Laser and Gaussian Beam Propagation and Transformation
`
`Javier Alda
`University Complutense of Madrid, Madrid, Spain
`
`INTRODUCTION
`
`Optical engineers and researchers working on optics deal
`with laser beams and optical systems as usual tools in
`their specific areas. The knowledge of the special cha-
`racteristics of the propagation of laser beams through
`optical systems has to be one of the keystones of their
`actual work, and the clear definition of their characteristic
`parameters has an important impact in the success of the
`applications of laser sources.[Hi' In this article, we will
`provide some basic hints about the characterization and
`transformation of laser beams that also deserve special
`attention in basic and specific text books (e.g., see Refs.
`[7—l3]). The Gaussian beam case is treated in the first
`place because of its simplicity.[14‘15' Besides, it allows to
`introduce some characteristic parameters whose definition
`and meanng will be extended along the following
`sections to treat any kind of laser beam. In between, we
`will show how the beam is transformed by linear optical
`systems. These systems are described by using the tools
`of matrix optics.[16’18'
`In the following, we will assume that laser beams have
`transversal dimensions small enough to consider them as
`paraxial beams. What
`it means is that
`the angular
`spectrum of the amplitude distribution is located around
`the axis of propagation, allowing a parabolic approxi-
`mation for the spherical wavefront of the laser beam. In
`the paraxial approach, the component of the electric field
`along the optical axis is neglected. The characterization of
`laser beams within the nonparaxial regime can be done,
`but it is beyond the scope of this presentation.[19’22' We
`will take the amplitudes of the beams as scalar quantities.
`This means that
`the polarization effects are not con-
`sidered, and the beam is assumed to be complete and
`homogeneously polarized. A proper description of the
`polarization dependences needs an extension of the for-
`malism that is not included here.[23’28' Pulsed laser beams
`
`also need a special adaptationlzg’n' of the fundamental
`description presented here.
`
`GAUSSIAN BEAMS
`
`Gaussian beams are the simplest and often the most de-
`sirable type of beam provided by a laser source. As we will
`
`Encyclopedia of Optical Engineering
`DOI: 10.1081/E—EOE 120009751
`Copyright © 2003 by Marcel Dekker, Inc. All rights reserved.
`
`see in this section, they are well characterized and the
`evolution is smooth and easily predicted. The amplitude
`function representing a Gaussian beam can be deduced
`from the boundary conditions of the optical resonator
`where the laser radiation is produced.[7’9‘33'34' The geo-
`metrical characteristics of the resonator determine the type
`of laser emission obtained. For stable resonators neglect-
`ing a small loss of energy, the amplitude distribution is
`self-reproduced in every round trip of the laser through the
`resonator. Unstable resonators produce an amplitude dis-
`tribution more complicated than in the stable case. Be-
`sides, the energy leaks in large proportion for every round
`trip. For the sake of simplicity, we restrict this first ana-
`lysis to those laser sources producing Gaussian beams. The
`curvature of the mirrors of the resonator and their axial
`
`distance determine the size and the location of the region
`showing the highest density of energy along the beam.
`The transversal characteristics of the resonator allow the
`
`existence of a set of amplitude distributions that are usu-
`ally named as modes of the resonator. The Gaussian beam
`is the lowest-degree mode, and therefore it is the most
`commonly obtained from all stable optical resonators.
`Although the actual case of the laser beam propagation
`is a 3-D problem (two transversal dimensions x,y, and one
`axial dimension z), it is easier to begin with the expla-
`nation and the analysis of a 2-D laser beam (one trans-
`versal dimension x, and one axial dimension 2). The
`amplitude distribution of a Gaussian laser beam can be
`written as:[7‘9‘34'
`
`(1)
`
`This expression describes the behavior of the laser beam
`amplitude as a function of the transversal coordinate x and
`the axial coordinate z. k = 271/}. is the wave number, where
`J, is the wavelength of the material where the beam pro-
`pagates. The functions R(z), 60(1), and (15(2) deserve
`special attention and are described in the following sub-
`sections. Before that,
`it is interesting to take a closer
`look at Fig.
