Off-axis mirror based optical system design forcircularization, collimation, and expansion of elliptical
laser beams
Mert Serkan,1,* Hulya Kirkici,1 and Hakan Cetinkaya2
1Department of Electrical and Computer Engineering, Auburn University, Auburn, Alabama 36849, USA2Department of Geodesy and Photogrammetry Engineering, Istanbul Technical University, Istanbul 34469, Turkey
Most lasers used in LIDAR systems are bulky andlarge-scale such as CO2 and Nd-YAG. Even thoughthese LIDARs are efficient tools for applications suchas environmental and atmospheric remote sensing,they are not easily transportable for commercial pur-poses [1]. Semiconductor lasers may on the otherhand have advantages over other lasers as the radi-ation source of LIDAR systems due to low cost, com-pactness, electronic compatibility, broad range ofwavelengths, and high PRFs (pulse repetition fre-quencies) [2]. However, they possess astigmatismand have elliptical beam profiles. Therefore, theselimitations must be overcome before they can be in-tegrated in LIDAR systems, where a highly colli-mated circular beam is necessary.
In general, edge emitting semiconductor lasers havedifferent origins and angles of divergence in the twotransverse directions [3]. Figures 1(a)–1(c) display thebeam in two transverse directions (x—perpendicular,y—parallel) as a function of propagation distance (z—optical axis). This property results in the inherent
astigmatism and elliptical beam shape. Astigmatismmay need to be addressed carefully, specifically for theoptical systems focusing the laser beam to a small spot[4–6]. On the other hand, for applications where beamexpansion is needed, astigmatism in �m levels can beneglected for collimated laser beams [7]. As seen in Fig.2, beam having same radii in both transverse direc-tions at a specific target plane does not mean that thebeam will further propagate with a circular beam pro-file. Thus, the divergence angles should be made equalby the intended beam shaper, ensuring that the beamradii in the two transverse directions are the same atan arbitrary target plane after propagating throughthe optical system.
One of the most efficient methods to transform anelliptical beam to a circular beam is to use cylindricallenses having different refractive powers in eachtransverse axis. In [8], a cruciform cylindrical lens,which is a combination of two separate cylindricallenses with their symmetry axes perpendicular toone another, is developed for this purpose. Anotherwidely used method is to use an anamorphic prismpair [9]. In this case, the beam entering the prism iscollimated and the prism pair serves to change thebeamwidth in one transverse direction to make thebeamwidth equal in both transverse directions, thus,
circularizing the beam. In another study, an asym-metrical glass prism is used as a beam-shapingscheme for a laser diode stack [10]. The beam shapingstudies in the literature are not specific to LIDAR,however most of them seem to employ lenses orprisms to reshape the beam.
In this paper, two design methods are presentedto reshape an elliptical semiconductor laser outputbeam in three aspects: circularization, collimation,and expansion. The systems are designed to establish
an output beam that could be used in a semiconductorbased LIDAR system, where a circular beam is de-sired. Furthermore, the beam is collimated and ex-panded to assure reasonable target resolution. Thesethree aspects can be managed simultaneously or sep-arately by several individual optical components.Both design methods consist of three individual mir-rors to accomplish the beam reshaping in this paper.In both design methods, the beam is circularized andexpanded simultaneously at first, and then colli-mated later. Circularization and expansion steps areaccomplished with two cylindrical hyperbolic off-axismirrors for the first design method whereas two cy-lindrical parabolic off-axis mirrors are used for thesecond design method. A rotationally symmetric off-axis parabolic mirror is employed for collimation stepin both design methods.
In the first design method, cylindrical mirror pa-rameters are attuned to expand the beam further inthe parallel transverse direction to have the samedivergence angle as in the perpendicular transversedirection. In the second method, they are attuned toreduce the beam divergence angle in the perpendic-ular transverse direction to have the same value as inthe parallel transverse direction. One should notethat the divergence of a beam cannot be eliminatedentirely in an optical system; thus, it is not possible tohave a perfectly parallel beam. In this paper, theterm “collimated beam” refers to a beam with a sig-nificantly reduced divergence angle, approaching thediffraction limit.
