Focusing non-truncated elliptical Gaussian beams

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FOCUSING

NON-TRUNCATED

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ELLIPTICAL GAUSSIAN

1988

BEAMS

Yajun LI Institute of Optical Sciences, National Central University, Chung-Ii. Taiwan 32054, ROC Received

3 May I988

Very simple expressions are obtained for the focal shift of focused gaussian beams with an elliptical cross section, like that produced by junction lasers. The beam is focused by a conventional circular thin lens with a radius which is much larger than the spot size of the beam.

2. Field distribution along the axis

1. Introduction

The advent of junction lasers has greatly expanded the need to handle elliptical gaussian beams. Propagation and diffraction of gaussian beams with an elliptical cross section has been the subject of several recent papers [ l-3 1. In this study, we are concerned with the focusing of a plane wave elliptical gaussian beam by a circular thin lens with a radius that is much larger than the spot size of the beam. The intensity profile of an elliptical gaussian beam may be expressed in the form [ 3,4]

Assume the elliptical gaussian beam is normally incident on a condenser lens of focal lengthfand the lens fills a circular aperture of radius a in an opaque screen (see fig. 1). We assume that (2.1)

Furthermore, the center 0 of the circular aperture is assumed to be on the axis of the beam. According to the Huygens-Fresnel principle [ 6 1, the field at any point P along the axis, not too close to the aperture plane, is given by (see eq. (2.18 ) in ref. [7])

(1.1)

~(x,~)=I~12exp[-2(x2/~~+~*/w~)l,

a u(p)

where A is a constant amplitude factor, w, and wv are the exp ( - 2) intensities along respectively the x and y axes in a rectangular coordinate system (x, y, z). Without loss of generally, we assume w,> w,. In a cylindrical coordinate system (p, 8, z), the plane wave elliptical gaussian beam may be expressed as I(P, O)= IAl2 Xexp[ -2p2(e2cos28+sin2B)/w~]

.

a*w,,

,

(1.2)

=

1 lJ.

exp( f(f+z)

2n

--HI

Uo(P, 0) .

II

0 0

xew

(2.2)

where A is the wavelength and z is the coordinate along the axis directed away from the screen and passing through the center of the aperture and the geometrical focus (z= 0). U,(p, t9) in the integrand on the right-hand side of eq. (2.2) is the field distribution of the incident beam, i.e.,

in which e=w.v/w.r(~l)

(1.3)

is the ellipticity of the beam. For gaussian beams from junction lasers, E is in the 2: l-4: 1 range [ 5 1. 0 030-4018/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )

xexp[

-p*(

&0s*e+sin2f9)/w2,]

.

(2.3)

To determine the axial field of a focused elliptical gaussian beam, we first perform the integration over B.V.

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Thir! Lens L

Y

‘Back f oral

+wx cEllIptical

point

Gaussian beam Spherical surface

reference centered

on F

Fin. 1. Geometry of the problem. w, and w, are the maior and the minor axes of the intensity smaller than the radius of the thin lens.

By introducing eq. (2.3) into eq. (2.2) some algebraic manipulations, we obtain

p.

and after

ellipse of the incident

beam. w, is much

Zn

s

B(u)=;

l-exp{-a’[(t’cos20+sin28)+itu]/w~}d0,

0

(tzcos20+sin20)+itu

(2.8) in which G=w’/Af

(w’=w,~w,.)

is the geometrical numbers G,=w:/Af

(=G/E)

mean

(2.5) of the gaussian

Fresnel

Under the condition of a >> wJ. [see eq. (2.1) 1, the exponential term in the numerator of the above equation is a small quantity in comparison with unity. For instance, when a > 3w,, we may obtain the following estimation: (exp{-a’[

(t2cos28+sin2r3)+ieu]/w;}


(=tG)

(2.6b)

associated to the minor and major axes of the elliptical gaussian beam. In eq. (2.4), a dimensionless parameter (2.7) has been introduced to specify the location of the observation point P on the axis. To this stage, attention is directed to the evaluation of the integral B( u) on the right-hand side of eq. (2.4 ), which is given by 318

I

(2.6a) -9)=

1.2x10e4.

(2.9)

Omission of the exponential term in eq. (2.8) leads to a closed solution of the diffraction integral (2.2)) i.e., ZJ(P)=F+(l-:)exp(ikf&) X[(l+itu)(l+iU/e)]-I”.

(2.10)

3. The focal shift According to eq. (2.10), the intensity = 1U(P) I2 along the axis is given by

Z(P)

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2

Z(P)=Z,

1 (1 +2KU2+U4)

( > I-

2

where 1,= (aG(A1 /f)’ metrical focus and

“2 ’

is the intensity

(3.1) at the geo-

(3.3) are determined by the combinations lowing two quantities:

“92 =;(I-&).

p=? ala2

K=(E2+EC2)/2=(w::+w;)/2w~w,~.