`l where we plot the irradiance pattern in
`terms of x and z. This irradiance is the square modulus of
`the amplitude distribution presented above. We can see
`
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`1000
`
`Laser and Gaussian Beam Propagation and Transformation
`
`He in amplitude, or He2 in irradiance with respect to the
`amplitude at
`the propagation axis. There exist some
`others definitions for the width of a beam related with
`
`some other fields.[36’38' For example, it is sometimes use-
`ful to haVe the width in terms of the full width at half the
`
`maximum (FWHM) values.[14' In Fig. 2, we see how the
`Gaussian width and the FWHM definitions are related. In
`
`Fig. 3, we calculate the portion of the total irradiance
`included inside the central part of the beam limited by
`those previous definitions. Both the 2-D and the 3-D cases
`are treated. For the 3-D case, we have assumed that the
`beam is rotationally symmetric with respect to the axis
`of propagation.
`Another important issue in the study of the Gaussian
`beam width is to know its evolution along the direction of
`propagation z. This dependence is extracted from the
`evolution of the amplitude distribution. This calculation
`provides the following formula:
`
`z).
`60(1) : coo 1+ —2
`T5600
`
`2
`
`(4)
`
`l as a
`The graphical representation is plotted in Fig.
`white line overimposed on the irradiance distribution. We
`can see that it reaches a minimum at z: 0, this being the
`minimum value of mo. This parameter, which governs the
`rest of the evolution, is usually named as the beam waist
`width. It should be noted that co(z) depends on 2., where 2.
`is the wavelength in the material where the beam is
`propagating. At each perpendicular plane,
`the z beam
`shows a Gaussian profile. The width reaches the mini-
`mum at the waist and then the beam expands. The same
`
`
`Gaussian beam amplitude and Irradiance
`1
`
`.0 on
`
`.00)
`
`
`
`19.u Irradiance
`
`
`
`(AU) 02 -
`
`-0,3
`
`-0.2
`
`Oil
`_€)
`4311
`2 coordinate (mm)
`
`0.2
`
`0.3
`
`Fig. 2 Transversal profile of the Gaussian beam amplitude at
`the beam waist (dashed line) and irradiance (solid line). Both of
`them have been normalized to the maximum value. The value of
`
`the width of the beam waist coo is 0.1 mm. The horizontal lines
`represent (in increasing value) the l/e2 of the maximum irra-
`diance, the l/e of the maximum amplitude, and the 0.5 of the
`maximum irradiance and amplitude.
`
`-(),4
`
`Irradiance
`
`.5cl
`
`mm)l
`
`
`
`O
`
`xcoordinate
`
`-100
`
`50
`I 0
`-50
`z coordlnate (mm)
`
`100
`
`Fig. 1 Map of the irradiance distribution of a Gaussian beam.
`The bright spot corresponds with the beam waist. The hyperbolic
`white lines represent the evolution of the Gaussian width when
`the beam propagates through the beam waist position. The
`transversal Gaussian distribution of irradiance is preserved as the
`beam propagates along the z axis.
`
`that the irradiance shows a maximum around a given
`point where the transversal size of the beam is minimum.
`This position belongs to a plane that is named as the beam
`waist plane. It represents a pseudo-focalization point with
`very interesting properties. Once this first graphical
`approach has been made, it lets us define and explain in
`more detail the terms involved in Eq. 1.
