Focusing of diode laser beams: a simple mathematicalmodel
Amir Naqwi and Franz Durst
A simplified mathematical model for the far field of a monomode diode laser is employed for easy but fairly ac-curate computations of the optical field in the focal region. The present treatment is concerned with laserjunctions significantly narrower than the wavelength. The field distribution in the plane perpendicular tothe diode junction is considered in detail. The results of computations are shown to agree well with themeasurements. Hence, the computational code is valuable for the designing of optical devices, such as diode-fiber couplings and laser Doppler anemometers. The present work is not concerned with design calculationsfor specific applications. Instead, it is intended to illustrate the general features of the proposed mathemati-cal model of monomode diode laser beams.
1. IntroductionDiode lasers are rapidly advancing and are, as light
sources, already being used in optical equipment ingeneral and optical measuring instruments in particu-lar. In nearly all applications, focused diode laserbeams are employed, requiring optical design ap-proaches which should be simple to apply but alsoshould be sufficiently accurate. The present workaddresses this need. A simple mathematical model ofa focusing diode laser beam is presented and subse-quently employed to predict the light intensity andphase distributions at various beam cross sections nearthe focus. The computed intensity distributions arein a good agreement with the corresponding measure-ments.
Analytical descriptions for the far field of the emit-ted light beam of a diode laser, based on Maxwell'sequations, are available in literature. Casey and Pan-ish' have given such expressions. These representa-tions agree well with the measured intensity patterns.However, owing to their mathematical complexity,they are seldom used for actual design calculations.Instead, a simple Gaussian model of the diode laserbeam has been used extensively in the studies concern-ing coupling efficiency between a diode and an optical
The authors are with Friedrich-Alexander Universitat, Erlangan-Nurnberg, Lehrstuhl fur Stromungsmechanik, D-8520 Erlangan,Federal Republic of Germany.
fiber.2-6 This model has, however, certain drawbacks.Although a Gaussian approximation is valid for thefield distribution parallel to the junction of a mono-mode diode laser,7 it does not appropriately representthe distribution normal to the junction. As shown byDumke, a Lorentzian distribution is a better approxi-mation of the field in the normal direction. The Lor-entzian approximation is valid for a variety of commer-cially available double heterojunction (DH)Gal-,AlAs lasers, whose active regions are as narrowas 0.1 Jsm for a typical emission wavelength of -0.8 ,gm.Furthermore, the divergence of the field normal to thejunction is generally so large that a lens placed in frontof the diode is not able to collect all the emissions; thebeam is truncated. This effect is not incorporated inthe above-mentioned Gaussian model.
The present model of the diode laser beam employs aLorentzian distribution in the direction normal to thejunction. The finite numerical aperture of the lens istaken into account by appropriately truncating theoptical field. In this way, this model is more realisticthan the Gaussian model. Nonetheless, it permits aconvenient computation of the focused field by meansof Fresnel integral. The computed distributions ofphase and intensity differ significantly from that of aGaussian beam. The calculated intensity field com-pares well with the measured data of a diode laserbeam.
II. Optical Field of a Diode LaserLight beams of diode lasers, with a stripe type junc-
tion geometry, are considered in this work. Figure 1shows a semiconductor junction and illustrates thelaser beam as it emerges from the junction. As thefigure indicates, the beam is elliptic in cross section
Fig. 1. Typical laser diode junction and the diverging laser beam.
and spreads significantly in the direction of propaga-tion. Typically, the intensity falls to 50% of the maxi-mum over an angle of 10° in the plane of the junctionand 30-40° normal to it. Normal to the junction, thebeam appears to originate from the exit plane of thediode. Whereas parallel to the junction, the origin liestypically 10-50 Am inside the diode.
The above description is valid for beams from mono-mode laser diodes with emitting power up to 40 mW.Higher power diodes exhibit several transverse modesleading to more complex beam profiles. Only newlydeveloped diode lasers with a multiple quantum wellstructure have a monomode beam with low divergenceat emitting power as large as 100 mW. These diodesare relatively more expensive. Hence, there is a con-tinued interest in diode lasers with double heterostruc-ture configuration, shown in Fig. 1.
The optical arrangement under consideration is de-picted in Fig. 2. One or two lenses are placed in frontof the laser diode, so that the diverging beam is trans-formed into a converging beam. In a laser Doppleranemometry system, the two-lens arrangement is usu-ally preferred. In these systems, the laser beam is splitinto two beams after the first lens. Using mirrors orlenses, the two beams are made to cross each other inthe focal regions. 9 For the present purpose, it sufficesto consider only one beam. Another application of theoptical arrangements shown in Fig. 2 is a diode-fibercoupling. In this case, a fiber end is placed in the focalregion.