Most imaging systems are designed with on-axiscomponents to prevent off-axis aberrations. In thispaper, however, we use an off-axis design approachand all the analytical equations are derived accord-ingly rather than using solely paraxial ray optics [11].We believe that the aberrations introduced by anoff-axis system designed for laser beam circulariza-tion and collimation can be tolerated to a degree aslong as the irradiance distribution of the originalbeam and the transformed beam are approximatelythe same.
Fig. 1. (a) Free-space radiation pattern of a semiconductor laser,showing a non-cylindrical symmetric beam profile. (b) Illustrationof propagation of a semiconductor laser beam. (c) Semiconductorlaser beam in two transverse directions (x, perpendicular; y, par-allel) as a function of propagation distance (z, optical axis).
Fig. 2. Beam profile plot as a function of propagation direction. Itis plotted with a MATLAB routine demonstrating a laser beamwith 46° divergence angle in the perpendicular transverse direc-tion and 12° divergence angle in the parallel transverse direction,assuming both the perpendicular and transverse angles originateat the same location at z � 0.
There is no single beam shaping method that can beemployed for a variety of applications [12]. For eachapplication, an appropriate beam shaping methodcan be developed and used. Optical design of beamshapers can be achieved using either physical or geo-metrical optics. It is customary to design an opticalsystem with geometrical ray optics instead of Gauss-ian beam analysis when ray optics is a good approx-imation to the Gaussian beam propagation. Aspresented below, we use geometrical ray optics todesign the optical systems, and use Physical OpticsPropagation tool of ZEMAX to verify the beam prop-agation. A detailed Gaussian beam analysis can befound in [13].
The amount of divergence or convergence of a laserbeam is measured by the divergence angle, which isthe angle subtended by the 1�e2 asymptotes of thebeam irradiance profile [14]. These asymptotes ap-proach the real beam in the far field, which is definedwith Rayleigh range z0 for a beam propagating inz-axis. The Rayleigh range is given by
z0 ��w0
2
�, (1)
where z0 and w0 are the Rayleigh range and the beamwaist radius respectively [3]. In the far field forz �� z0, the waves approximate to two 1�e2 rays em-anating from a point source centered at the waist[15], as shown in Fig. 1(c). The rays traced over theseasymptotes are referred to as “1�e2 rays” in this pa-per. We assume that the diffraction losses are notsignificant in this study since the aim of the design isto expand a �m size beam emitted from the source(semiconductor laser) to a 10s cm range at the outputof the beam shaping optics [16].
In addition to the assumptions above, we also cal-culate a dimensionless parameter, �, which providesa quantitative measure of the diffraction effects andthe validity of the geometrical ray optics design[17,18]. The dimensionless parameter � is defined as
� �2�w0wf
d�, (2)
where w0, wf, d, and � are the incoming beam radius,the beam radius at the target plane, the distancebetween the optical component and the target plane,and the wavelength of the laser beam respectively.For simple output geometries, such as a circle, when� � 4 the beam shaping is not possible, when 4� � � 32 the diffraction effects should be included inthe calculations, and when � � 32 the diffractioneffects are negligible and the geometrical optics is avalid approximation [19]. In general, for � �� 1 thediffraction effects are negligible [20]. Based on thiscriteria, � should be calculated for each optical sur-face to validate the geometrical optics approach. Inthis paper, we calculate � values for all the surfaces.Based on these values, geometrical optics approach isused in the beam shaping optics design calculations.
B. Design Approach
Two design examples for the first design method andone design example for the second design method arepresented in this paper. Both design methods consistof three aspheric mirrors as shown schematically inFig. 3. The symbols i, j, and k �� 1, 2, 3, 4, 5� denotethe rays from the laser source to the first mirror, fromfirst mirror to second mirror, and from second mirrorto third mirror respectively. A detailed analysis ofaspheric surfaces and the conic sections are given in[21] and [22] respectively. Also, in order to reduce
Fig. 3. Illustration of the general design approach.
Fig. 4. x-z plane illustration of the first design method.
Fig. 5. y-z plane illustration of the first design method.
aperture diffraction effects while minimizing the mir-ror sizes, only the portions of the mirrors where ap-proximately 98% of the beam irradiance is reflectedare assumed to be the surface diameter. A detaileddiscussion on aperture diffraction effects can befound in [23].