`
`Width
`
`This is probably one of the most interesting parameters
`from the designer point of ViCW.[14’35’36I The popular
`approach of a laser beam as a “laser ray” has to be
`reviewed after looking at the transversal dependence of
`the amplitude. The ray becomes a beam and the width
`parameter characterizes this transversal extent. Practic-
`ally, the question is to know how wide is the beam when it
`propagates through a given optical system. The exponen-
`tial term of Eq. 1 shows a real and an imaginary part. The
`imaginary part will be related with the phase of the beam,
`and the real part will be connected with the transversal
`distribution of irradiance of the beam. Extracting this real
`portion, the following dependences of the amplitude and
`the irradiance are:
`
`2
`
`
`m, z) o< exp [ 60%]
`
`I(x,z) : |T(x,z)lzo<exp[ )]
`
`
`
`(2)
`
`(3)
`
`where the function co(z) describes the evolution along the
`propagation direction of the points having a decrease of
`
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`

`Laser and Gaussian Beam Propagation and Transformation
`
`1001
`
`Integrated irradiance
`
`Radius of Curvature
`
`\l0
`
`CD0
`
`
`
`
`
`IntegratedIrradiance(%)
`
`_\ O
`
`
`O0
`
`0.2
`
`0.4
`
`0.6
`
`0.8
`
`1
`
`1.2
`
`1.4
`
`1.6
`
`1.8
`
`2
`
`transversal coordinate (in units of to)
`
`Integrated irradiance for a 2-D Gaussian beam (black
`Fig. 3
`line) and for a rotationally symmetric 3-D Gaussian beam. The
`horizontal axis represents the width of a l-D slit (for the 2-D
`beam) and the diameter of a circular aperture (for the 3-D beam)
`that is located in front of the beam. The center of the beam
`
`coincides with the center of the aperture. The size of the aperture
`is sealed in terms of the Gaussian width of the beam at the plane
`of the aperture.
`
`amount of energy located at the beam waist plane needs to
`be distributed in each plane. As a consequence, the ma-
`ximum of irradiance at each z plane drops from the beam
`waist very quickly, as it is expressed in Fig. l.
`
`Eq. 4 has a very interesting behavior when z tends to 00
`(or 7 oo ). This width dependence shows an oblique
`asymptote having a slope of:
`
`J.
`60 g tan 60 : TC—COO
`
`(5)
`
`where we have used the paraxial approach. This pa-
`rameter is named divergence of the Gaussian beam. It
`describes the spreading of the beam when propagating
`towards infinity. From the previous equation, we see that
`the divergence and the width are reciprocal parameters.
`This means that larger values of the width mean lower
`values of the divergence, and vice versa. This relation has
`even deeper foundations, which we will show when the
`characterization of generalized beams is made in terms of
`the parameters already defined for the Gaussian beam
`case. Using this relation, we can conclude that a good
`collimation (very low value of the divergence) will be
`obtained when the beam is wide. On the contrary, a high
`focused beam will be obtained by allowing a large di-
`vergence angle.
`
`Divergence
`
` Radius of curvature
`
`Following the analysis of the amplitude distribution of a
`Gaussian beam, we now focus on the imaginary part of
`the exponential function that depends on x:
`
` kx2
`“pl ‘2R<z>
`
`]
`
`(6)
`
`where k is the wave number and R(z) is a function of z.
`The previous dependence is quadratic with x. It is the
`paraxial approach of a spherical wavefront having a ra-
`dius R(z). Therefore this function is known as the radius
`of curvature of the wavefront of the Gaussian beam. Its
`
`dependence with z is as follows:
`
`R(Z) : Z 1+
`
`7-5603
`
`2
`
`(7)
`
`When z tends to infinity, it shows a linear variation
`with z that
`is typical of a spherical wavefront
`that
`originated at z=0;
`i.e., coming from a point source.
`However, the radius of curvature is infinity at the beam
`waist position. This means that at the beam waist, the
`wavefront is plane. A detailed description of the previous
`equation is shown in Fig. 4. The absolute value of the
`radius of curvature is larger (flatter wavefront) than the
`corresponding point source located at
`the beam waist
`along the whole propagation.
`
`
`
`
`
` 200 -
`
`r
`
`150
`
`100-
`
`/
`
`
`
`I
`l
`l
`—5-4-3-§-‘1012345
`
`z coordinate (in units of 2R)
`
`Fig. 4 Radius of curvature of a Gaussian beam around the
`beam waist position. The beam reaches a minimum of the
`absolute value of the radius at a distance of +zR and i ZR from
`the beam waist. At
`the beam waist position,
`the radius of
`curvature is infinity, meaning that the wavefront is plane at the
`beam waist. The dashed line represents the radius of curvature of
`a spherical wavefront produced by a point source located at the
`point of maximum irradiance of the beam waist.
