Optical system for variable resizing of round flat-top distributions George Nemeúa, John A. Hoffnagleb a
ASTiGMAT, 1457 Santa Clara St. Ste. 6, Santa Clara, CA 95050, USA;
[email protected] b IBM Almaden Research Center, 650 Harry Rd., San Jose, CA 95120, USA;
[email protected] ABSTRACT
An optical system is described which is capable of variable resizing of a round flat-top light distribution of a highly coherent and collimated laser beam at a certain, fixed working distance. It contains cylindrical and spherical optics. First-order optics is used to analyze the system, and the second-order moments method is used to describe the beam properties. The analysis indicates the existence of an "image-mode" regime, allowing a dynamic range, or a zoom range of about (13 – 15) : 1 for resizing the diameter of the round, flat-top distribution in the target plane, by rotating one cylindrical element. Experiments using as incoming beam a collimated Ar-laser TEM00 gaussian beam reshaped to a collimated flat-top with 6.8 mm in diameter, confirm the approach, by obtaining at the target plane spot sizes with diameters continuously variable between 1.0 mm and 13.6 mm and with flat-top profiles. Keywords: beam shaping; flat-top beams; Fermi-Dirac beams; gaussian beams; variable-size imaging; matrix optics; cylindrical optics; second-order moments
1. INTRODUCTION Laser beam shaping 1-3 is the technique used to transform a certain laser beam irradiance profile, usually as it is obtained directly from a laser, into a desired profile, more appropriate to a specific application 3. Many industrial or medical laser applications require having a beam with an almost uniform irradiance profile, usually called a flat-top (or top-hat) profile. Also, sometimes, it is desirable that in the target plane, i.e., the plane where the laser delivers its power or energy to the component to be machined (in an industrial application) or to the tissue to be irradiated (in a medical or biological application), to have an adjustable spot size. Such a capability allows to adjust the appropriate power density or energy density at the target with respect to the specific application and also to the target shape and size. Laser beams with flat-top profiles can be directly obtained from some lasers with a relatively low spatial coherence, at least in one transverse direction, as, for example, happens in most excimer lasers. An alternative is to obtain such profiles from highly spatially coherent lasers (transverse single-mode lasers) with near-gaussian profiles by using specially designed optical systems to change the laser beam profile – beam shapers4-6. Such beams maintain their flat-top profile with only small degradation over some distance (tens of centimeters to meters) in free-space propagation5. On the other hand, sometimes it is sufficient to have a flat-top profile only in the spot on the target, where the laser application is performed. In these cases a spot with a flat-top profile can be obtained by spatially sampling and superimposing the incoming laser beam (for example, from an excimer laser, or from a high power diode stack) into a desired spot shape and size, a technique called homogenizing the beam. Another technique to obtain spots with a flat-top profile is to image a certain transverse plane of a beam with a good flat-top profile with the desired (sometimes adjustable) optical magnification and at the desired working distance, where the target is placed. This latter approach is used in this work. Here we describe a zoom-type optical system able to resize a round flat-top light distribution of a highly coherent and collimated laser beam at a certain, fixed working distance. It contains cylindrical and spherical optics. First-order optics is used to analyze the system, and the second-order moments method is used to describe the beam properties. The analysis indicates the existence of an "image-mode" regime, allowing a dynamic range, or a zoom range (defined as the ratio between the maximum and the minimum spot diameter while preserving the flat-top profile) of about (13 – 15) : 1. The adjusting element is the rotation angle of one cylindrical optical element7. Experiments using as incoming beam a collimated, Ar+-laser TEM00 gaussian beam, reshaped to a collimated flat-top with 6.8 mm in diameter, confirm the
Laser Beam Shaping VII, edited by Fred M. Dickey, David L. Shealy, Proceedings of SPIE Vol. 6290, 629008, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.682815
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approach. Section 2 is a short background of matrix treatment of optical systems and of laser beams, necessary to understand the "image-mode" property of the new optical system. In Section 3 we give the structure of the optical system and analyze its properties. Section 4 gives the experimental setup used to test the optical system with incoming gaussian beam and with incoming flat-top (Fermi-Dirac type) beam. Section 5 gives the main results of the experiment and discusses them. Section 6 concludes the paper.