Laser source is placed with an angle, called thefield angle, on the optical axis. Field angle is definedas the angle between the center of the beam and theoptical axis (z-axis). This prevents the on-axis obscu-ration, but keeping the field angle as small as possi-ble limits the off-axis aberrations.
C. Design Method I
The first two mirrors are cylindrical hyperbolic mir-rors. The first mirror is convex and the second one isconcave, specifically used to circularize and expandthe elliptical laser beam. Each is defined by its ownfocal length in the parallel transverse direction whilethey are optically flat in the perpendicular transversedirection. Figures 4 and 5 display the position of themirrors with respect to each other on the x-z and y-zplanes.
The hyperbola function is illustrated in its localcoordinate system in Fig. 6. In order to circularize anelliptical laser beam, two conditions must be satis-fied: (1) the two transverse divergence angles of thebeam must be equal after the reflection from thesecond mirror and (2) the source points for both trans-
verse directions must coincide at the same point oforigin after reflecting from the second mirror, as seenin Figs. 4 and 5.
Referring to Fig. 7, the second condition results inEq. (3):
�A, E� � �B, D� � �D, E�. (3)
The path lengths [A, B], [B, D], and [D, E] in thisfigure and in Eq. (3) are the rays representing thecenter of the beam incident on the center of eachreflecting surface. These rays are referred to as the“chief rays” throughout this paper. We also know,from the primary property of hyperbolas, that thedifference between the distances from an arbitrarypoint on the hyperbola curves to the two foci alwaysequals 2a, as seen in Fig. 6. Therefore, referring toFigs. 6 and 7, the following equation for the firsthyperbolic mirror is derived:
�B, D� � �C, D� � 2a. (4)
The mirror parameters for the second hyperbolahave the same meaning except they are named d, e,and f respectively. Also, the equations derived aboveare the same for the second hyperbolic mirror and aregiven as
�A, E� � �C, E� � 2d. (5)
If Eqs. (4) and (5) are substituted in Eq. (3), weobtain
a � d. (6)
Fig. 6. Illustration of the hyperbola function.
Fig. 7. Optical path of the “chief ray” of the beam.
The third mirror is chosen as a rotational symmet-ric concave parabola to collimate the diverging laserbeam. The parabola function is characterized by onlyone parameter, g, and is shown in Fig. 8 in its localcoordinate system. The rays in the parallel and per-pendicular directions have a common source point atthe virtual focal point of the second mirror, as seen inFig. 3. This source point is also the focal point of therotationally symmetric parabola serving to collimatethe laser beam in both transverse directions. In ad-dition, the virtual focal point of the first mirror mustbe chosen to coincide with the real focal point of thesecond mirror for the intended reflection of the beamat the mirrors.
Table 1 shows the general hyperbola and parabolaequations [Eqs. (7)–(9)], the distance between the la-ser source and the first mirror vertex [Eq. (7)], andthe distances between the vertexes of each mirrorrespectively [Eqs. (8) and (9)]. As stated earlier, thebeam divergence angle in the parallel transverse di-rection (y-axis) is expanded and kept constant in theperpendicular direction. Therefore, the mirror pa-rameters are calculated and presented only in the y-zplane. Before any ray tracing calculations, the localcoordinates are transformed to a global coordinatesystem where the source (the laser output beam) islocated at (0, 0, 0) and is chosen as the reference pointof the global coordinate system. Using these equa-tions and coordinate transformation, mirror coordi-nate equations are derived and presented in Table 2[Eqs. (10)–(12)]. Table 3 shows the equations [Eqs.(13)–(15)] of the slopes of the rays reflected from eachsurface.
It should be noted that the laser output beam sizeis on the order of a few �m, and that the design
presented here is expected to expand this beam to acollimated circular beam of 10 cm after reflectionfrom the third (parabolic) mirror. With this and otherconditions in mind, the mirror sizes are chosen sothat 98% of power is reflected by the system. Thiscorresponds to the beam size equal to 1.4 times the1�e2 rays [23]. These rays will be called “1.4 rays”throughout this paper.