(3.2)

In fig. 2, we have plotted the normalized intensity Z(P) /I, as a function of the normalized distance z/ funder the conditions of G= 1 and e= 1, 2, .... 6, respectively. It is seen from this figure that for large values of E the curves become more symmetrical about the focal point (i.e., the point of z=O), approaching the behavior predicted by the classical theory of light focusing. This phenomenon can be explained by the fact that the gaussian Fresnel number G,. associated with the major axis increases with the increasing values of e [see eq. (2.6b) 1. It is also clear from fig. 2 that for small values of E the maximum axial intensity I,,, does not occur at the focal point, but is shifted toward the aperture by a distance Af This focal shift can be determined by the root of the first derivative of Z(P). After a straightforward calculation, omitted here, we obtain from dZ( P) /dz= 0 the following cubic equation: u3fa2u2+a,

u+ao

(3.3)

co,

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where a, = K/zG, a, = K and a0 = 1/nG. According to the theory of cubic equations [ 81, solutions of eq.

q=----

---2

6

a0

a?

2

27

1988

of the fol-

(3.4a)

(3.4b) It is also known that a cubic equation may have either one real root or all roots real according to the sign of the discriminatory number: D=p3+q2 =&-$(1+$)]+$(1+&). (3.5) Different solutions of eq. (3.3) correspond to different axial intensities as we will see in the next section.

4. Discussion To determine the boundary between the regions of D>O and D
c

l/I,

z/f I

-0.8

I

b

-0.6

-0.4

-0.2

Fig. 2. Normalized on-axis intensity versus the normalized distance and different values of the ellipticity 6 of the incident beam.

0

0.2

along the axis under the condition

0.4 of gaussian

Fresnel number

G=

I

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terms of u,, we find at once from eq. (2.7) the relative focal shift Af If= D)O

u,l ( ZG - u, ) .

(3.7)

For the special case of a circular (conventional) gaussian beam ( t = 1)) eqs. (3.1) and (3.3) reduce, respectively, to the forms

i

2

Z(P)=Z, 06-

( > 1-s

Ju*+ 1

(3.8)

and

u3+ (l/nG)u’+u+(

l/aG)=O

.

(3.9)

0.4 -

This cubic equation can be easily factorized as (u + 1/rrG) (u* + 1) = 0. Substitution of the real root - 1/ nG into eq. (3.7) yields

02-

Af/f=-l/(l+n*G*).

IEz2.414 I,,

0 1

3

2

,

,

4

5

(3.10)

Again, substituting the real root into eq. (3.8), we intensity the relative excess AZ/Z, obtain = (I,,,,, -IF) /Z, of the maximum intensity Z,,,,Xover the intensity IF at the geometrical focus, i.e.,

6 -E

AZ/Z, = 1/n*G* . Fig. 3. Regions in which the discriminatory number Dz 0 and D are emerged from the axial intensities.

region in fig. 3 we have D > 0, and outside of this region are points of D < 0. The physical implications of the numerical results demonstrated in fig. 3 is explained as follows. 4.1. One real root and a pair of complex conjugate

roots (D > 0) Under the condition of D>O, only one intensity peak can be observed in the axial intensity distribution as demonstrated by the curves in fig. 2, for which G = 1 and E= 1, 2, . ... 6, respectively. Points in the unshaded region in fig. 3 correspond to a distribution of one intensity peak. According to the theory about the solution of cubic equations, the location of this intensity peak is given by u, = (d, +&) -a*/3 where 320

(3.6)

,

d, = (q+D’/2)“3

and

d2= (q-D’/2)‘/3.

In

(3.11)

The above two formulas agree with the results obtained by Goubau, van Nie and Kogelnik in the years immediately following the invention of laser beams (see, e.g., the introductory section of ref. [ 71). 4.2. The maximum flat intensity distribution along

the axis This type of axial intensity distribution is worthy of note, because it reduces the accuracy requirement in focusing the beam onto a target. The maximum flat distribution may be obtained when eq. (3.3) has triplet roots. Substituting the particular values of G= l/n and K= 3 into eq. (3.3), we find the following cubic equation: u3+3u2+3u+1=(u+1)3=o,

(3.12)

which has triple roots at u= - 1. The corresponding values of the focal shift and intensity excess are given by

Af/f= -0.5 respectively.

and

AZ/Z,=$,

(3.13)

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z/f=-0.873

I

1

I -10

-0.8

1

- 0.6

- 0.6

Fig. 4. Illustration of the evolution maximum flat distribution.

in axial intensity

I

I

I

-0.127

-0.2

distribution

0

as the gaussian

It is not difficult

to solve this equation

I

I

0.2

0.6

0.6

Fresnel number

G changes. Conditions

1988

z/f

for obtaining

the

the root t=(3+2$)“*=2.414. The intensity curves shown in fig. 4 are plotted under the condition G= 1In. In the middle of this figure is the curve of maximum flat distribution. The other curves demonstrate the evolution in the axial intensities. It is interesting to see that the curves on the lower part of fig. 4 and labeled t=3, 4, .... 6 have two intensity

(3.14)

1=0 .

