Characteristic of laser diode beam propagationthrough a collimating lens

Xu Qiang,* Han Yiping, and Cui ZhiweiSchool of Science, Xidian University No. 2 Tai Bai South Road, Xi’an, 710071 China

*Corresponding author: qxu@xidian.edu.cn

Received 18 September 2009; accepted 4 December 2009;posted 5 January 2010 (Doc. ID 117381); published 20 January 2010

A mathematical model of a laser diode beam propagating through a collimating lens is presented. Wavepropagation beyond the paraxial approximation is studied. The phase delay of the laser diode wave inpassing through the lens is analyzed in detail. The propagation optical field after the lens is obtainedfrom the diffraction integral by the stationary phase method. The model is employed to predict the lightintensity at various beam cross sections, and the computed intensity distributions are in a good agree-ment with the corresponding measurements. © 2010 Optical Society of America

OCIS codes: 140.3300, 140.2020.

1. Introduction

Because of its compactness, lightness, and low cost, alaser diode (LD) is often used as a light source inmany scientific and engineering applications. Butthe laser diode has certain properties peculiar to it-self: the diode cavity produces a wide-angle outputfrom the narrow cavity dimension and a smaller-angular output from the broader cavity dimension.Typically, the intensity falls to 50% of the maximumover an angle of 10° in the plane of the junction and30°–40° normal to it.Because of the beam characteristics of a laser

diode, some elements, such as conventional refrac-tive optical elements, are used to obtain a collimatedbeam tomeet the requirements of many applications.Because of their simpleness and sufficient accuracy,shorter-focal-length, high-numerical-aperture objec-tives would most often be used as beam collimators.For analysis of a LD beam precisely collimated by

the lens, the paraxial approximation theory no long-er holds, and the wave propagation beyond the para-xial approximation has been little discussed untilnow, so new theoretical analysis is needed. We willdiscuss this collimated light wave beyond the pa-

raxial approximation here [1–7]. The present workaddresses this need. A mathematical model of a laserdiode beam passing through the collimating lens ispresented and subsequently employed to predictthe light intensity at various beam cross sections.The computed intensity distributions are in a goodagreement with the corresponding measurements.

2. Laser Diode Beam Propagation through theCollimation Lens

Figure 1 show an LD beam propagating though a col-limation lens. A ray from the LD entering the lens atcoordinates ðx; yÞ on one face emerges at approxi-mately the same coordinates on the opposite face,if there is negligible translation of the ray withinthe lens. The lens delays an incident wavefront byan amount proportional to the thickness of the lensat each point.

Referring to Fig. 1, let the maximum thickness ofthe lens be Δ0, and let the thickness at coordinatesðx; yÞ be Δðx; yÞ. Then the total phase delay of thewave at coordinates ðx; yÞ in passing through the lensmay be written as [8]

ϕðx; yÞ ¼ knΔðx; yÞ þ k½Δ0 −Δðx; yÞ�¼ kΔ0 þ kðn − 1ÞΔðx; yÞ; ð1Þ0003-6935/10/030549-05$15.00/0

© 2010 Optical Society of America

20 January 2010 / Vol. 49, No. 3 / APPLIED OPTICS 549

where n is the index of refraction of the lensmaterial,k is the wavenumber, knΔðx; yÞ is the phase delay in-troduced by the lens, and k½Δ0 −Δðx; yÞ� is the phasedelay introduced by the remaining region of freespace between the planes M and N. The complexfield E2ðx; yÞ across the plane N immediately behindthe lens is then related to the complex field E1ðx; yÞincident on the plane M immediately in front of thelens by

E2ðx; yÞ ¼ exp½ikΔ0 þ ikðn − 1ÞΔðx; yÞ�E1ðx; yÞ: ð2Þ

In Fig. 1 the radius of curvature of the left-hand sur-face of the lens is R1, while the radius of curvature ofthe right-hand surface of the lens is R2. The totalthickness function is seen to be

Δðx; yÞ ¼ Δ0 − R1

1 −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

x2 þ y2

R21

s !