For both the optical setups shown in Fig. 2, theoptical field at cross section 0 (immediately after thelast lens) may be expressed in the following form:
uO(xOyO) = uox(xo)uoY(yo), (1)
where2= O 21 A
u0y(y0) = exp-I- _ - I (2)L _woJ 2 ROY
exp __uOX(x 0) = { 0 +2 Ron)
0
if 1x01 S I
otherwise.
Fig. 2. Optical arrangements for the focusing of a diode laser beam.
According to Eq. (1), the optical field satisfies aseparability condition, i.e., it may be expressed as aproduct of two functions each dependent only on oneof the two transverse coordinates (x0 or yo). The sepa-rability condition is generally valid for the divergingfield emanating from the laser diode.10 The separablenature of the field is preserved during transmissionthrough a thin spherical lens whose transfer function isgiven as follows:
expp[i k (x2 + y2)]
Separability is also preserved during passage through athick lens, if it is not strongly decentered with respectto the beam. In the applications of present interest(diode-fiber coupling and anemometry), at least one ofthe above requirements for preservation of separabil-ity is satisfied.
The amplitude distributions of the field along the xand y-axes are taken as those of Lorentzian and Gauss-ian functions, respectively. The Lorentzian approxi-mation for the field normal to the junction is based onthe assumptions that the active layer is infinitesimallysmall and that the obliquity factor is negligible. Thefirst assumption predicts narrower beam whereas thesecond assumption overestimates the width of the fielddistribution. The two effects seem to cancel eachother quite well for the kind of layers under consider-ation. Hence, the Lorentzian model turns out to be agood approximation.
For the present study, the Lorentzian profile is trun-cated at the radius of the first lens. The field ampli-tude is normalized so that its value at the center is 1.The parameters w0 and y are the 1/e-width of theGaussian distribution and the half width of the Lor-entzian distribution, respectively. These parametersmay be evaluated with the help of manufacturers' dataon divergence of the emitted beam parallel and per-pendicular to the diode junction. The results present-ed in Figs. 3-5 are based on the values of y deducedfrom the full angle at half maximum of intensity speci-fied by the manufacturers.
In Eqs. (2) and (3), the phase distribution of anaberration-free system is represented by two radii ofcurvature Ro, and Roy, which may be evaluated using
20 April 1990 / Vol. 29, No. 12 / APPLIED OPTICS
Berm c'sss-sec tan (mm)
Fig. 3. Intensity profile of a collimated beam: comparison be-tween measurement and the truncated Lorentzian model. (Laserdiode: Sharp LT024; Collimator: Melles Griot 06GLCO02, 8-mm
focal length, 0.5 numerical aperture.)
ComputedF +10 mm
Ir
0 200 400 600
Cross-section (microns)
-±10 mm
. 0
-6.5 mm
--8.5 mm
-12.5 mm
--14.5 mm
-16.5 mm
-- 18.5 mm
mm
I ' !0 200 400 600
Cross-section (microns)
Fig. 4. Intensity distribution normal to the junction for a focusingbeam of laser diode: comparison between measurement and com-
putation.
Cross-sectional Distance (micron)
(a)
1.1
1.0
_ 0.9
._10a*0
' 0.8
.0
0.7
0.6
0.5
0 5 10 15 20 25
Cross-sectional Distance (micron)
(b)
5 10 15 20 cCross-sectional Distance (micron)
(c)
1.1
1.0
_ 0.9
a._
*0
't 0.8
a.
0.7
0.6
0.5
0 5 10 15 20 25
Cross-sectional Distance (micron)
(d)
Fig. 5. Wavefront distortions: comparison between a truncated Lorentzian beam a Gaussian beam, and a top-hat beam. (Laser diode:Hitachi HL8314; First lens: 7-mm focal length, 0.35 numerical aperture; Second lens: 100-mm focal length.)
thin lens formula. In certain applications, such as adiode-fiber coupling using a ball lens,3 6 it is not appro-priate to ignore the aberration effects. In such cases, ahigher-order term may be included in the phase func-tion. It would generally suffice to append the follow-ing term to the phase of the field in Eq. (3):
( kx 2
t2Rx
where E may be evaluated by ray tracing methods.In Fig. 3, the truncated Lorentzian model of Eq. (3)
is compared with the measured intensity profile of acollimated beam (see the upper diagram in Fig. 2).The broadness of the actual intensity distribution isfully represented in the model. However, the fluctua-tions with higher frequency are not included. Thesefluctuations would not affect the field distribution inthe central region of a focused spot. They are likely toaffect only the outer lobes in the focal plane, which areusually not crucial.