`
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`

`1002
`
`Rayleigh Range
`
`The width, the local divergence, and the radius of cur-
`vature contain a special dependence with 600 and 2.. This
`dependence can be written in the form of length that is
`defined as:
`
`2
`T5600
`
`ZR :
`
`(8)
`
`This parameter is known as the Rayleigh range of the
`Gaussian beam. Its meaning is related to the behavior of
`the beam along the propagating distance. It is possible to
`say that the beam waist dimension along z is zR. The width
`at z=zR is x/E larger than in the waist. The radius of
`curvature shows its minimum value (the largest curvature)
`at z =zR. From the previous dependence, we see that the
`axial size of the waist
`is larger
`(with quadratic de-
`pendence) as the width is larger. Joining this dependence
`and the relation between the width and the divergence, we
`find that as the collimation becomes better,
`then the
`region of collimation becomes even larger because the
`axial extension of the beam waist is longer.
`
`Guoy Phase Shift
`
`There exists another phase term in Eq. 1. This term is
`q5(z). This is known as the Guoy phase shift. It describes a
`TE phase shift when the wavefront crosses the beam waist
`region (see pp. 682—685 of Ref. [7]). Its dependence is:
`
`q5(z) : tanil (i)
`
`ZR
`
`(9)
`
`This factor should be taken into account any time the
`exact knowledge of the wavefront
`is needed for the
`involved applications.
`
`3-D GAUSSIAN BEAMS
`
`In “Gaussian Beams,” we have described a few pa-
`rameters characterizing the propagation of a 2-D beam.
`Actually,
`these parameters can be extended to a rota-
`tionally symmetric beam assuming that the behavior is the
`same for any meridional plane containing the axis of
`propagation. Indeed, we did an easy calculation of the
`encircled energy for a circular beam by using these
`symmetry considerations (see Fig. 3). However, this is not
`the general case for a Gaussian beam.[3941' For example,
`when a beam is transformed by a cylindrical lens,
`the
`waist on the plane along the focal power changes, and the
`other remains the same. If a toric, or astigmatic, lens is
`used, then two perpendicular directions can be defined.
`Each one would introduce a change in the beam that will
`
`Laser and Gaussian Beam Propagation and Transformation
`
`be different from the other. Even more, for some laser
`sources,
`the geometry of the laser cavity produces an
`asymmetry that is transferred to a nonrotationally sym-
`metric beam propagation. This is the case of edge-emit-
`ting semiconductor lasers, where the beam can be mo-
`deled by having two 2-D Gaussian beam propagations.[42'
`In all
`these cases, we can define two coordinate sys-
`tems: the beam reference system linked to the beam sym-
`metry and propagation properties, and the laboratory refe-
`rence system.
`The evolution of the simplest case of astigmatic
`Gaussian beams can be decoupled into two independent
`Gaussian evolutions along two orthogonal planes. The
`beams allowing this decoupling are named as orthogonal
`astigmatic Gaussian beams. Typically, these beams need
`some other parameters to characterize the astigmatism of
`the laser, besides the parameters describing the Gaussian
`evolution along the reference planes of the beam refe-
`rence system. When the beam reaches the waist in the
`same plane for the two orthogonal planes defined within
`the beam reference system, we only need to provide the
`ellipticity parameter of the irradiance pattern at a given
`plane. In some other cases, both orthogonal planes de-
`scribing a Gaussian evolution do not produce the waist at
`the same plane. In this case, another parameter describing
`this translation should be provided. This parameter is
`sometimes named as longitudinal astigmatism. Although
`the beam propagation is located in two orthogonal planes,
`it could be possible that these planes do not coincide with
`the orthogonal planes of the laboratory reference system.
`An angle should also be given to describe the rotation of
`the beam reference system with respect to the laboratory
`reference system.
`When the planes of symmetry of the beam do not
`coincide with the planes of symmetry of the optical
`systems that the beam crosses,
`it is not possible to de-
`couple the behavior of the resulting beam into two
`orthogonal planes. This lack of symmetry provides a new
`variety of situations that are usually named as general
`astigmatism case.[39'
`
`Orthogonal Astigmatic Beams
`
`In Fig. 5, we represent a 3-D Gaussian beam having the
`beam waist along the direction of x and the direction of y
`in the same z plane. In this case, the shape of the beam
`will be elliptic at every transversal plane along the pro-
`pagation, except for two planes along the propagation
`that will show a circular beam pattern. The character-
`istic parameters are the Gaussian width along the x and
`y directions.