I

1

I

Next, we turn to a consideration of the ellipticity of the incident beam, which gives rise to the maximum flat distribution. By equating eq. (3.2) to 3, we find that c4-6e2+

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and obtain

I/IF

t

2.0

MAX

Transition

-Rapid

I

8

1

-1.0

- 0.8

- 0.6

I - 0.4

-

I - 0.2

Fig. 5. Illustration of the growth and decline of the relative strength the effect of focal shift.

i

t

I

1

1

0

0.2

0.4

0.6

of the two intensity

peaks. An explanation

of the rapid

z/f

* transition in 321

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peaks of the same height. Variations in the relative strength of these two peaks lead to rapid transition of the point of the principal maximum intensity along the axis [ 91. This is the subject that we will discuss in the following section.

G=l/?t=0.318

1988

-r.

4.3. Rapid transition of the point of maximum intensity It is known that when the discriminatory number D < 0, all the three roots of cubic eq. (3.3) are real. One of them has been designated as u, and shown in eq. ( 3.6 ). Location of the intensity maximum was specified by u, when D>O, i.e., when the intensity curve has one peak only. But now, under the condition of D < 0,u, becomes the parameter that specifies the location of the intensity minimum as one may readily discover by an examination of the curves demonstrated in fig. 4. The other two in the three real roots of cubic eq. (3.3) are given by u,,z= 4 (d, +d,) - $a2 f ti,/?(d,

-d2)

.

-0.6

-0.8

(3.15)

Here d, and d2 are complex conjugates when D
Z,,,, >Zm2

when G> l/n,

(3.16a)

(b)

I,,,, =Zmz

when G= l/n,

(3.16b)

(c)

Z,,,,
when G< l/n.

(3.16~)

0

These conditions provide information about how to pick up the principal maximum intensity from the two peaks. Numerical results about the relative focal shift Af /f and intensity excess AZ/Z, are displayed by the curves in fig. 6. The curves of the focal shifts [fig. 6(a)] are discontinuous at the point G= l/n when t > 2.414. The reason is that when the quantitative limit G = 1/ 7cis exceeded, the intensity peak Z,,,, takes the position of the principal maximum in the axial intensity distributions. Since the intensity peak Z,,,, lies in a region near the aperture, whereas I,,,, is near 322

0.5

1.0

1.5

2.0 -G

Fig. 6. The relative focal shift Af /f [ (a) ] and the corresponding relative excess AI/I,= (I,,, - I,)/Z, of the maximum intensity I max along the axis, over the intensity 1, at the geometrical focus [ (b) 1,as a function of G with the ellipticity 6 as a parameter.

the focal plane [see fig. 5 1, a sudden increase in the focal shift is observed. This phenomenon is denoted in this study by the term of rapid transition in the effect of focal shift. The appearance of two intensity peaks in the axial intensities may be explained as follows. It is seen from eqs. (2.6a) and (2.6b) that the quantitative difference between the gaussian Fresnel

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numbers G, and G, increases with the increasing values of the ellipticity E of the incident beam. According to eq. ( 3.10 ), we may obtain from G, and G, two different focal shifts, which correspond respectively to the two intensity peaks Z,,,, and Z,,,z as shown in figs. 4,5. The above argument was supported, in an inferior degree of accuracy, by the numerical results obtained from eqs. (2.6) and (3.10). For instance, under the conditions of G = 1/n and E= 4, we obtain for eqs. (2.6a) and (2.6b) that G,=G/t= 1147~ and G,= cG=4/n, respectively. Substitution of these data into eq. (3.10) yields the relative focal shifts Af/f = - 16/17= -0.941, =-l/17=-0.059 and whereas the actual locations of the two intensity peaks as given by eq. (3.15) are at z/f= -0.127 and z/ f= -0.873 [see fig. 41, respectively. The discrepancy between these two groups of data can be explained by the fact that eq. ( 3.10) is a formula for circular gaussian beams, whereas figs. 4,5 contain results concerning elliptical gaussian beams. Moreover, since the interactions between the two intensity peaks were neglected in the above argument, appli-

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I988

cation of this argument to explain the rapid transition in the focal shift is not possible. Finally we wish to mention that the focal shift shown in fig. 6 (a) may attain - 0.95f; diffraction integral (2.2) may be valid to a distance zx -0.95f only when (a/f)<3w,,. This restriction can be greatly released if the region, in which the effect of focal shift is considered, is reduced [7]. References [ I] Y. Li and E. Wolf, Optics Lett. 7 ( 1982) 256. [2]R.Simon,OpticsComm.55 (1985)381. [3] Y.P. Kathuria, Mod. Optics 34 (1987) 1085. [4] A.B. Marchant, Appl. Optics 23 (1984) 670. [5] T.H. Zachos, Appl. Phys. Lett. 12 (1968) 318. [ 61 M. Born and E. Wolf, Principles of optics, 6th Ed. (Pergamon Press, Oxford, 1980) sec. 8.2. [ 71 Y. Li and E. Wolf, Optics Comm. 42 ( 1982) I5 1. [8] M. Abramowitz and LA. Stegun, Handbook of mathematical functions (Dover Pub. Inc., 1972) sec. 3.8.2. [9]Y.Li,J.Opt.Soc.Am.A3(1986) 1761. [IO] Y. Li, Optik 64 (1983) 207.

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