Introduction to
`
`Optics and Lasers
`in Engineering
`
`
`
`Gabriel Laufer
`
`University of Virginia
`
`CAMBRIDGE
`UNIVERSITY PRESS
`
`
`
`ASML1512
`
`ASML 1512
`
`

`
`CAMBRIDGE UNIVERSITY PRESS
`
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`
`© Cambridge University Press 1996
`
`This publication is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
`
`First published 1996
`
`A catalogue remralfbr tbispublicatian 2': availablefiom t/J: Britt‘:/1 Library
`
`Library afCong're.r: Cataloguing in Publication Data
`Laufer, Gabriel
`
`Introduction to optics and lasers in engineering I Gabriel Laufer.
`p.
`cm.
`
`Includes bibliographical references and index.
`ISBN 0-521-45233-3 (he)
`
`I. Lasers in engineering.
`TA367.5.L39
`1996
`62.1.36’6 - dczo
`
`2. Optics.
`
`I. Title.
`
`K
`
`_
`
`95-44046
`CIP
`
`‘
`
`.
`
`ISBN 978-0-521-45233-5 Hardback
`ISBN 978-0-521-01762-6 Paperback -/
`
`Cambridge University Press has no responsibility for the persistence or
`accuracy of URLs for external or third-parry internet websites referred to in
`this publication, and does not guarantee that any content on such websites is,
`or will remain, accurate or appropriate. Information regarding prices, travel
`timetables, and other factual information given in this work is correct at
`the time of First printing but Cambridge University Press does not guarantee
`the accuracy of such information thereafter.
`
`

`
`13.3 Transformation of a Gaussian Beam by a Lens
`
`449
`
`the transformation of Gaussian beams by lens. Such equations are expected to
`describe the size of the waist behind a lens or its location relative to the lens.
`
`Similarly, equations that describe the oscillation of Gaussian beams inside la-
`ser cavities are needed for cavity design. Although the formation of Gaussian
`
`beams in a resonator logically should be considered before examining beam
`transformation by lenses, the latter information is needed to analyze the cavi-
`ties and will therefore be presented next.
`
`13.3 Transformation of a Gaussian Beam by a Lens
`
`The beam parameter q(z) (eqn. 13.12) and the phase-shift parameter P(z)
`(eqn. 13.15) fully characterize Gaussian laser beams. Therefore, the propaga-
`tion of a laser beam through optical media, or its transformation by lenses or
`mirrors, can be generally modeled by evaluating q(z) and P(z) at the input and
`
`output planes of an optical element. However,.in the absence of absorption
`losses (or gain), the real and imaginary confionents of the complex beam pa-
`rameter q(z) can be.used to fully specif-y P(z) (see) Problem 13.2). Therefore,
`the propagation through nonlossy media can be described by simply evaluating
`q(z) at each point. Similarly, transformation by an optical element, such as a
`lens, can be described by evaluating q. and q2, the beam parameters at the in-
`put and output planes of that element (Kogelnik 1965a). The relation between
`q, and q; for such transformation must then be specific to that element. Beam
`propagation through a train of optical elements and through the separating
`media between them can be described by applying consecutively the transfor-
`mation function of the various elements.
`
`The transformation of radiation by optical elements can also be analyzed
`
`by geometrical optics, where the paths of geometrical rays is drawn. Therefore,
`it can be expected that some of the techniques of geometrical optics will also be
`applicable to the analysis of propagating laser beams. In particular, the method
`of ray transfer matrices (Section 2.8) is adopted here to describe the transfor-
`mation of Gaussian beams by lenses and mirrors. This method is extremely
`useful for describing the transformation by several consecutive elements or an
`oscillation between the mirrors of a laser cavity. Similarly to geometrical op-
`
`tics, each element will be described by a 2 x2 matrix (Figure 13.5). However,
`unlike geometrical optics, where the matrix is used to compute r and r’ (eqn.
`2.17), here it will be used to calculate the real and imaginary components of q;
`in terms of the two components of 41,.
`To develop the equation for the transformation of a Gaussian beam by a
`lens, consider first as an analogy the transformation by a lens of a spherical
`wave with a radius of curvature R1. The curvature is defined as positive if it is
`concave when viewed from the left side of the front. For radiation emitted by an
`
`ideal point source, this curvature for propagation to the right is simply the dis-
`tance to the source R, = s,. After being focused by a lens of focal length f, the
`wave is transformed into a new spherical wave with a curvature —R; propagat-
`ing toward a point at a distance s2 = —R2 away. Therefore, by replacing s, and
`
`
`
`