`together the evolution of the
`In Fig. 6a, we plot
`Gaussian widths along the two orthogonal planes where
`the beam is decoupled. The intersection of those planes is
`
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`

`Laser and Gaussian Beam Propagation and Transformation
`
`1003
`
`beam were rotated with respect to the laboratory refe-
`rence system, an angle describing such a rotation would
`also be necessary.
`
`Nonorthogonal Astigmatic Beams.
`General Astigmatic Beams
`
`The variety of situations in the nonorthogonal case is
`richer than in the orthogonal one and provides a lot of
`information about the beam. General astigmatic Gaussian
`beams were described by Arnaud and Kogelnikmgl by
`
`(a)
`
`2
`
`l
`
`Gaussian Width Evolution
`.
`l
`.
`l
`.
`l
`.
`
`1.8-
`
`-
`
`A E
`
`1.5
`Ev m-
`<1).4
`(U l2?
`
`
`
`
`
` pDD xlandycoordinate(mm)
`
`
`
`—
`
`7
`
`a
`
`,
`
`—
`
`—
`-
`
`EE
`
`C m
`
`><
`
`1
`8U 0.87
`>‘
`'5 DE-
`
`011-
`DZ—
`
`rlUDIJIJ
`
`7800
`
`VEDIEI
`
`110D
`2EIIIJ
`f]
`VZEIIEI
`VAD‘IJ
`2 coordinate (mm)
`
`500
`
`800 mill]
`
`
` (b) Radius of curvature Evolution
`800 7
`
`m:1E
`
`
`BUD
`
`Eldfl
`
`110D
`20D
`I]
`EDT]
`ADD
`z coordinate (mm)
`
`50D
`
`BD‘D
`
`Fig. 6 Evolution of the Gaussian width (a) and the radius of
`curvature (b) for an orthogonal astigmatic Gaussian beam having
`the following parameters: coo,C = 0.1 mm, coo}. = 0.25 mm, A = 633
`nm, and 360 mm of distance between both beam waists. The
`black curve corresponds with the y direction and the gray curve
`is for the x direction. The intersection in (a) represents the
`position of the points having circular patterns of irradiance. The
`intersection in (b) represents those planes showing a spherical
`wavefront. The plots shows how both conditions cannot be
`fulfilled simultaneously.
`
`
`
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`Gaussian width evolution
`
`-D5 ycoordinate
`(mm)
`
` JDU
`
`x coordinate (mni)
`
`_
`-2UD
`z coordinate (mm)
`
`Fig. 5 A 3-D representation of the evolution of the Gaussian
`width for an orthogonal astigmatic Gaussian beam. The Gaus-
`sian beam waist coincides at the z = 0 plane and the beam refe-
`rence directions coincide with the laboratory reference direc-
`tions. The sizes of the waists are coo,C = 0.07 mm, coo}. : 0.2 mm,
`and 22632.8 nm.
`In the center of the beam, we have re-
`presented the volume of space defined by the surface where 1/62
`of the maximum irradiance is reached. It can be observed that
`
`the ellipticity of the irradiance pattern changes along the pro-
`pagation and the larger semiaXis changes its direction: in the
`beam waist plane, the large semiaXis is along the y direction and
`at 300 mm, it has already changed toward the x direction.
`
`the axis of propagation of the beam. The evolution of the
`beam along these two planes is described independently.
`There are two different beam waists (one for each plane)
`with different sizes. To properly describe the whole 3-D
`beam, it is necessary to provide the location and the size
`of these two beam waists. The locations of the inter-
`
`section points for the Gaussian widths correspond with the
`planes showing a circular irradiance pattern. An interest-
`ing property of this type of beam is that the ellipticity of
`the irradiance profile changes every time a circular irra-
`diance pattern is reached along the propagation, swapping
`the directions of the long and the short semiaxes.
`In Fig. 6b, we represent the evolution of the two radii
`of curvature within the two orthogonal reference planes.
