariant with respect to T.
a 0 = [ det ( A + D ) + ( B 12 – B 21 ) ( C 12 – C 21 ) ] ⁄ 4. The quantities θ1, 2 are real if t
a 1 ≤ 2,
a1 – 1 ≤ a0 ≤ a1 ⁄ 4 . 2
(5)
± λj
In this case, the eigenvalues are mutually phase –* ± conjugate: λ j = λ j , where j = 1, 2. In the case where all slack inequalities in (5) are replaced with strict inequalities, all eigenvalues are different, and T possesses ± a complete set of eigenvectors Y j , j = 1, 2: ±
±
±
TY j = λ j Y j . OPTICS AND SPECTROSCOPY
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OPTICAL BEAMS WITH COMPLEX ASTIGMATISM IN THE CASE
These vectors can be chosen to be a mutually complex conjugate as well: – + Y = Y * , j = 1, 2. (6) j
j
The eigenvectors corresponding to different values of j (and νj) are skew-orthogonal. One can easily verify this property with the aid of the simplecticity condition (1). In each pair of complex conjugate vectors, one vector is positive and the other one is negative; by choosing the signs of θj, one can obtain positive values for the + vectors Y j . Each pair of eigenvectors specifies an invariant plane of the matrix T. Of the six planes of this type, four are Lagrangian, and only one plane, which represents a + linear envelope of the vectors Y 1, 2 , is positive. The positiveness of an arbitrary linear combination of the + vectors Y 1, 2 follows from relation (6) and the skeworthogonality of eigenvectors. Thus, in the case of noncoincident eigenvalues, the plane L is unambiguously determined. The matrix H is determined by formula (4), where the components of positive eigenvectors of the matrix T may be taken as r1, 2 and α1, 2. CONSTRUCTION OF THE FUNDAMENTAL MODE OF AN OPTICAL CAVITY IN THE CASE OF COINCIDENT EIGENVALUES OF ITS RAY MATRIX The situations taking place in the case where some eigenvalues of the matrix T coincide with one another qualitatively differ from the situation considered above. First of all, one should bear in mind that when the root subspaces contain both eigenvectors and adjoint vectors, this leads to the loss of stability. Because of this, we shall restrict further analysis to the case where all root subspaces are of the proper type. The presence of proper subspaces with dimension exceeding unity may lead to the appearance of entire families of invariant Lagrangian planes, including positive planes, each of them characterized by its own solution of equation (3). Let us list the cases where multiple eigenvalues may appear. − 2 , a0 = 1): λ = ν is a qua(1) ν1 = ν2 = ν = ±1 (a1 = + druple eigenvalue. T = νE4 (E4 is the 4 × 4 unit matrix). In this case, an arbitrary symmetric matrix H with a positive imaginary part is a solution of equation (3), L is an arbitrary Lagrangian plane. (2) ν1 = 1, ν2 = –1 (a1 = 0, a0 = –1), λ1, 2 = ν1, 2. The matrix T has two non-Lagrangian skew-orthogonal, two-dimensional proper subspaces V1, 2. One may take as L a linear envelope of two arbitrary positive vectors belonging to V1 and V2, respectively. ±
(3) ν1 ≠ ±1, ν2 = ±1 (|a1| – 1 = a0 ≠ ±1), λ 1 = e
± iθ 1
≠
±
±1, λ2 = ν2. The matrix T has two eigenvectors Y 1 and OPTICS AND SPECTROSCOPY
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a non-Lagrangian two-dimensional proper subspace V, which is skew-orthogonal to them and belongs to the eigenvalue λ2. One may take as L a linear envelope of +
the vector Y 1 and an arbitrary positive vector belonging to V. (4) ν1 = ν2 = ν ≠ ±1 (a0 = a 1 /4 ≠ 1): θ = θ1 = ±θ2. The matrix T has two proper Lagrangian planes V1, 2 ± belonging to the eigenvalues λ 1 = e±iθ; V2 = V *1 , and the complex conjugation transforms vectors of one plane to vectors of the other one. Note that in the first three cases, at least one eigenvalue is equal to ±1, and a cavity is found at the boundary of the stability domain. This situation also takes place for two-dimensional cavities at the boundary of stability. The fourth case is of a different nature and occurs for coincident eigenvalues different from ±1. Depending on the character of proper subspaces, this coincidence may lead (or not) to the appearance of a family of solutions of equation (3) with positive definite imaginary parts. Let us dwell on this case in more detail. Let us explain the statement that the planes V1, 2 are Lagrangian. For example, let us prove the skew-orthog+ onality for an arbitrary pair of vectors Y 1, 2 ∈ V1. Indeed, by virtue of (1) we have 2
