`Optics and Lasers
`in Engineering
`
`
`
`'-
`
`\_
`
`Gabriel Laufer
`
`University of Virginia
`
` CAMBRIDGE
`
`UNIVERSITY PRESS
`
`ASML 1418
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`CAMBRIDGE UNIVERSITY PRESS
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`© Cambridge University Press 1996
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`This publication is in copyright. Subject to statutory exception
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`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
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`First published 1996
`
`A catalogue recardfir tin’:publication is zzvaihblefiom t/ye. Britirb Library
`
`Library afCongre.r: Cataloguing in Publication Dam
`Laufer, Gabriel
`
`Introduction to optics and lasers in engineering/ Gabriel Laufer.
`‘ p.
`cm.
`_
`Includes bibliographical references and index.
`ISBN 0-521-45233-3 (he)
`I. Lasers in engineering.
`TA367.5.L39
`I996
`6z1.36’6 - dczo
`
`2. Optics.
`
`I. Title.
`
`ISBN 978-0-521-452335 Hardback
`ISBN 978-o~52I-o1762~6 Paperback /
`
`5
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`95-44046
`CIP V’
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`
`13.3 Transformation of a Gaussian Beam by a Lens
`
`449
`
`the transformation of’Gauss'ian 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 bylenses, the latter information is needed to analyze the cavi-
`ties and will therefore be presented next.
`
`
`
`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 q,.
`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 R,. 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, x s,. After being focused by a lens of focal length f, the
`wave is transformed into a new spherical wave with a curvature «R2 propagat-
`ing toward a point at a distance s2 = —R2 away. Therefore, by replacing s, and
`
`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. I-Iowever,.in the absence of absorption
`losses (or gain), the real and imaginary components of the complex beam pa-
`rameter cm) can be .used to fully specify P(z) (sad Problem 13.2). Therefore,
`the propagation through nonlossy media can be described by simply evaluating
`q(z) at each point. Similarly, transformation byan optical element, such as a
`lens, can be described by evaluating qyand qz, the beam parameters at the in- ‘
`put and output planes of that element (Kogelnik 1965a). The relation between
`q, and qz 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.
`
`
`
`450
`
`13 Propagation ofLaser Beams
`
`at
`
`<12
`
`'2
`
`
`
`Input
`plane
`
`Output
`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.
`
`s; with R, and -R2, the lens equation (eqn. 2.10) can also be used to describe
`the curvatures of the incident and transmitted beams:
`
`(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 = r/r’. Since r2 and r; at the output plane of an optical element
`can be calculated by the ray transfer 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 input’p1ane.~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
`
`(13.27)
`
`(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
`1
`l
`/\
`
`4:
`
`R1
`
`WW?
`
`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:
`
`
`
`
`
`
`
`
`
`
`
`._.,_.,......_.......u..t,............,.._,,._._...,,,,,.-,.-,_,.,.,,,M.,..,,
`
`13.3 Transformation ofa 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
`
`-3- = —’—--1-.
`(12
`41
`f
`
`(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:
`
`
`
`q,
`
`/~ ~
`
`:
`
`g
`
`(13.29)
`
`X
`7
`.
`— Cq]
`This is the ABCD law. It has been demonstrated ti) accurately describe paraxial
`propagation of Gaussian beams through optical elements of interest, including
`thick lenses, spherical mirrors, and media with varying refraction indices.
`V
`To illustrate the utility ofthe 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 = 2:, C = 0,
`and D =1 (Table 2.1). After inserting these elements into (13.29), the follow-
`ing transformation of q, is obtained:
`
`Q2 :7 q,+z.
`
`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 d, 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 fromthe lens; the third ele-
`, ment is the lens with a focalglength 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
`
`(1-"5-)a+(d=-5’-f—”~+d2)
`
`(13.31)
`
`Owing to the selection of the input and output planes at the waists, where .
`NR, = 1/R2 = 0, q, and q; can be reduced to
`2
`'
`2
`,7l'W1
`,7l‘W2
`== -«I»-—— and
`= —1——
`A
`A
`
`42
`
`‘I1
`
`(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:
`
`dl '‘f __ W12
`d2_f ~;;2,— and <d.—/xdz-f)=f’~fo’.
`where
`
`(13.32)
`
`I 7fW]W2
`fo--
`A
`
`(Kogelnilc 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 A, 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 difference 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 ofthe 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.
`
`
`
`
`
`.__V.».3,»vam-«.r;.v.\y.~¢,vn-34):;vi:-eve‘:/;'IrN'~"‘V~*""‘f'P"%I‘j"’
`
`13.3 Transformation ofa Gaussian Beam by a Lens
`
`453
`
`Although most system de_signsinvolve an optimization between the Rayleigh
`range and the spot size, the beam radius is usually the first parameter to be calcu-
`lated. Other parameters, such as dz, are only secondary and can be determined
`after'selecting'Van optimal lens. Therefore, the following more useful equation '
`‘ for w; was obtained by eliminating dz between the two equations (13.32):
`1__1
`'_d.’1.2rw.’
`'
`~
`wg --W176
`+75 T
`(13.33)
`
`_
`
`known parameters (A, w,, d,) of the incoming laser beam and in terms of the
`focal length of the selected focusing lens.
`It is evident from (13.33) that to tightly focus a beam, the lens must have a
`= short focal length. This is hardly surprising. However, a less expected result is
`that for Gaussian beams, the spot size at the focus depends on the spot size of
`the incident beam. With the same focusing lens, the spot size of the focused
`beam may increase or decrease when the radi/us of the incident beam increases
`(Problem 13.5). Usually, to obtain small W2,‘ the spot size at the waist of the in-
`coming beam must be made as large-‘as possible; as w, increases, the first term
`of (13.33) diminishes «while the second term increases until it dominates (13.33),
`which can now be reduced to:
`
`.
`
`J“
`W; ‘*5 Tr-y-6: .
`
`Therefore, for high irradiance or for fine spatial resolution, a Gaussian beam
`must be focused by a lens with a short focal length and the spot size of the in-
`coming beam at its waist must be expanded until the beam fills the clear aper-
`ture of the lens.
`
`An alternative presentation of this result can be obtained by introducing
`the divergence angle 0, of the incoming beam. Using (13.21) to replace w, in
`(13.34), we have
`
`(13.35)
`_
`W2 2 f0,.A
`Thus, for tight focusing, the far-field divergence angle of the income beam
`must be reduced. Hypothetically, with 9 ~+O (i.e., when w, —+eo), the spot size
`at the focus can be reduced to a point.
`When the diameter of the incoming beam is comparable to the diameter of
`the focusing lens, (13.34) may also be expressed in terms of the lens f/# (Ver-
`deyen 1989, p. 94) as follows:
`
`W2 = 3,;*—<//#>.
`
`(13.36)
`
`where f/# -~= f/2w,. This result shows a linear dependence between the lens f/#
`and the spot size of the focused beam.
`Equations (13.34) and (13.35) can be inverted to describe the cone angle
`02 obtained by focusing an incident beam with a spot size of w,. The result is
`
`
`
`454
`
`13 Propagation of Laser Beams
`
`identical to the cone angle projected by Example 13.] 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 f, 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 absorption;,‘scatter_in§, or reflections at interfaces do not
`discriminate among the various transyerse 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 effects 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