2. SUMMARY OF OPTICAL SYSTEMS AND BEAMS – MATRIX TREATMENT We begin with the basic concepts necessary to understand the results of the paper. They are: rays, optical systems, and beams. We use the simplest approach of paraxial rays and beams, and of idealized (aberration-free, lossless, passive, centered, and normally positioned to beam axis) optical systems8,9. In the following the z axis represents the optical axis of the beam and of the optical systems, and the x and y axes are the transverse, horizontal and vertical axes, respectively. A (paraxial) ray is represented as a 4x1 column matrix with real elements containing the two transverse spatial coordinates x(z), y(z) of its intercept of the z = constant plane, and the two paraxial angles u, v representing the angles between the ray and its projections on the yOz and xOz plane (in this order), respectively. The ray is considered in free space, and therefore u and v do not depend on z. Here paraxial means that all the angles between the rays and their projections are conventionally considered d S/10 | 0.31 rad, or approximately 18 degrees9. The ray R is represented here by its transpose RT (1x4 row matrix): RT = (x(z) y(z) u v)
(1)
In the sequel, for simplicity, the explicit z dependence is dropped; z is the evolution parameter of the beam, i. e., the beam is evaluated for each plane z = constant. An optical system, which can be made of spherical lenses, cylindrical lenses (possibly rotated about the optical axis), and sections of free spaces, is represented by the 4x4 real matrix S, called the ray transfer matrix of the optical system: A11 S = A21 C11 C21
A12 A22 C12 C22
B11 B21 D11 D21
B12 B22 D12 D22
= A B C D
(2)
where A, B, C, D, are real 2x2 matrices, A and D with elements without physical dimensions, B with elements having dimension of length, and C with elements with dimension of reciprocal length. Such optical systems are called ABCD-type optical systems. The matrix S is symplectic, meaning the following mathematical property holds: S J ST = J where J is the following 4x4 real matrix: J =
(3) 0 I , with the properties J1 = JT = – J; J2 = – I, and where 0 is the –I 0
2x2 zero matrix and I is the 4x4 or the 2x2 identity matrix, respectively. From (3) it follows uniquely that det S = 1 (and not –1); det stands for the determinant of the matrix. The symplectic condition (3) is equivalent to the following three conditions involving the 2x2 matrices A, B, C, D, ultimately reduced to six scalar conditions: ADT – BCT = I; ABT = BAT; CDT = DCT. This reduces the maximum number of independent elements of the matrix S to ten. Any real, symplectic matrix S of the form (2) and fulfilling (3), and which has its A, B, C, D submatrices with the appropriate physical dimensions mentioned above, is physically obtainable using only a finite number of thin lenses (spherical and cylindrical) and free spaces. The ray transfer property of the optical system specifies the ray at the output of the system, Rout (sometimes written with subscript 2, R2) from the ray at its input Rin (sometimes written with subscript 1, R1), through the linear relation:
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Rout = S Rin (or, similarly, R2 = S R1)
(4)
By cascading (putting in series) two optical systems S1 and S2 such as the input plane of S2 coincides with the output plane of S1, a ray traversing first the system S1 an then the system S2, the equivalent optical system between the input of S1 and the output of S2 has the following matrix, S: S = S2 S1
(5)
The basic two optical systems in the three-dimensional space are the free-space, SF, a one-parameter optical system, and the (rotated simple astigmatic) thin lens, SL, a three parameter optical system10. Their matrices are: SF = I dI 0 I
(6)
with d – the length of the free-space section, and: SL= I 0 , with C = CT = C11 C12 = –1/f11 –1/f12 = – cos2(D)/fx – sin2(D)/fy – sin(D)cos(D)(1/fx – 1/fy) –1/f12 –1/f22 – sin(D)cos(D)(1/fx – 1/fy) – sin2(D)/fx – cos2(D)/fy C I C12 C22
(7)