`
`450
`
`13 Propagation ofLaser Beams
`
`Q1
`
`02
`
`'1
`
`
`
`lnput
`plane
`
`liutput
`plane
`
`Figure 13.5 Schematic presentation of a transforming optical element, showing
`the ray and beam parameters at the input and output planes and the ray transfer
`matrix of that element.
`
`32 with R, and —R2, the lens equation (eqn. 2.10) can also be used to describe
`the curvatures of the incident and transmitted beams:
`
`
`
`
`
`:3_,3.5.-r_'-’
`
`2
`
`,5.
`_?
`
`I
`V_
`2
`] -
`
`"
`
`'
`
`(13.27)
`
`(see Problem 13.3).
`A more general expression that describes the transformation of a spherical
`wave by any optical element can be obtained by assuming that, for paraxial
`propagation, R z r/r’. Since r2 and r5 at the output plane of an optical element
`can be calculated by the ray transfier matrix_whe_n r, and r{ are known, the cur—
`vature of the spherical wave at the output plane can also be determined by that
`matrix from the curvature in the inpu.t’plane.-It-can be shown (Problem 13.3)
`that, for paraxial propagation through an optical component with ray transfer
`matrix elements A, B, C, and D, the transformation of a spherical wave is
`
`R _ AR.+B
`2’ CR,+D
`
`(Kogelnik 1965a).
`To extend this result for the analysis of Gaussian laser beams, consider
`the transmission of a Gaussian beam by a thin, nonabsorbing lens. The beam
`parameter at the input plane of the lens (eqn. 13.12) is
`1l.A
`—=—+i———
`q]
`R]
`‘IFWI2 .
`For a thin lens, the radius of the beam immediately past the lens is w, = 'w., and
`the only effect of that lens on the beam is to transform its radius of curvature
`from R, to R2:
`
`’-“
`
`,.
`
`

`
`13.3 Transformation of a Gaussian Beam by a Lens
`
`451
`
`Thus, similarly to the transformation of spherical waves by a lens, the trans-
`formation of the beam parameter by a lens is
`
`— = ———.
`
`(13.28)
`
`Although the analogy between the beam parameter q and the radius of curva-
`ture of the wavefront R was shown here only for a transformation by a thin
`lens, it was generalized (Kogelnik 1965b) to describe transformation by arbi-
`trary elements. For an optical component with ray transfer matrix elements
`A, B, C, and D, the transformation of the beam parameter similar to (13.27) is:
`
`92
`
`l
`
`1
`
`This is the ABCD law. It has been demonstrated tb accurately describe paraxial
`propagation of Gaussian beams through optical elements of interest, including
`thick lenses, spherical mirrors, and media with varying refraction indices.
`To illustrate the utility of the ABCD law for deriving the transformation by
`elements other than a thin lens, consider the propagation through a distance z
`in free space. The matrix elements for this propagation are A = 1, B = z, C = 0,
`and D = 1 (Table 2.1). After inserting these elements into (13.29), the follow-
`ing transformation of q. is obtained:
`
`(I2 = G: +z.
`
`(13-30)
`
`Although this is not a new result (compare it to eqn. 13.6), it demonstrates that
`the ABCD law is not restricted to transformations by thin elements.
`The ABCD law can also be used to describe the propagation of a Gaussian
`beam from its waist where its radius is w. to a lens located a distance all away,
`through that lens and then to the new waist a distance dz away, where the radius
`is w, (Figure 13.6). When modeled by ray transfer matrices, this transformation
`
`
`
`Figure 13.6 Transformation of a Gaussian beam by a lens.
`
`