+ t
Y 1 JY 2 = Y 1 T JTY 2 = ( TY 1 ) J ( TY 2 ) +i
+
+t
t
= e
2iθ
+
+t
+
+
Y 1 JY 2 = 0,
because e2iθ ≠ 1. The fact that V2 is a Lagrangian plane is proved in a similar way. Moreover, one can find, for an arbitrary vector from V1, the vector that is skew-orthogonal to the first one and belongs to V2, and their linear envelope that is invariant with respect to T represents a Lagrangian plane. Each of these planes, as well as V1, 2, is characterized by its own symmetric solution of equation (3). Let us show the validity of the following alternative: either (a) the planes V1, 2 are of fixed sign, i.e., θ2 = θ1, ±
±
λ 2 = λ 1 , and then one of them (for definiteness, V1) represents a single positive plane; or (b) the planes V1, 2 ±
−
have no fixed sign, θ2 = –θ1, λ 2 = λ +1 , and each positive vector belonging to V1 is characterized by its own positive plane. Let Y 1 ∈ V1 be a certain positive vector. Then Y 1 = +* Y 1 ∈ V2 is a negative vector. Let Y2 ∈ V2 be a vector that is skew-orthogonal to +
–
+
Y1 : t
+
Y 2 JY 1 = 0,
(7)
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Then the vector Y 2* ∈ V1 is skew-orthogonal to Y 1 . –
+
–
The vectors Y 1 , Y 1 , Y2, and Y *2 are linearly independent. (a) If Y2 is negative, Y *2 is positive. Let us show that in this case the whole plane V1 is positive. To do this, let us make sure that an arbitrary linear combination of the + vectors Y 1 and Y *2 is a positive vector: ( c 1 Y 1 + c 2 Y 2* ) * J ( c 1 Y 1 + c 2 Y 2* ) +
+
t
= c 1 Y 1 JY 1 + c 2 Y 2 JY *2 > 0 2
–t
+
2
t
because of (7). Therefore, the plane V1 is positive, and V2 is negative. In this case, L = V1, and the matrix H is unambiguously determined. (b) If the plane V1 is not positive, Y2 is positive and an arbitrary linear combination of the vectors Y2 is positive as well:
+ Y1
and
( c 1 Y 1 + c 2 Y 2 )* J ( c 1 Y 1 + c 2 Y 2 ) +
+
t
t
= c 1 Y 1 JY 1 + c 2 Y *1 JY 2 > 0 2
–t
+
2
–
because of the skew-orthogonality of the vectors Y 1 +
and Y2 and of the vectors Y *2 and Y 1 . Therefore, a lin+
ear combination of the vectors Y 1 and Y2 is the desired invariant positive Lagrangian plane. It is obvious that, in this case, one can take a linear envelope of an arbitrary positive vector belonging to V1 and a vector from V2 that is skew-orthogonal to the first one (and positive as well) as L. Each choice of this kind is characterized by its own symmetric matrix H with a positive definite imaginary part satisfying equation (3). Note that some of the vectors belonging to V1, 2 are neither positive nor negative; i.e., they are skew-orthogonal to their complex conjugates. Nevertheless, one can show that V1, 2 cannot only consist of such vectors, so that the subspace V1 is not negative and always contains +
the positive vector Y 1 . Finally, let us present a simple criterion that the matrices T satisfy in the case under consideration. As noted above, ν1, 2 are eigenvalues of the matrix (T + T–1)/2, and in the case where they are equal and adjoint vectors are absent, this matrix is equal to νE4. In view of (1), from this it follows that the blocks B and C are bound to be symmetric and A + Dt = 2νE. EXAMPLES Let us use the simplest examples to illustrate the cases considered above. From the analysis of threedimensional beams with complex astigmatism, it fol-
lows that additional parameters that are independent of the geometrical optics characteristics of a cavity also appear in the case of separable variables when the ray matrix of a three-dimensional cavity is described in terms of ray matrices for two-dimensional cavities. Because of this, to make our study more illustrative and avoid cumbersome calculations, the matrices A, B, C, and D will be chosen to have the diagonal form (the A 0 case of simple astigmatism), say, A = x ; i.e., 0 Ay T is formed of two independent ABCD matrices Tx = Ax Bx A B ± and Ty = y y . In this case, λ 1 and Cx Dx Cy Dy ±