The system variables are the field angle, the pa-rameters a and b of the first hyperbolic mirror, theparameters d and e of the second hyperbolic mirror,and the parameter g of the parabolic mirror. Thebeam (waist) radius wf at the parabolic mirror is aconstant depending on the particular application andchosen to be 10 cm as the design constraint in thispaper. The parameter g of the parabolic mirror de-pends on the parameters of the first two hyperbolicmirrors. The parameters c and f of the hyperbolicmirrors depend on a, b, d and e of the two hyper-bolic mirrors and can be determined by the followingequations:
c � �a2 � b2, (16)
f � �d2 � e2. (17)
When all the equations are substituted, a complexequation system containing the system parameters isdeveloped. A MATLAB code, which simulates the raytracing through the system with preset values of theparallel divergence half angle ���, the perpendiculardivergence half angle ���, the field angle (�), and theexpanded beam waist radius �wf� is written. Based onthe design constraint, the beam divergence angleequals to 2� after reflection from the second mirror,where the beam is defined with the representative1�e2 rays k2 and k4, as shown in Fig. 3. This providesan analytical relationship between mk2 and mk4 equa-
Table 2. Global Coordinate Equations of the Mirrors
Coordinates (in global coordinate system) Eq.
Hyperbolic mirror-1 z1 �a2mi
2c � ab2�mi2 � 1
b2 a2mi2 � c y1 � miz1 (10)
Hyperbolic mirror-2 z2 �de2�mj
2 � 1 d2mj2f
e2 d2mj2 � 2c f y2 � mj�2c z2� (11)
Parabolic mirror z3 � 2�c �g��mk
2 � 1 1�
mk2 f� y3 � mk�z3 � 2�f c� (12)
Table 3. Equations of the Slopes of the Rays Reflected from EachSurface
Reflected Ray Slope Eq.
Hyperbolic mirror-1 mj �mi�cb2 � ab2�mi
2 � 1�
cb2 ab2�mi2 � 1 2a2mi
2c(13)
Hyperbolic mirror-2 mk �mj�fe2 de2�mj
2 � 1�
fe2 � de2�mj2 � 1 2d2mj
2f(14)
Parabolic mirror mr � 0 (15)
Table 4. Parameters of the First Example Designed with the FirstMethod
�� �� � a (cm) b (cm) d (cm) e (cm) g (cm) wf (cm)
tions, determined by substituting these particularrays in Eq. (14). Then, the first and second mirrorparameters are determined by iteration. Anotherconstraint is that wf is chosen at the beginning de-pending on the particular application (wf � 10 cm inthis paper), as stated earlier. This provides an ana-lytical relationship between y3 values of k2 and k4rays. This equation is used to derive the g parameterof the third mirror. Table 4 shows these values for thefirst example, namely 10°–30° beam.
These calculated parameters are input in the raytracing MATLAB code as a second step to check if thesystem is feasible for the 1.4 rays. A MATLAB gen-erated plot tracing 1.4 rays for the system with theparameters above can be seen in Fig. 9. In this figure,the second mirror seems to obscure a portion of thebeam reflected from the third mirror. One can in-crease the field angle to reduce the mirror obscura-tion. However, this might increase the aberrationlevel. Displacing the parabolic mirror around the vir-tual point source at the focal point of this mirror isanother method to prevent obscuration on the secondmirror. We use this method to prevent obscurationproviding an option to keep the overall system sizesmall rather than changing the system parametersthat may increase the system size.
As a second example, namely 12°–46° beam, a laserbeam profile with the divergence angles of 12° and46° is chosen and this laser beam is circularized,collimated, and expanded. This beam profile is a spe-cific profile of a commercially available laser that canbe used in a possible LIDAR system. The presentand calculated parameters for a beam propagatingthrough the same system are given in Table 5. Asseen, the circularization and collimation techniquepresented here can be used for any divergence anglepair.
D. Design Method II
The second design method consists of three off-axiscylindrical parabolic mirrors and satisfies the sametwo conditions as in the first method to be able toreshape an elliptical laser beam. The first two mir-rors in this method are parabolic in perpendiculardirection and flat in parallel direction. The third oneis a rotationally symmetric parabolic mirror with areal focal point that coincides with the virtual focalpoint of the second mirror collimating the beam inboth transverse directions as in the first method. Fig-ures 10 and 11 show the beam shaping system in theperpendicular transverse direction—optical axis (x-z)and the parallel transverse direction—optical axis(y-z) planes respectively. It is seen in Fig. 10 thatthe rays are reflected parallel from the first mirrorin the x-z plane. This also means that mj equals zero.The variables in this system are the mirror parame-ters g1, g2 and g3, the parallel and perpendicular di-vergence angles (� and �), and the field angle (�) ofthe laser.