`
`452
`
`13 Propagation ofLaser Beams
`
`includes three elements: two of these elements are the flat media with n =1
`(Table 2.1) that represent the propagation to and from the lens; the third ele.
`ment is the lens with a focal length of f. The combined ray transfer matrix
`of these elements was previously calculated (eqn. 2.18). Therefore, using the
`ABCD law, the transformation of the beam may be written as
`
`<x—s>q1+<ax~a+az>
`
`(13.31)
`
`-_-,2
`
`:r.-='«es._—.—,_-~.-,4_
`
`Owing to the selection of the input and output planes at the waists, where
`1/R, = l/R2 = 0, q, and q; can be reduced to
`
`1rw2
`q, = -1-7-'— and
`
`1rw2
`qz = -1-T2
`
`(cf. eqn. 13.12). Although q. and q; are now purely imaginary, (13.31) still in-
`cludes real and imaginary components. By separating and equating the imagi-
`nary and real parts at both sides of this equation, one obtains:
`
`d1"f _ W3
`d2_f - -W7
`2
`where
`
`_ ‘KWIW2
`A
`
`fo
`
`and
`
`(d1“f)(d2‘f) =f2"fo2.
`
`(13-32)
`
`
`
`=i*-._-..‘-'.'3:.’-hp‘
`‘-iv-.
`
`(Kogelnik and Li 1966).
`With the two equations of (13.32), the spot size of the focused beam w;
`and the distance dz to its waist ‘from a lens with a focal length f can be calcu-
`lated when z\, w], and d, are known. Although this result can be viewed as the
`counterpart to the lens equatjon (eqfi. 2.10), one should note that focusing a
`Gaussian laser beam may be significantly different from the results of geomet-
`rical optics. This is illustrated in Problem 13.4, where the location of the waist
`of a focused beam is seen to fall short of the geometrical focus of that lens.
`The dilference between the two results may be attributed to diffraction effects
`that are neglected in geometrical optics. Thus, the location of the waist coin-
`cides with the geometrical focus of the lens only when b, >> f, that is, when
`diffraction is indeed negligible.
`The two parameters solved by (13.32) — the location of the waist of the fo-
`cused beam and its radius - determine other important beam parameters such
`as the peak irradiance and the Rayleigh range. Alternatively, these equations
`may be used to select a focusing lens for particular beam-shaping needs. For
`applications such as laser-assisted cutting or surface heat treatment, where high
`irradiance is required, the lens is selected primarily to minimize wz. By contrast.
`for illumination, the beam is expected to remain collimated over a prescribed
`length and the lens is selected to extend the Rayleigh range by increasing wz.
`
`
`
`..35j;_.‘_'
`
`" _
`
`

`
`

`
`454
`
`13 Propagation of Laser Beams
`
`identical to the cone angle projected by Example 13.1 of the previous section,
`where focusing was considered using concepts of geometrical optics.
`Although the analysis here was limited to transformations of axially sym-
`metric beams by lenses with spherical symmetry, a beam can have independent
`parameters in two perpendicular planes. Thus, with the use of a cylindrical
`lens, the beam cross section can be shaped into an ellipse with an effective width
`along one axis that is different from its width along the other axis. The axes of
`the ellipse and the beam propagation axis define the two orthogonal planes. As
`the beam propagates, it may have two divergence angles; the shallower diver-
`gence is in the plane that includes the wider effective width and the fast diver-
`gence is in the plane that includes the narrower width. Such disparity in the
`beam properties is necessary to shape laser beams into light sheets. With cylin-
`drical focusing, the beam parameters q. and q; can no longer be defined by a
`single application of the ABCD law (eqn. 13.29). Instead, two beam parame-
`ters at the output plane of the cylindrical lens must be evaluated separately,
`using its two focal lengths fl and f2 (see Research Problem 13.1).
`
`13.4 Stable and Unstable Laser Cavities
`
`The equations of the transformation of laser beams by a lens, together with
`the ABCD law, can now be used to identify criteria for stable oscillation of the
`fundamental mode in a laser cavity. When a cavity is properly designed and
`aligned, random field distributions anywhere within it must gradually evolve
`until they give way to the least lossy distribution — the Gaussian mode (Fox and
`Li 1961). Thus, during an initial transition period, the field inside the cavity can
`vary from transit to transit. Such variations may include an increase in the am-
`plitude until 155 is reached (eqn. 12.24), or evolution of the transverse distri-
`bution until the mode that has the lowest diffraction losses is fully developed.
`During this transition period, the irradiance is typically low,‘ and most loss
`mechanisms such as absorptionfscattering, or refiections at interfaces do not
`discriminate among the various transverse modes and are therefore extraneous
`to the selection of a stable field distribution. (Note that, in high-gain lasers, the
`field may develop rapidly and such nonlinear efl‘ects as saturation or bleaching
`may participate in shaping the transverse mode.) Thus, initially the primary
`loss mechanism that discriminates among transverse modes is diffraction, and
`the mode that presents the lowest diffraction losses survives to the detriment of
`other potentially competing modes. One such stable distribution is the Gauss-
`ian mode. After steady state is established, the distribution of the field, as well
`as its gain, remains unchanged during consecutive cavity transits. This condi-
`tion was used to find the amplitude of the electric field and the irradiance (eqn.
`12.24) and will also be used here to establish the condition for steady-state oscil-
`lation of the Gaussian mode.
`
`Figure 13.7 illustrates a laser cavity designed to support steady oscillation
`of a Gaussian beam. This cavity is shown empty - the gain medium and other
`intracavity devices were omitted for simplicity. In the absence of nonlinear