þ R2

1 −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

x2 þ y2

R22

s !: ð3Þ

For the collimated beam, the facet of the LD is placedat the geometrical focus of the lens. L is the distancefrom the facet of the LD to plane M before the lens. fis the focal length of the lens. Plane H is the object’sprincipal plane. With the relationship [9]

lH ¼ f − L ¼ −r1Δ0

nðr2 − r1Þ þ ðn − 1ÞΔ0ð4Þ

the field distribution of an LD on the plane M isdescribed as [10,11]

E1ðx; yÞ ¼ ALrexpðikrÞ

rΓ2

Γ2 þ x2exp

�−y2

Ω2

�; ð5Þ

where

A ¼ u02iλp

ffiffiffiπq

r;

Γ2 ¼ p2

k2r2;

Ω2 ¼ 4q

k2r2;

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ L2

q:

Here p and q are constants relating to the waveguidestructure of the LD. Substitution of Eq. (5) intoEq. (2) yields the field distribution of the LD beamafter the lens:

E2ðx; yÞ ¼ Af − lH

rexpðikrÞ

rΓ2

Γ2 þ x2exp

�−y2

Ω2

�× exp½ikΔ0 þ ikðn − 1ÞΔðx; yÞ�: ð6Þ

3. Laser Diode Beam Propagation after theCollimation Lens

The optical field immediately after the lens was de-scribed in the previous section. Further propagationof this fieldmaybeadequately representedby calcula-tion of the Rayleigh–Sommerfeld diffraction integral.

A solution to the Helmholtz equation can be ex-pressed in terms of its boundary values (z > 0),

E3ðx; y; zÞ ¼−12π

ZZs

E2ðx0; y0Þ∂

∂z

�expðikRÞ

R

�dx0dy0;

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx − x0Þ2 þ ðy − y0Þ2 þ z2

q; ð7Þ

where E3ðx; y; zÞ is the diffracted field distributionand E2ðx0; y0Þ is the source field distribution onthe plane N after the lens.

By a Taylor series expansion, one can obtain

R ≈ ρþ ðx20 þ y20 − 2xx0 − 2yy0Þ=ð2ρÞ;

ρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

q;

E3ðx; y; zÞ ¼ −izλρ

expðikρÞρ

ZZs

E2ðx0; y0Þ

× exp½ikðx20 þ y20 − 2xx0 − 2yy0Þ=ð2ρÞ�× dx0dy0 ð8Þ

for large k ð104 mm−1Þ, exp½ikðx20 þ y20 − 2xx0 − 2yy0Þ�oscillates rapidly, with no contribution to the inte-gral, except near stationary phase, so the station-ary-phase method is employed in order to find theasymptotic expansion of the diffraction integral [12].

The corresponding diffraction integral is approxi-mated by [12]Fig. 1. LD beam propagation though a collimation lens.

550 APPLIED OPTICS / Vol. 49, No. 3 / 20 January 2010

Uðx; y; zÞ ¼ZZD

f ðx0; y0Þ exp½ikgðx0; y0Þ�dx0dy0

≈2πσ

kffiffiffiffiffiffiffijHjp f ðxs; ysÞ exp½ikgðxs; ysÞ�; ð9Þ

where

H ¼ ∂2g

∂x20

∂2g

∂y20−

�∂2g

∂x0∂y0

�2

ð10Þ

σ ¼

8>>>>><>>>>>:

1 ðif H < 0Þi

�if H > 0; ∂

2g∂x20

����xs;ys

> 0

�

−i

�if H > 0; ∂

2g∂x20

����xs;ys

< 0

� : ð11Þ

Here xs, ys are stationary phase points and

gðx0; y0Þ ¼ ðx20 þ y20 − 2xx0 − 2yy0Þ=ð2ρÞ: ð12Þ

Letting

∂g∂x0

¼ x0 − xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p ¼ 0;