where fx, fy represent the focal lengths of two plano-cylindrical lenses with optical powers along the axes x and y, respectively (positive focal lengths means converging lenses), and Dis the common angle with which each cylindrical lens of above is rotated about the z axis with respect to x and y axis, respectively. The beam is represented by the second-order moments matrix, or the beam matrix, P. The second-order moments of any combination of the coordinates x, y, u, v, are obtained by an averaging procedure, denoted < >8-11: P = = T
=
W M MT U
(8)
Eq. (8) is the definition of P, and it is written in its 4x4 matrix form, or in its 2x2 submatrices form, respectively. The matrix P depends on the specific z = const. plane where its elements are evaluated, through the x(z) and y(z) dependence of the ray's coordinates; however, for simplicity, we do not explicitly represent this dependence. From Eq. (8) it follows that the matrix P is real, positive definite11, and symmetric, or: P > 0; PT = P (equivalent to: WT = W; UT = U; MT M)
(9)
Therefore, W and U are symmetrical matrices, with maximum three independent elements, while M is non-symmetrical with maximum four independent elements. It follows that P has also maximum ten independent elements, but for different reasons than the matrix S. The 2x2 submatrices of P have elements with the following physical dimensions: W - length squared, M - length, and U - paraxial angle squared, no dimension. An approximate physical interpretation of the three submatrices W, M, and U is: W is a matrix with elements proportional to the squared transverse size of the beam; U has its elements proportional to the squared angular divergences of the beam, and M is a matrix with its symmetrical part related to the radii of curvature of the average phase front of the beam, and with its antisymmetrical part related to the orbital angular momentum carried by the beam. The beam transfer property of S, or the beam "propagation" property of P through the optical system S is (combining Eqs. (4) and (8)): Pout = SPinST (or similarly, with subscripts 2 and 1, respectively: P2 = SP1ST)
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(10)
Simple example of a beam and its transformation ("propagation") Here we give a simple example of a rotationally symmetric beam with no angular momentum (called also stigmatic beam) and its transformation (or "propagation") through a general ABCD-type optical system. Because the beam is stigmatic, its four submatrices are all proportional to the identity matrix8,9: W = WI; M = MI; U = UI
(11)
Such a stigmatic beam has the following main spatial parameters9: diameter D(z), at a specific z = const. plane, divergence (full angle) T, Rayleigh range zR, and beam propagation ratio M2, respectively: D(z) = 4W1/2(z); T = 4U1/2; zR = D0/T M2 = (S/4)D0T/O
(12)
For simplicity, we apply the beam with its waist (denoted by the subscript 0) at the input of a general ABCD-type optical system, and, therefore, we have at the input plane: W = W0; M = 0; U = U0. The following is the beam at the output plane of the optical system (obtained by applying Eq. (10)): W2 = AAT W0 + BBTU0 M2 = ACT W0 + BDTU0 U2 = CCT W0 + DDTU0
(13)
We have now all necessary relations to analyze the optical system providing a variable spot size by imaging a specific transverse plane from an incoming stigmatic, round beam, with a specific profile (gaussian, flat-top, or possibly other type of profiles).
3. OPTICAL SYSTEM FOR VARIABLE SPOT RESIZING 3.1. Schematic of optical system The schematic of the optical system analyzed here is given in Fig. 17,12. This system is part of a larger family of systems with similar functions we call VariSpot7. cAr U a)
— cAr
(— a)
2W
OflSaI — iW biSUG
to
onuq abo; o(a)
ucowiud PC SW Do
(3)
(.1)
Fig. 1. Schematic diagram of the optical system. The incoming beam has its waist at the input plane of the system. (1) – input plane; (2) – output ("quasi-image") plane; D0 – incoming beam waist diameter; D(D) – diameter of the adjustable-size round spot at the output plane; f – focal length of plano-cylindrical lenses (positive and negative); f0 – focal length of the spherical lens; D – relative angle of rotation between cylindrical lenses.