`The analytical description of the wavefront of the beam is
`given by a 3-D paraboloid having two planes of sym-
`metry. The intersection of the two evolutions of the radii
`of curvature describes the location of a spherical
`wavefront. It is important to note that a circular irradiance
`pattern does not mean a spherical wavefront for this type
`of beams. To completely describe this beam, we need the
`values of the beam waists along the beam reference di-
`rections, and the distance between these two beam waists
`along the propagation direction. This parameter is some-
`times known as longitudinal astigmatism. If the whole
`
`

`

`1004
`
`Laser and Gaussian Beam Propagation and Transformation
`
`adding a complex nature to the rotation angle that relates
`the intrinsic beam axis (beam reference system) with the
`extrinsic (laboratory references system) coordinate sys-
`tem. One of the most interesting properties of these beams
`is that the elliptic irradiance pattern rotates along the
`propagation axis. To properly characterize these Gaussian
`beams, some more parameters are necessary to provide a
`complete description of this new behavior. The most
`relevant is the angle of rotation between the beam re-
`ference system and the laboratory reference system,
`which now should be provided with real and imaginary
`parts. In Fig. 7, we show the evolution of the Gaussian
`width for a beam showing a nonorthogonal astigmatic
`evolution. An important difference with respect to Fig. 5,
`besides the rotation of axis,
`is that the nonorthogonal
`astigmatic beams shows a twist of the l/e2 envelope that
`makes possible the rotation of the elliptical irradiance
`pattern. It should be interesting to note that in the case
`of the orthogonal astigmatism,
`the elliptic irradiance
`pattern does not change the orientation of their semi-
`axes; it only swaps their role. However, in the general
`astigmatic case, or nonorthogonal astigmatism, the rota-
`tion is smooth and depends on the imaginary part of the
`rotation angle.
`A simple way of obtaining these types of beams is by
`using a pair of cylindrical or toric lenses with their cha-
`racteristic axes rotated by an angle different from zero or
`90°. Although the input beam is circular,
`the resulting
`beam will exhibit a nonorthogonal astigmatic character.
`The reason for this behavior is related to the loss of
`
`symmetry between the input and the output beams along
`each one of the lenses.
`
`Gaussian width evolution
`
`
`ABCD LAW FOR GAUSSIAN BEAMS
`
`ABCD Matrix and ABCD Law
`
`Matrix optics has been well established a long time
`ago.[16’18‘43‘44' Within the paraxial approach, it provides
`a modular transformation describing the effect of an
`optical system as the cascaded operation of its compo-
`nents. Then each simple optical system is given by its
`matrix representation.
`Before presenting the results of the application of the
`matrix optics to the Gaussian beam transformation, we
`need to analyse the basis of this approach (e.g., see
`Chapter 15 of Ref. [7]). In paraxial optics, the light is
`presented as ray trajectories that are described, at a given
`meridional plane, by its height and its angle with respect
`to the optical axis of the system. These two parameters
`can be arranged as a column vector. The simplest mathe-
`matical object relating two vectors (besides a multiplica-
`tion by a scalar quantity) is a matrix. In this case,
`the
`matrix is a 2 X 2 matrix that
`is usually called the
`ABCD matrix because its elements are labeled as A, B,
`C, and D. The relation can be written as:
`
`(at; W
`
`where the column vector with subindex 1 stands for the
`
`input ray, and the subindex 2 stands for the output ray.
`An interesting result of this previous equation is
`obtained when a new magnitude is defined as the ratio
`between height and angle. From Fig. 8,
`this parameter
`coincides with the distance between the ray—optical axis
`intersection and the position of reference for the des-
`cription of the ray. This distance is interpreted as the
`radius of curvature of a wavefront departing from that
`intersection point and arriving to the plane of interest
`where the column vector is described. When this radius of
`
`curvature is obtained by using the matrix relations, the
`following result is found:
`
`AR] "B
`R :—
`
`2
`CRIWD
`
`(11)
`
`This expression is known as the ABCD law for the
`radius of curvature. It relates the input and output radii
`of curvature for an optical system described by its
`ABCD matrix.