Similar to the first design, the mathematical equa-tions of the surfaces of this design method are derivedin their local coordinate systems and then modifiedto function accurately in a global coordinate system
Fig. 9. MATLAB plot demonstrating the representative beampropagation in the y-z plane for the first design method first ex-ample.
Fig. 10. x-z plane illustration of the second design method.
Fig. 11. y-z plane illustration of the second design method.
Table 5. Parameters of the Second Example Designed with the FirstMethod
�� �� � a (cm) b (cm) d (cm) e (cm) g (cm) wf (cm)
where the laser beam radiates from a point source at(0, 0, 0). These equations are presented in Tables 6and 7 [Eqs. (18)–(24)]. The distance between the lasersource and the first mirror vertex [Eq. (19)] and thedistances between the vertexes of each mirror respec-tively [Eqs. (20) and (21)] are also presented in Table6. In this method, different from the first method, thebeam divergence angle in the perpendicular trans-verse direction (x-axis) is compressed and kept thesame in the parallel transverse direction. Therefore,the mirror parameters are calculated and presentedonly in the x-z plane. Incident and reflected ray slopesin the x-z plane are directly related to the divergenceangles of the beam because the field angle is in the y-zplane. Hence, the slope equations in Table 7 are pre-sented for the “upper” 1�e2 ray that represents thebeam.
As in the first design method, the distances be-tween the mirrors depend on the mirror parametersand the field angle. As seen in Fig. 10, to ensure thatthe beam reflected from the first mirror propagates asa collimated beam to the second mirror. The Rayleighrange of the beam must be high compared to thedistance between the two mirrors. The beam waist inthe perpendicular direction after reflecting from thefirst mirror is defined by wi, as seen in Fig. 10. wi
must be chosen such that a long Rayleigh range isensured. On the other hand, in order to design a morepractically feasible system, the mirrors should bechosen as small as possible setting an upper limit forwi. As a result, wi is chosen to be 1 cm in this design.Once the coordinate transformation is done, usingthe design constraints for wi and wf, the mirror pa-rameters g1, g2, and g3 are determined by using Eqs.(22)–(24). Based on the results of the first design, weonly used a laser beam with 46° divergence angle in
the perpendicular transverse direction and 12° diver-gence angle in the parallel transverse direction forthis design method. The preset and calculated pa-rameters of this example are given in Table 8. Ob-scuration problem does not exist for this designexample. In case of obscuration, rotation and tilt ofany mirror is also an option in this design method asin the first design method.
3. ZEMAX Implementation
A. System Implementation and Ray Tracing
Once the analytical calculations, MATLAB simula-tions, and symbolic equation solutions are completed,the system variables determined by these methods,are used in ZEMAX simulations. ZEMAX Optical De-sign Program is a software tool for optical design. Itcontains features and tools to design, optimize, andanalyze any optical system [24,25].
We trace the rays weighted with Gaussian apodiza-tion representing the laser beam, and propagate thephysical beam through the three design examplesstudied in this paper to analyze the output beam foreach of them. Using the results of the ZEMAX sim-ulations, we further optimize the system parameterspreviously calculated analytically and with MATLABcodes.
