Laser Physics Chapter 2

7/28/2019 Laser Physics Chapter 2

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2

Beam-Like Solutions of the Wave Equation

References

1. A. Siegman, Lasers (University Science Boks, Mill Valley, 1986), Chapters 16,17, 19.

2. P. W. Milonni and J. H. Eberly (John Wiley and Sons, Inc., Hoboken, NJ,2010), Chapter 7.

For monochromatic1

elds E (r

, t ) = E (r

)e it

, where E (r

) is any of the Cartesiancomponents of the complex vector eld, the corresponding atomic response can bewritten as P at = at ()e it . To avoid proliferation of symbols, we have denotedthe space dependent part by the same symbol as the total eld. Using this in thewave equation (1.10), we nd that the space dependent part E (r ) satises

2 + 2 (1 + at + i / ) E (r ) = 0 . (1.10*)

For at = 0 = we get the homogeneous wave equation

2 + 2 E o(r ) = 0 , (2.1)

with a solution E (r ) = E oeik r , where the magnitude of the propagation constant is

(real and ) given by

k = = nc

, (2.2)

and n is the refractive index of the host medium. Combining this space dependencewith the time dependence e it , we see that the solution E (r , t ) = E oei (k

r t ) , rep-resents a plane wave propagating in the direction = k / |k | in a loss-free medium.We now explore beam like solutions of the wave equation.

Using this, we write the solution to the full wave equation (1.10*) as E (r ) =E o(r )eik

r , which has a space dependent amplitude E o(r ) that satises the equation

2 + k2 ( at + i / ) E (r ) = 0 . (2.3)

1 We can always decompose a time-dependent eld into its Fourier frequency components each withtime dependence of the form e

i t


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60 Laser Physics

2.1 Gaussian Beams

Beam-like solutions should have the following characteristics:

(i) a predominant direction of propagation, and

(ii) a nite transverse cross-section (nite extent in directions perpendicular tothe direction of propagation).

Finitetransverse size

Transverse sizechanges slowly

Predominant directionof propagation

z

x y

FIGURE 2.1

A beam has a dominant direction of propagation and nite extent in directionsperpendicular to the direction of propagation.

Since a beam has nite transverse size, wave di ff raction will cause its cross-sectionto change as the beam propagates. However, we can still speak of a predominantdirection of propagation if the di ff ractive change of its cross-section is not too rapid

(to be made more precise shortly) as the beam propagates. Let us now see how wecan construct a solution with these characteristics.

From the introduction we recall that a monochromatic plane wave propagatingin the z-direction will have the form

E (r ) = E oei (kz t ) , (2.4)

where the propagation constant k = n/c = 2 n . Using this plane wave solutionas a guide, we now look for solutions that correspond to a superposition of planewaves propagating in directions making small angles with the z-axis, we can write

the propagation vectors as k = e x kx + e yky + e zkz with k2 k2x + k2y + k2z = n22/c 2and kx , ky k, kz . For such elds kz k2 k2x k2y k (k2x + k2y)/ 2k2 and wecan write the space dependent part of the eld as

E (r ) =1

(2 )2 dkx dky E (kx , ky)eik x x+ ik y y+ ik z z ,=

eikz

(2 )2 dkx dky E (kx , ky)eik x x+ ik y y i (k2x + k2y )z/ 2k ,

eikz E o(r ) . (2.5)


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Beam-Like Solutions of the Wave Equation 61

The amplitude |E (kx , ky)| of off -axis plane wave components decreases rapidly forincreasing values of |kx | , |ky | . This means the dominant direction of beam propa-gation is along the zaxis. Comparing Eq. (2.5) with the expression for a planewave (2.4), we notice that unlike a plane wave, the eld amplitude E o for a paraxialbeam depends on spatial coordinates. Using this in Eq. (1.10*), we nd that theequation satised by E o(r ) is

2 + 2

z2+ 2 ik

z+ k2 ( at + i / ) E o(r ) = 0 , (2.6a)

where 2 = 2

x2+

2

y2. (2.6b)

Since the amplitude E (kx , ky) is dominated by small values of kx and ky , eldamplitude E o varies slowly with z. This means the following inequalities hold:

1

k

E o

z=

2

E o

z|E o| , (2.7a)

1k

2E o z2

=

2

z E o z

E o z

. (2.7b)

The rst inequality says that the change in beam prole over distances of the orderof a few wavelengths

E o ( r ) z are a small fraction of |E o(r )|. The second inequality

states that the rate at which the beam prole evolves does not signicantly changeover distances of the order of a few wavelengths, z

E o ( r ) z

E o ( r ) z . In view of

the inequalities (2.7), the dominant variation of E o with z in Eq.(2.6) is describedby the rst derivative term. Neglecting the 2E o(r )/ z2 term compared to the E o(r )/ z term in Eq. (2.6), we nd that the equation governing the variation of E o(r ) with z is

E o(r ) z

=i

2k2 E o(r ) + i

k at2

E o(r ) k2

E o(r ) ,

or E o(r )

z=

i2k

2 E o(r ) + ik at

2E o(r )

k2

+ at E o(r ) . (2.8)

This is one of the basic equations of this course, which tells us that the eld ampli-tude changes slowly with z due to (i) di ff raction (represented by the 2 term), (ii)interaction with the atomic medium resulting in a phase shift (represented by the at term), and (iii) dissipation of electromagnetic energy (represented by the con-ductivity term) because of Joules heating and absorption by the atoms (representedby the at term).