`
`The Complex Radius of Curvature, q
`
`For a Gaussian beam, it is possible to define a radius of
`curvature describing both the curvature of the wavefront
`
`
`
`!:‘!:‘4:61
`C‘ M
`
`
`
` C 4:.ycoordinate(mm)
`
`D M
`
`
`{,5
`_1
`-2uu
`n
`El 5'
`x coordinate (mm)
`‘
`
`Fig. 7 Evolution of the Gaussian width for the case of a
`nonorthogonal astigmatic Gaussian beam. The parameters of this
`beam are: 2:63 nm c00X=0.07 mm, c003,:02 mm, and the
`angle of rotation has a complex value of 51225071150. The
`parameters, except for the angle, are the same as those of the
`beam plotted in Fig. 5. However, in this case, the beam shows a
`twist due to the nonorthogonal character of its evolution.
`
`
`
`
`
`
`
`
`
`Copyright©2003byMarcelDekker.Inc.Allrightsreserved.
`
`fl
`
`MARCEL DEKKER, INC.
`270 Madison Avenue, New York, New York 10016
`
`

`

`
`
`
`
`Copyright©2003byMarcelDekker,Inc.Allrightsreserved.
`
`
`
`
`
`Laser and Gaussian Beam Propagation and Transformation
`
`1005
`
`Invariant Parameter
`
`
`
`When a Gaussian beam propagates along an ABCD op-
`tical system,
`its complex radius of curvature changes
`according to the ABCD law. The new parameters of the
`beam are obtained from the value of the new complex
`radius of curvature. However,
`there exists an invariant
`parameter
`that
`remains the same throughout ABCD
`optical systems. This invariant parameter is defined as:
`
`Input Plane Optical Output Plane
`System
`
`2.
`60600 : —
`TC
`
`(16)
`
`Fig. 8 The optical system is represented by the ABCD ma-
`trix. The input and the output rays are characterized by their
`height and their slope with respect
`to the optical axis. The
`radius of curvature is related to the distance between the
`
`intersection of the ray with the optical axis and the input or
`the output planes.
`
`and the transversal size of the beam. The nature of this
`
`radius of curvature is complex. It is given by:[9‘34'
`
`_ : _ a l
`l
`l
`.
`W)
`R(z)
`
`
`2.
`7m>(z)2
`
`(12)
`
`If the definition and the dependences of R(z) and co(z)
`are used in this last equation, it is also possible to find
`another alternative expression for the complex radius of
`curvature as:
`
`QQ):Z+RR
`
`on
`
`the phase
`By using this complex radius of curvature,
`dependence of the beam (without
`taking into account
`the Guoy phase shift) and its transversal variation is
`written as:
`
`Its meaning has been already described in “Diver-
`gence.” It will be used again when the quality parameter
`is defined for arbitrary laser beams.
`
`Tensorial ABCD Law
`
`The previous derivation of the ABCD law has been made
`for a beam along one meridional plane containing the
`optical axis of the system that coincides with the axis of
`propagation. In the general case, the optical system or the
`Gaussian beam cannot be considered as rotationally
`symmetric. Then the beam and the system need to be
`described in a 3-D frame. This is done by replacing each
`one of the elements of the ABCD matrix by a 2 X 2 matrix
`containing the characteristics of the optical system along
`two orthogonal directions in a transversal plane. In the
`general case,
`these 2 X2 boxes may have nondiagonal
`elements that can be diagonalized after a given rotation.
`This rotation angle can be different
`in diagonalizing
`different boxes when nonorthogonal beams are treated.
`Then the ABCD matrix becomes an ABCD tensor in the
`form of:
`
`exp [i
`
`
`ka
`
`J
`
`(14)
`
`P :
`
`Once this complex radius of curvature is defined, the
`ABCD law can be proposed and be applied for the
`calculation of the change of the parameters of the beam.
`This is the so-called ABCD law for Gaussian beams (see
`Chapter 3 of Ref. [18]):
`
`AC),
`BM
`,0,
`A”
`A)')' B)'X BXX
`A)“
`CM ny DH 1),).
`C)?“
`C)')' D)'X Dy)’
`
`(17)
`
`where, by symmetry considerations, AW=Ayx and is the
`same for the B, C, and D, boxes.