λ 2 are the eigenvalues of Tx and Ty, respectively. (1) Tx = Ty = ±E. A cavity is found at the boundary of the stability domain. H [the solution of equation (3)] is an arbitrary symmetric matrix with a positive imaginary part. Consequently, a Gaussian beam concentrated in this cavity near the optical axis is characterized by three arbitrary parameters that are not determined by the cavity. (2) Tx = E, Ty = –E. A cavity is also found at the boundary of the stability domain. In this case, H is the diagonal matrix 1 ⁄ qx 0 H = , 0 1 ⁄ qy where qx and qy are arbitrary complex numbers from the lower half plane (Imqx, y < 0). A concentrated Gaussian beam is characterized by two arbitrary parameters that are independent of the cavity parameters. In the examples considered below, we assume that Tx is the ABCD matrix with eigenvalues (1 1/q)t, (1 1/q*)t belonging to the eigenvalues e±iθ, Imq < 0: 2i 1 1 --- = --- + ---------2 , q R kw
1 1 2i ------ = --- – ---------2 . q* R kw
(3) Ty = ±E. A cavity is found at the boundary of the stability domain. H is a diagonal matrix, qx = q and qy is an arbitrary number from the lower half plane (Imqy < 0). A concentrated Gaussian beam is characterized by one arbitrary parameter that is not determined by a cavity. (4a) Ty = Tx. A cavity is found inside the stability domain; astigmatism is absent. H = (1/q) × E. A concentrated Gaussian beam is unambiguously determined by the geometrical optics parameters of a cavity (arbitrary parameters are absent). OPTICS AND SPECTROSCOPY
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125
M2
–1
(4b) Ty = T x . A cavity is found inside the stability domain, but a concentrated Gaussian beam is not reconstructed in a unique way. It depends on an arbitrary parameter that is not determined by the cavity. Let us consider this example in more detail. The subspace V1 is a linear envelope of the vectors (1 0 1/q 0)t and (0 1 0 1/q*)t, and V2 is a linear envelope of the vectors (1 0 1/q* 0)t and (0 1 0 1/q)t. Within a factor, the positive vectors from V1 have the form (1 ζ 1/q ζ/q*)t, |ζ| < 1, and the vectors from V2 that are skeworthogonal to them have the form (ζ 1 ζ/q* 1/q)t. The matrix H is found from the equation
γ
x
z x z
z
O
M1
1 ⁄ q ζ ⁄ q* 1ζ = H , ζ ⁄ q* 1 ⁄ q ζ1
M3
(8)
x Figure.
from which it follows that 1 ζ 1 1 H = --- E + --- – ------ --------------2 ζ – 1 q q q* 1 – ζ – 1 ζ
For |ζ| = 1, ζ ≠ ±1, the eigenvalues of this matrix simultaneously vanish, so that the matrix H becomes purely real: If ζ = eiµ,
2 1 2i 1 = ---E + ---------2 --------------2 1 + ζ – 2ζ . R kw 1 – ζ – 2ζ 1 + ζ 2
1 2 1 H = ---E – ---------2 ----------- cos µ – 1 . R kw sin µ – 1 cos µ
Thus, each value of the complex parameter ζ lying inside the unit circle is characterized by its own symmetric solution of equation (3) with a positive definite imaginary part. In spite of the fact that the matrices A, B, C, and D are not diagonal in the case under consideration, H is no longer a matrix of this kind, with one exception, ζ = 0. The eigenvectors of the matrix H have the form (1 1)t and (1 –1)t, and the corresponding eigenvalues are given by 4 Im ζ 2i 1 – ζ 1 1 1 1 ζ --- – --- – ------ ------------ = --- + ---------2 -----------------2 + ---------2 -----------------2 1 + ζ R kw 1 + ζ q q q* kw 1 + ζ 2
and 4 Im ζ 2i 1 – ζ 1 1 1 1 ζ --- + --- – ------ ----------- = --- – ---------2 ----------------2 + ---------2 ----------------. 2 R kw 1 – ζ q q q* 1 – ζ kw 1 – ζ 2
The eigenvectors of the matrix H determine the orientation of the principal axes of the elliptic section of a Gaussian beam. In the case under consideration, they are rotated through 45° with respect to the transverse axes of the coordinate system. The same vectors are also eigenvectors for the real and imaginary parts of the matrix H, and the real and imaginary parts of the eigenvalues of the matrix H serve as eigenvalues for ReH and ImH. One can see from the form of eigenvalues that the matrix Im H is indeed positive definite for |ζ| < 1. OPTICS AND SPECTROSCOPY