In ZEMAX lens data editor, the cylindrical surfacesare defined with “Toroidal” or “Biconic” surface typeswhereas the rotationally symmetric surfaces are de-fined with “Standard” surface types. Since none of thedesign examples studied in this paper has any phys-ical aperture stops other than the mirror sizes, the“Stop Surface” option of ZEMAX is chosen to be at thevertex of the first lens surface in the lens data editor.Radius and conic constants of the surfaces and thick-ness values between each surface are needed in thiseditor. In order to determine the values of these vari-ables, the mirror parameters a, b, d, e and g aretransformed to radius (R) and conic constant (K) foreach mirror separately using the equations below:
Table 6. General Mirror and Ray Equations
Surface Equation for All Mirrors (in local coordinates) ¡
Rays are traced in parallel and perpendicular direc-tions separately. Input values of the “Entrance PupilDiameter” and the “Apodization Factor” in the GeneralSystem Dialog Box of ZEMAX specify the geometricalrays representing the laser beam. The value of the“Entrance Pupil Diameter” is calculated separately forboth directions. The “Apodization Factor” refers to therate of decrease of the beam amplitude (or intensity) asa function of radial pupil coordinate, and is the same inboth directions. The beam amplitude is normalizedto unity at the center of the pupil in ZEMAX. Theamplitude at the other points in the entrance pupil isgiven by
A��� � eG�2, (27)
where G and are the “Apodization Factor” and thenormalized pupil coordinate at the “Stop Surface”respectively. is equal to zero at the center of the“Stop Surface” and 1 at the margin of the “Stop Sur-face”. Therefore, if G is equal to 1, then the beamamplitude falls to the 1�e value (intensity falls to the1�e2 value) at the margin of the “Stop Surface”. Aspresented in Subsections 2.A and 2.C, we adjustedthe mirror diameters to reflect the 1.4 rays corre-sponding approximately to 98% power reflectance.Therefore, the value of G is calculated as G � 1.96using Eq. (27). For accurate geometrical ray repre-sentation of the beam, the value of the “EntrancePupil Diameter” is then calculated using the equationbelow:
EPD �2wst
�1�G, (28)
where EPD and wst are the “Entrance Pupil Diameter”and the beam radius at the “Stop Surface” respectively.
For a fundamental TEM00 Gaussian beam, the spotsize at a particular propagation distance and far fielddivergence angle are determined by
w�z� � w0�1 � z2�z02, (29)
�w�z�
z �w0
z0�
�
�w0, z �� z0, (30)
where w�z�, w0, z0, � are the spot radius at a specificdistance, the beam waist radius, the Rayleigh range,and the far-field divergence angle respectively [3]. Asa result, the wst and the EPD are calculated for bothdirections separately using Eqs. (1), (28), and (29).
B. The Dimensionless � Parameter Calculations
The parameter �, defined in Subsection 2.A, is calcu-lated using ZEMAX Analysis—Calculations—RayTrace tool for each optical surface in both transversedirections. This tool simplifies the task of determin-ing the path length of the chief ray propagating be-tween each surface for such off-axis systems. Thechief ray is traced by setting the value of the “Px” and“Py” (the normalized pupil coordinates in the x and yaxes respectively) to zero in the settings menu of thistool window. The “Stop Surface” “Semi-Diameter”value must be set automatically by ZEMAX in thelens data editor allowing 1�e2 rays to pass throughthis surface. The path lengths from one surface to thenext surface are displayed in this analysis window.
Table 9. � Parameter Calculations of Each Studied Example Separately for Perpendicular and Parallel Transverse Directions
These path length values provide the parameter d inEq. (2) for each mirror. As long as the � �� 1 condi-tion is satisfied separately in both transverse direc-tions for the first and second mirrors, the geometricalrays are valid to use for design and analysis purposes.Geometrical rays reflected from the third mirror arenot used for design or analysis purposes so that we donot need to calculate the � parameter for this colli-mating mirror. Calculated values of � for the first andsecond mirrors in each design example are presentedin Table 9. These values validate the geometrical rayoptics approach for these examples.
C. Output Beam Assessment
The wavelength and the total power of the laser beamare chosen to be 870 nm and 30 W based on the com-mercially available laser mentioned earlier. The beamdiameter of the laser beam after collimation at thethird mirror is chosen to be 20 cm as stated earlier.Using Eqs. (1), (29), and (30), the beam characteristicvalues for 10°–30° and 12°–46° laser beams are calcu-lated analytically and given in Table 10 (where wf, z0f,f are the expected beam waist, the expected Rayleighrange, and the expected divergence half angle of thefinal beam circularized and collimated beam respec-tively). We also use Analysis—Physical Optics Prop-agation tool to assess the output beam using eachdesign example. Results of ZEMAX simulations foreach design example are presented in the Section 4.
4. Results and Discussion
“Physical Optics Propagation” tool results for the firstdesign method 10°–30° beam example, the first de-sign method 12°–46° beam example, and the seconddesign method 12°–46° beam example are displayed
in Figs. 12, 13, and 14, respectively. In these figures,the input beam images are determined at the lasersource exit. The output beam images are determinedafter reflection from the collimating parabolic mirror(the “exit plane”), 0.5 km away from the exit plane,and 1 km away from the exit plane respectively. Theimage sizes are 4 �m 10 �m for the input beamsand 40 cm 40 cm for the output beams for eachexample.