`For a Gaussian beam in the 3-D case, we will need to
`expand the definition of the complex radius of curvature
`to the tensorial domain.[45 I The result is as follows:
`
`
`c0526
`sin2 6
`+
`
`(Ix
`
`qy
`
`Q" :
`
`I
`
`,
`
`— 5m 26 —i—
`2
`(Ix
`Qy
`
`(I
`
`1)
`
`I
`I
`,
`I
`— 5m 26 —i—
`2
`(Ix
`qy
`
`+
`
`cos2 6
`
`‘1)
`
`sin2 6
`
`(IX
`
`(18)
`
`l i
`
`1
`C771)—
`q]
`1
`‘12 A,,B_
`‘11
`
`
`
`i
`
`(12
`
`”B
`C 77D or
`(II
`
`
`
`(15)
`
`The results of the application of the ABCD law can be
`written in terms of the complex radius of curvature and
`the Gaussian width by properly taking the real and ima-
`ginary parts of the resulting complex radius of curvature.
`
`MARCEL DEKKER, INC.
`270 Madison Avenue, New York, New York 10016
`
`fl
`
`

`

`1006
`
`Laser and Gaussian Beam Propagation and Transformation
`
`where 6 is the angle between the laboratory reference
`coordinate system and the beam reference system. If the
`beam is not orthogonal,
`then the angle becomes a
`complex angle and the expression remains valid. By
`using this complex curvature tensor, the tensorial ABCD
`law (see Section 7.3 of Ref. [18]) can be written as:
`
`Their characteristic parameters can be written in terms
`of the order of the multimode beam.[48’53' When the beam
`
`is a monomode of higher-than-zero order, its width can be
`given by the following equation:
`
`a)” : 600V 2n + l
`
`(20)
`
`(21)
`
`
`Q? : f
`
`(19)
`
`for the Hermite—Gauss beam of n order, and
`
`where A, B, C, and D with bars are the 2 X 2 boxes of the
`ABCD tensor.
`
`60pm : com/2p + m +1
`
`ARBITRARY LASER BEAMS
`
`As we have seen in the previous sections, Gaussian beams
`behave in a very easy way. Its irradiance profile and its
`evolution are known and their characterization can be
`
`made with a few parameters. Unfortunately, there are a lot
`of applications and laser sources that produce laser beams
`with irradiance patterns different
`from those of the
`Gaussian beam case. The simplest cases of these non-
`Gaussian beams are the multimode laser beams. They
`have an analytical expression that can be used to know
`and to predict the irradiance at any point of the space for
`these types of beams. Moreover,
`the most common
`multimode laser beams contain a Gaussian function in the
`
`core of their analytical expression. However, some other
`more generalized types of irradiance distribution do not
`respond to simple analytical solution. In those cases, and
`even for multimode Gaussian beams, we can still be
`interested in knowing the transversal extension of the
`beam,
`its divergence in the far field, and its departure
`from the Gaussian beam case that is commonly taken as a
`desirable reference. Then the parameterization of arbit-
`rary laser beams becomes an interesting topic for
`designing procedures because the figures obtained in this
`characterization can be of use for adjusting the optical
`parameters of the systems using them.[46‘47'
`
`Multimode Laser Beams
`
`The simplest cases of these types of arbitrary beams are
`those corresponding to the multimode expansion of laser
`beams (see Chapters l6,
`l7, 19, 20, and 21 of Refs.
`[7,34]). These multimode expansions are well determined
`by their analytical expressions showing a predictable
`behavior. Besides, the shape of the irradiance distribution
`along the propagation distance remains the same. There
`exist two main families of multimode beams: Laguerre—
`Gaussian beams, and Hermite—Gaussian beams. They
`appear as solutions of higher order of the conditions of
`resonance of the laser cavity.
`
`for the Laguerre—Gauss beam of p radial and m azimuthal
`orders, where 600 is the width of the corresponding zero
`order or pure Gaussian beam. For an actual multimode
`beam, the values of the width, the divergence, and the
`radius of curvature depend on the exact combination of
`modes, and need to be calculated by using the concepts
`defined in “Generalized Laser Beams.” In Fig. 9, we
`have plotted three Hermite—Gaussian modes