∂g∂y0

¼ y0 − yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p ¼ 0;

ð13Þ

we find stationary phase points

xs ¼ x; ys ¼ y; ð14Þ

and ∂2g∂x20

¼ ∂2g∂y20

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þy2þz2

p , ∂2g∂x0∂y0

¼ 0; so

H ¼ ∂2g

∂x20

∂2g

∂y20−

�∂2g

∂x0∂y0

�2¼ 1

x2 þ y2 þ z2¼ 1

ρ2 ð15Þ

and H > 0, ∂2g∂x20

����xs;ys

> 0;,

σ ¼ i: ð16Þ

Substitution of Eqs. (14)–(16) into Eq. (9) yields

E3ðx; y; zÞ ¼ Aðf − lHÞz

ρr2Γ2

Γ2 þ x2exp

�−y2

Ω2

�× exp½ikΔ0 þ ikðn − 1ÞΔðx; yÞ�× exp½ikðrþ ρÞ − ikðx2 þ y2Þ=ð2ρÞ�; ð17Þ

I3ðx; y; zÞ ¼�Aðf − lHÞz

ρr2Γ2

Γ2 þ x2exp

�−y2

Ω2

��2: ð18Þ

Relations (17) and (18) invoke the approximationthat the distance from the source to the observation

point is much larger than the wavelength as well asthe size of the source. These approximations are rea-sonable for laser diodes, as beam shaping is usuallyattempted at distances significantly larger than thewavelength or the size of the waveguide.

4. Experimental Procedure

To examine the validity of the theoretical results de-scribed above, the intensity profile of a beam emittedfrom a laser diode was measured through a scan of a200 μm⌀ pinhole with a photodiode in the x–z, y–zalong the lines parallel to the facet of the chip.The uncertainty of measurements is 1%. Here thediode is a V-channeled substrate inner-stripe GaAlAslaser (Sharp LT022MCO), the peak wavelength is780nm, and the current is set to 60mA. The beamparameter k=p ¼ 2:24, k=q ¼ 10:36.

For parameters of the collimating lens herewe usedr1 ¼ 14:087mm, r2 ¼ 6:016mm, Δ ¼ 3:073mm, n ¼1:6132, f ¼ 7:2990mm, and L ¼ 5:9mm.

Open circles in Figs. 2–5 show the measured nor-malized intensity Iðx; 0; zÞ and Ið0; y; zÞ, respectively,for the LD. The solid curves represent the profilescalculated from Eq. (18).

For the profiles in Figs. 2 and 3, the collimatinglens was not used. D is the distance between thefacet of the LD and the observation plane: D ¼ 5:9;9:9; 13:9mm. (D ¼ 5:9mm is the distance betweenthe focus and the left vertex of the lens for Figs. 4and 5.) These figures show that the LD beam hasan elongated elliptical cross section and spreads sig-nificantly in the direction of propagation. The calcu-lated profiles in Fig. 2 agree well with the measuredones in the x–y plane. In Fig. 3, comparison is madebetween the measured and calculated profiles in they–z plane; the calculated profiles agree with the mea-sured ones in the central portion of the profile, andthere is a discrepancy in the low-intensity values re-gion. A cause for the discrepancy may be that the

Fig. 2. Measured and theoretical intensity profiles in the x–zplane without the collimating lens.