Ill. Focusing of the Optical FieldThe optical field immediately after the focusing lens
has been described in the previous section. Furtherpropagation of this field may be adequately represent-ed by the Fresnel formula of diffraction, which pre-serves the separable nature of the field. At a distancez from the lens, the optical field is described as
u(x,y,z) = exp(-ikz)ux(x,z)uY(yz). (4)
The functions ux and uY represent field distributionsalong the respective transverse directions. A detaileddescription of these functions is given below.
A. Focusing of the Field Parallel to the JunctionAccording to the Fresnel formula, the distribution
along y-axis may be expressed as follows:
uY(yz)= - j uoy(yO) exi{-i k (Y - Yo)J dyo.
On substituting the value of uoy from Eq. (2),
UY('Z)= i ex 1 Yo 2 k YA +y -yo)2 doX e - -O - -+ dy 0 .Cy~~z = / zJep-w) 2LOY z 2
(5)
The above integral may be solved analytically in thesame manner as the corresponding integral for aGaussian beam with a circular cross section (see e.g.,Siegman'). The solution is conveniently expressed inthe following form:
u(y,z) = -4 exp{- () + i[ AY - ( )]}'
1 1 W0 4 \ o I
Ry W2 2 (kWO)2
(Ply /20~yl
(8)
(9)
where
sin'iy- 2ROyz,
cos'y - kw2(Roy + z),
and the range of is (0,27r).As expected, the solution for a 1-D Gaussian distri-
bution function is similar to that of a circular crosssection Gaussian field, with the following differences:
The amplitude changes inversely with the squareroot of the waist size w, instead of waist size itself.
Added phase shift ckAy changes through the focalregion by an amount 7r/2, instead of r.
B. Focusing of the Field Normal to the JunctionFor the x-distribution of the field, the Fresnel for-
mula takes the following form:
+1 y2 k F o (X IX)uX(x"z) = ~ X -t T2 +X2exP t-j 2 [ + jdxo.
(10)
Special numerical methods are employed to evaluatethe above integral. Diffraction integrals are generallyevaluated by dividing the range of integration intosmall intervals and using a linear or parabolic approxi-mation of the amplitude and phase over each interval.Although this procedure is now well optimized,' 2 it isnot the most efficient method for computing the inte-gral in Eq. (10). The simple analytical form of thisintegral allows one to use complex analysis and asymp-totic series expansions to perform an efficient compu-tation. A complete algorithm is given by Muller andNaqwi.13 Only the essential concepts are highlightedbelow.
At the geometrical focal plane (z = -Ro), the inte-gral in Eq. (10) is simply a Fourier transform of thefield immediately after the last lens. It may be com-puted using standard techniques. For either side ofthe geometrical focal plane, it is convenient to expressthis distribution as follows:
u( )=(I + i) kx[- (R +2 )] X G if z < _Rox,UX(x,z) = expl -- i I2a~z L \Rox+zJ 1G' if z > -Rox,
(11)
where G* is the complex conjugate of G, which isdefined as below:
(6)
where w, Ry, and 4bAy are the waist half width, radius ofcurvature and the added phase shift, respectively.These parameters depend upon the initial conditionsand the distance z:
=(2R0 z)2 + [kw2(R 0y + z)]2
7 + (+ _ exp( i d (12)
where al, =-e(l + j3) and '2 = a(1-). The parame-ters a and A are defined as follows:
The above formulation is useful even within a fewwavelengths of the geometrical focal plane. If thenumber of oscillations of the integrand in Eq. (12) issmall, the integral may be computed directly by meansof numerical quadrature. Otherwise, this integral isexpressed in terms of the functions of the form G(0,)and G(6,c), where is a large number. The functionG(0,) is evaluated by extending the integral analyti-cally into the complex plane and integrating along thepath of steepest descent, which lies along the radialline at 0 = -4. The other function G(O,) may beexpanded into an asymptotic series using integrationby parts. So that,
G(6,o;3) = ay2{[P(6) sin( ) + Q(5) cos(2)]
+ i[P(6) cos( 2)) Q(W) sin(2)]} (14)
where
P(6) = -al(a) + a3(b) -(15)
Q(M) = a2(b) - a4(b) + * .-
The terms ai are defined as below:A
a1 = 2 -=
a2 = | d ()| 'etc.,
where
A (1) = _ _ _ __ _ _ (16)a'Y2 + (2 + C/#)2
The above relations may be used for surveying alarge region on either side of the geometrical focalplane.