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±1, the matrix 1 ζ entering in (8) ζ1 becomes degenerate, and one of the eigenvalues of the matrix H tends to infinity. In this case, the imaginary part of the second eigenvalue vanishes as before. For ζ
EXAMPLE OF A CONCRETE CAVITY FOR THE CASE 3b Consider the three-mirror cavity shown in the figure. In each arm, the z-axis is directed along the optical axis, the x-axis lies in the plane of the cavity circuit, and the y-plane is perpendicular to this plane. M1 and M3 are flat mirrors, and M2 is an elliptic mirror whose radii of curvature rx, y are related to the optical length of the cavity circuit l by the expression r x cos γ = r y ⁄ cos γ = l, where γ is the angle of incidence of the axial ray on the mirror M2. In this case, the matrices Tx, y calculated at the point 0 found at the optical distance l/2 from the mirror M2 have the form T x = 0 –l ⁄ 2 , 2⁄l 0
Ts = 0 l ⁄ 2 ; –2 ⁄ l 0
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i.e., Ty = Tx. For such matrices, q = –il/2, q* = –q = il/2, θ = –π/2 (i.e., the eigenvalues are equal to − + i for Tx and ±i for Ty), 1/R = 0 (the beam waist is the at the origin of coordinates), 1/q = 2i/l, and 1/q* = –2i/l. The family of matrices H, which represent the solutions of equation (3), has the form 2 2 H = ( 2i ⁄ l ) ( 1 ⁄ ( 1 – ζ ) ) 1 + ζ – 2ζ 2 – 2ζ 1 + ζ
.
The eigenvalues of such matrices corresponding to the eigenvalues (1, 1)t and (1, –1)t are equal to ( 2i ⁄ l ) [ ( 1 – ζ ) ⁄ ( 1 + ζ ) ] = ( 2 ⁄ l ) [ 2Im ζ + i ( 1 – ζ ) ] ⁄ 1 + ζ 2
2
and ( 2i ⁄ l ) [ ( 1 + ζ ) ⁄ ( 1 – ζ ) ] = ( 2 ⁄ l ) [ – 2 Im ζ + i ( 1 – ζ ) ] ⁄ 1 – ζ , 2
2
respectively. Note that in the general case, equation (3) has no explicit solutions, and therefore, numerical calculations are required. CONCLUSIONS Thus, in contrast to two-dimensional cavities for which the coincidence of eigenvalues of the ray matrix may take place only at the boundary of stability (when these values are equal to +1 or –1), the three-dimensional case allows one more possibility. If the blocks B and C of the cavity ray matrix T are symmetric, and A + Dt = 2νE, |ν| < 1, the matrix has doubly degenerate
complex conjugate eigenvalues e±iθ, where cosθ = ν. In this case, equation (3) has two isolated symmetric solutions determined by proper subspaces of the matrix T and, moreover, a continual family of symmetric solutions, each of them being determined by its Lagrangian plane that is invariant with respect to T and represents a linear envelope of a pair of skew-orthogonal eigenvectors, one of them belonging to the eigenvalue eiθ, and the other one belonging to the value e–iθ. The beams concentrated near the cavity axes are characterized by the solutions with a positive definite imaginary part. If the imaginary part of one of the isolated solutions is positive definite, the imaginary part of other solutions is not positive definite, and the fundamental mode is unambiguously determined. Otherwise, one can separate out in the continual family a subfamily of solutions with a positive imaginary part, which leads to the degeneration of the fundamental mode, and a beam becomes dependent on a parameter that is not determined by the cavity. This parameter can be changed only by external actions, such as the pumping, an active medium, etc. Thus, the technique developed in the paper makes it possible to control this parameter, which can be used for controlling astigmatism of beams in optical cavities. REFERENCES 1. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in the Problems on Diffraction of Short Waves (Nauka, Moscow, 1972). 2. I. V. Golovnin, A. I. Kovrigin, A. N. Konovalov, et al., Kvantovaya Élektron. (Moscow) 22, 461 (1995). 3. V. I. Arnol’d, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989). 4. M. J. Bastiaans, Optik. 88, 163 (1991).
Translated by A. Kirkin
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