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The conditions that the system of Fig. 1 to work properly are: the incoming beam to be stigmatic and well collimated (i.e., its Rayleigh range to be >> all other distances and focal lengths); the optical distance between cylindrical lenses to be negligible (ideally zero); the distance between the spherical lens and the cylindrical lenses is, to first order, irrelevant, but it should be small enough for compactness; the working distance (from the spherical lens to the target plane, or the "quasi-image" plane) is essentially f07. 3.2. Optical system analysis Here we analyze the optical system of Fig. 1. Eqs. (13) give the beam matrix elements at the output plane of any ABCDtype optical system for a stigmatic incoming beam with its waist positioned at the input plane of that optical system. Out of the three 2x2 submatrices W, M, and U, only the matrix W is directly measurable by measuring irradiance profiles and the same matrix has also its elements at a finite distance, z = constant10. If that matrix at the output plane is of the type: W2 = W2I,
(14)
then the spot in the output plane is round. Because we are not interested in the spot characteristics before or after the output plane (which is also the target plane) we need to calculate only the matrix W2. From the first Eq. (13) it follows that we need to know the submatrices A and B of the whole optical system. They can be obtained by multiplying the four elementary 4x4 matrices representing the optical elements of Fig. 1, in reverse order, as Eq. (5) states. By doing so and replacing the results in the first Eq. (13), a submatrix of type (14) is obtained, with: W2 = W2(D) = [(f02/f2) sin2(D)] W0 + (f02) U0
(15)
which can be written also as: W2(D) = W0 [(f02/f2) sin2(D) + f02/z2R]
(16)
This equation expresses the second moment W2 at the output plane as a function of the second moment W0 of the incoming beam (at waist), the incoming beam Rayleigh range zR, and the parameters of the optical system of Fig. 1, f, f0, and D. It clearly shows the possibility to adjust the size of this matrix element by varying D. From this matrix element the diameter of the adjustable spot at the output plane, D(D), is obtained as a function of the incoming beam waist diameter, D0, as: D(D) = D0 [(f02/f2) sin2(D) + f02/z2R]1/2 = Dm[1 + sin2(D)/sin2(DR)]1/2
(17)
with some of the following notations: Dm = D2(D = 0) = D0 f0/zR = Tf0 DM = D2(D = S/2) = D0 [(f02/f2) + f02/z2R]1/2 D0 f0/f sin(DR) = f/zR
(18)
where the approximate equality of the second equation is valid for f/zR > B2U0 (or, similarly A2 >> B2/z2R)
(22)
Here >> can be conventionally interpreted as "10 - 100 times greater than", depending on the degree of accuracy we would like to have the following approximate relation fulfilled: W2 Q-IM | A2W0
(23)
We call this condition a "quasi-image regime" for the optical system and the incoming beam (together), because the condition expressed by Eq. (22) is beam dependent (and, correspondingly, the use of the additional subscript Q-IM in Eq. (23)). Quasi-image regime for VariSpot Applying the "quasi-image regime" Eq. (22) for the optical system analyzed here, we get: [(f02/f2) sin2(D)] W0 >> (f02) U0
(24)
from which the relevant output quantity in this regime is obtained: W2 | (f02/f2)sin2(D)W0
(25)
or, using the diameters of the beam spots, D(D) | D0(f0/f)sin(D)
(26)
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Eq. (26) gives the adjustable size output diameter as a function of incoming beam waist diameter and the optical system parameters in the "quasi-image regime" of the VariSpot. Note that this regime is equivalent to the "angular far-field" regime for which 1