IV. Properties of the Focused FieldThe focusing properties of the 1-D Gaussian distri-
bution are similar to those of a circular Gaussian beam,which are well documented in literature.11' 1 4'15 Hence,only the focusing characteristics of the truncated Lor-entzian distribution are considered here. The follow-ing discussion is concerned with the amplitude andphase distribution in the focal region.
A. Amplitude DistributionIn Fig. 4, computations based on the truncated Lor-
entzian model are compared with the scans of intensityprofiles. These profiles correspond to focusing of thecollimated field of Fig. 3 by means of a 200-mm focallength lens. The computed results are in good agree-ment with the measured data in exhibiting fluctua-tions in intensity distribution. Such fluctuations arenot predicted by the Gaussian model. The computedfocal spot size also agrees with the measured one.However, the measured fluctuations before the focal
spot are stronger than the computed ones. The oppo-site is true for the locations beyond the focal spot.This discrepancy may be attributed to phase aberra-tions which are not included in the present computa-tions.
The computed results exhibit an asymmetric inten-sity distribution around the focus and also a focal shift.As many recent studies have pointed out,16-2 0 the loca-tion of maximum intensity is displaced a short dis-tance from the geometrical focus, towards the focusinglens.
B. Phase DistributionAn examination of the amplitude distribution alone,
indicates that the fluctuations of the field becomesignificant only at a distance of order ZR from the focalspot, where ZR is defined as the ratio of the square ofthe spot size to the wavelength. This observation mayjustify the use of Gaussian model in the close neighbor-hood of the focus. It is, in fact, a practice in laserDoppler anemometry applications to use the measuredfocal spot size to fit a Gaussian model to the near focusfield. The phase behavior in the neighborhood of thefocus is then estimated from the Gaussian model. Toexamine the validity of this approach, the near focusphase distributions predicted by various models areconsidered here. Also examined are the individualcontributions of truncation and the Lorentzian form tothe peculiar phase behavior of the truncated Lorent-zian distribution.
Figure 5(a) shows the intensity profile at a focusedspot produced by a truncated Lorentzian beam. Alsoshown for comparison are the intensity profiles of aGaussian beam and another beam with a top-hat dis-tribution at the focusing lens. The top-hat beam hasthe same aperture as the Lorentzian beam, whereas theGaussian beam has a focal spot size comparable withthe Lorentzian model. Intensity profiles for eitherdistribution are practically unaltered over distances of±100 gm around the focus. However, a noticeablechange in phase distribution occurs. As illustrated inFigs. 5(b) and (d), deviation of phase fronts from aplanar structure is significantly larger for the truncat-ed Lorentzian distribution than the Gaussian distribu-tion. Furthermore, unlike the Gaussian beam, thetruncated Lorentzian model does not allow the phasefronts to be represented by a single radius of curvaturein the neighborhood of the focus.
The above example clearly shows that the phasebehavior of a truncated Lorentzian distribution is sig-nificantly different from that of a Gaussian distribu-tion. Even in the near field of the focus, it may not beappropriate to use a Gaussian approximation. It isalso clear from Figs. 5(a)-(c) that the focused field of atruncated Lorentzian distribution is substantially dif-ferent from that of a truncated uniform beam. Hencethe peculiar phase behavior of the model beam, shownin Fig. 5(b), is caused by both the truncation of thebeam and its Lorentzian form.
A mathematical model of a diode laser beam, basedon Gaussian and Lorentzian distributions, is present-ed. This model is appropriate for monomode diodelasers, with active layers thinner than the wavelength.In the present model, the optical field is truncated toaccount for the limited numerical aperture of the opti-cal system, used for collimating and focusing the beam.
Intensity and phase distributions in the focal regionare computed using this model. The results are signif-icantly different from those obtained with a simpleGaussian model of the elliptic cross-section beam.The present model shows a better agreement with themeasured intensity field.
This model is useful in obtaining realistic estimatesof the coupling efficiency between a laser diode and anoptical fiber. It is also valuable in estimating theuniformity of fringes in the measuring volume of adual-beam laser Doppler anemometer employing a di-ode laser as the light source.
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18. J. J. Stamnes and B. Spjelkavik, "Focusing at Small AngularApertures in the Debye and Kirchhoff Approximations," Opt.Commun. 40, 81-85 (Dec. 1981).
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