The Rayleigh range for each output beam can be de-termined with Physical Optics Propagation Analysis—Irradiance tool either. Using the beam waist valueof the output beam for w0 value in Eq. (30): ����w0, the far-field divergence half angle for eachdesign example is also calculated and given with theRayleigh range values in Table 11. These values aregiven separately for the two transverse directionsbecause there is no perfect circularization. Thus, thistable also summarizes the circularization and colli-mation effectiveness for each design example com-paring together with Table 10. The results presentedin these tables reveal that the beams are circularizedand collimated satisfactorily.
As stated earlier, in the first design, the beam re-flected at the second mirror propagates to the colli-mating parabolic mirror with a divergence angle thatis equal to the input beam perpendicular divergenceangle. Thus, one can expect that the level of aberra-tions introduced by the optical system to increasewith the increasing values of �, as comparison ofFigs. 12 and 13 imply. These results reveal that thefirst design method cannot be efficiently used in thefar-field region for large � values. This problem isless significant in the second design method becausethe beam reflected at the second mirror propagates to
Fig. 12. First design method 10°–30° beam example for the stated image sizes: (a) input �4 �m 10 �m�, (b) output �40 cm 40 cm�,(c) output at 0.5 km distance �40 cm 40 cm�, (d) output at 1 km distance �40 cm 40 cm�.
Fig. 13. First design method 12°–46° beam example for the stated image sizes: (a) input �4 �m 10 �m�, (b) output �40 cm 40 cm�,(c) output at 0.5 km distance �40 cm 40 cm�, (d) output at 1 km distance �40 cm 40 cm�.
the collimating parabolic mirror with a divergenceangle that is equal to the input beam parallel diver-gence angle. Given that generally � is low comparedto �, the aberrations introduced by the secondmethod are less significant in the second method.This can be seen in Fig. 14.
5. Summary
Elliptical profile of a semiconductor laser outputbeam is a drawback for many applications requiringlaser sources such as LIDAR systems. In this paper,two different design methods, based on hyperbolic�parabolic off-axis mirrors, are presented to circular-ize, collimate, and expand the elliptical beam. Firstmethod uses two cylindrical hyperbolic mirrors tocircularize and expand the beam simultaneouslywhereas the second one uses two cylindrical parabolicmirrors for the same purpose. The circularized andexpanded beam is collimated with a rotationally sym-metric parabolic mirror in both methods. The maindifference between the two methods is that the firstone further increases the beam divergence angle inthe parallel transverse direction to be equal to theangle in perpendicular direction while the second onereduces the divergence angle in the perpendiculardirection to be equal to the angle in the other direc-tion. The design approach is based on geometrical rayapproach as long as it is valid. It is quantitativelyshown that this approach is valid for both methodscalculating the dimensionless � parameter. The de-sign equations are derived analytically and presentedin the paper. A MATLAB code is written to solve theequations system for the parameters of the mirrors.ZEMAX simulation results are presented to analyzethe output beam for both design methods. The output
beam images, which are derived by physically prop-agating the beam through the optical systems, revealthat the elliptical input beam is circularized, ex-panded, and collimated effectively and the results areconsistent with analytical calculations. It is furthershown that second design method provides more sat-isfactory results.
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Fig. 14. Second design method 12°–46° beam example for the stated image sizes: (a) input �4 �m 10 �m�, (b) output �40 cm 40 cm�, (c) output at 0.5 km distance �40 cm 40 cm�, (d) output at 1 km distance �40 cm 40 cm�.
Table 11. Output Beam Results of all Studied Design Examples
10°–30°Beam First
Design Method
12°–46°Beam First
Design Method
12°–46°Beam Second
Design Method
wf� 9.94 cm 10.10 cm 10.12 cmwf� 10 cm 10.36 cm 10.34 cmz0f� 35.70 km 36.49 km 38.56 kmz0f� 36.12 km 37.46 km 36.79 km�f� 2.79 �rad 2.74 �rad 2.74 �rad�f� 2.77 �rad 2.67 �rad 2.68 �rad