20 January 2010 / Vol. 49, No. 3 / APPLIED OPTICS 551

field on the LD facet is not purely Gaussian along they axis.For the profiles in Figs. 4 and 5, the collimating

lens was used; the facet of the LD was placed atthe focus of the lens, z is the distance between pointO (the right vertex of the lens) and the observationplane in Fig. 1. z ¼ 10:0; 20:0; 30:0mm. In Fig. 4,It is shown that the calculated curve is fitted quitewell with the measured data in the x–z plane; the dis-crepancy in the low-intensity values region wascaused by the limiting aperture of the lens. For a dif-ferent observation plane, the collimated beam inten-sity profiles are little changed. In Fig. 5, comparisonis also made between the measured and calculatedprofiles in y–z plane after collimating. For the differ-ent observation plane, the collimated beam intensityprofiles are similar. The difference between the re-sults of the model and the measurements is less than

4% if the observation angle θ ≤ 30° in the x–z planeand the observation angle θ ≤ 10° in the y–z plane.

5. Conclusions

A mathematical model of laser diode beam propaga-tion through the collimating lens is presented. Byanalysis of the phase delay of the LD beam wavein passing through the lens and use of the stationaryphase method for the diffraction integral of the non-paraxial optical field, this model can obtain realisticexpression of a LD beamwith large divergence angle;the model is also able to predict the proper resultwhen the field distribution from the LD is differentfrom the expression given in expression (5). The mod-el is appropriate for LD beams collimated beyond theparaxial approximation. Intensity distributions of aLD are computed by this model. The present modelshows good agreement with the measured intensityfield. This model is useful in obtaining realistic esti-mates of collimating a laser diode beam.

This study is supported by the National NaturalScience Foundation of China (grant 60371019).

References

1. R. Katayama and Y. Komatsu, “Blue/DVD/CD compatibleoptical head,” Appl. Opt. 47, 4045–4054 (2008).

2. W. Sun, K. Liu, and J. Tien, “Laser expander design of highlyefficient Blu-ray disc pickup head,” Opt. Express. 17, 2235–2246 (2009).

3. K. Otsuka and S. Chu, “Generation of vortex array beams froma thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34, 10–12 (2009).

4. Q. Zheng, Y. Yao, and B. Li, “13:2W laser-diode-pumpedNd:YVO4 /LBO blue laser at 457nm,” J. Opt. Soc. Am. B26, 1238–1242 (2009).

5. N. Coluccelli, G. Galzerano, and D. Parisi, “Diode-pumped sin-gle-frequency Tm : LiLuF4 ring laser,” Opt. Lett. 33, 1951–1953 (2008).

Fig. 3. Measured and theoretical intensity profiles in the y–zplane without the collimating lens.

Fig. 4. Measured and theoretical intensity profiles in the x–zplane after the collimating lens.

Fig. 5. Measured and theoretical intensity profiles in the y–zplane after the collimating lens.

552 APPLIED OPTICS / Vol. 49, No. 3 / 20 January 2010

6. X. Chen, X. Zhang, and Q. Wang, “Highly efficient diode-pumped actively Q-switched Nd : YAG–SrWO4 intracavityRaman laser,” Opt. Lett. 33, 705–707 (2008).

7. M. Itonaga, F. Ito, K. Matsuzaki, K. Matsuzaki, S. Chaen, K.Oishi, T. Ueno, and A. Nishizawa, “Single objective lens hav-ing numerical aperture of 0.85 for a high density optical disksystem,” Jpn. J. Appl. Phys. 41, 1798–1803 (2002).

8. J. Goodman, Introduction to Fourier Optics (McGraw-Hill,1996), pp. 101–104.

9. M. Born and E. Wolf, Principle of Optics (Cambridge Univ.Press, 1999), pp. 171–174.

10. X. Zeng and A. Naqwi, “Far-field distribution of double hetero-structure laser diode beams,”Appl. Opt. 32, 4491–4494 (1993).

11. S. Nemoto, “Experimental evaluation of a new expression forthe far field of a laser diode beam,” Appl. Opt. 33, 6387–6392(1994).

12. J. Stamnes, “Waves, rays, and the method of stationaryphase,” Opt. Express 10, 740–751 (2002).

20 January 2010 / Vol. 49, No. 3 / APPLIED OPTICS 553