Neglecting the loss and atomic interaction terms in Eq. (2.8) we nd that theenvelop function E o(r ) satises

2

x2 +

2

y2 + 2 ik

zE o(r ) = 0 . (2.9)


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62 Laser Physics

This equation is known as the paraxial wave equation. By writing this equation as

E o(r ) z

=i

2k 2E o(r )

x2+

2E o(r ) y2

, (2.10)

we see that this equation expresses the fact that the beam prole changes withpropagation because of di ff raction (due to nite extent in transverse dimensions).

If the transverse beam prole does not depend on spatial coordinates, we recoverthe plane wave result E o = const.

2.1.1 Fundamental Gaussian Solution

The paraxial wave equation has many known solutions. These will be introducedin due course. The simplest of these has circular cylindrical symmetry about thedirection of propagation and is given by ( = x2 + y2 )

E o(, z) = Aw0

w(z)ei

k 22q ( z )

i (z) (2.11a)

w(z) = w0 1 + ( z/z R )2 (2.11b)zR =

12

kw20 = w20n

(2.11c)

1q (z)

=1

R(z)+ i

2kw2(z)

(2.11d)

R(z) = z +z2Rz

= z 1 +z2Rz2

(2.11e)

(z) = tan 1 zzR(2.11f)

To see the physical meaning of various terms in this equation, let us rst considerthe time averaged intensity I (, z) 12 0ncEE given by

I (, z) = I 0w0

w(z)

2

exp 22

w2(z), (2.12)

where I 0 = 12 0nc|A|2. This has the form of a gaussian distribution in the variable

. For this reason this beam is called a gaussian beam. In terms of the total powerof the beam

P = 2

0d

0dI (, z) = I 0w202

14

= I o w2o

2(2.13)

we can express the peak intensity I o = 2 P/ w2o and the intensity as

I (, z) =2P

w2(z)e 2

2 /w 2 (z) . (2.14)

Figure (2.2) shows the intensity prole of the beam as a function of distance (along the x

axis) measured from the z

axis. The intensity attains its peak value


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0

0.5

1

x / w

1/ e

1/ e 2

= w

= w / 22

equivalent"top hat" beam

I / I max

2 1 0 1 2

FIGURE 2.2

Intensity prole of a circular symmetric gaussian beam along the x (or y) axis( = x).

I max = 2 P/ w2(z) at the center of the beam ( = 0) and falls to 1 /e 2 14% of this value when = w(z). w(z) is referred to as the 1 /e 2 intensity radius or simplyas beam spot size (radius). The intensity of a gaussian beam falls o ff rapidly as

increases beyond the spot radius w(z).The form of the peak intensity 2 P / w2(z) suggests another measure of beam size.If we were to imagine a circular cylindrical beam of uniform intensity and the sametotal power as the Gaussian beam, the radius of such a beam will be

wTH = w/ 2 . (2.15)Such a beam with uniform intensity over its cross sectional area is referred to as atop hat beam because the intensity distribution of such a beam has the shape of a top hat [See Fig. (2.2)]. We may also refer to wTH = w/ 2 as the 1/e intensityradius.There are other measures of beam size. For example, we can use a criterion basedon power the transmitted by an aperture. A circular aperture of radius a placed atthe center of a gaussian beam will transmit a fraction of power

P TP

=2

w2 2

0d

a

0de (2

2 /w 2 ) = 1 e (2a 2 /w 2 ) . (2.16)

This fraction as a function of aperture radius is plotted in Fig. 2.3.An aperture of radius wTH will transmit only 63% power while an aperture of

radius w will transmit 86% power. An aperture of radius w/ 2 will pass 99%


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64 Laser Physics

0

0.5

1

0 0.5 1 1.5 2 2.5 3a /w

63%

86%

99%

P T

P

a = w

a =

w

/ 2

a =

2 . 3

w

a = w

/ 2

R i p p l e

1 7 %

R i p p l e

1 %

FIGURE 2.3

Fractional power of a gaussian beam transmitted by a circular aperture of radius centered on the beam.

power. We nd that considerably larger apertures than those with radius w areneeded to pass the power gaussian beam of spot size w.

Beyond optimizing power transmission of a gaussian beam, we may also want tominimize di ff raction ripples which will signicantly distort the intensity distribu-tion of the transmitted beam. Such ripples will be present whenever sharp edgedcircular apertures are used even if they pass a large fraction of the total power. Asharp circular aperture of radius w/ 2, which passes 99% of the total power, willcause ripples with intensity variations of 17% in the near eld and a peak intensityreduction of the same amount in the far eld. To keep di ff raction ripples down to1% in the beam transmitted by a sharp edged circular aperture we must employ anaperture of radius 2.3w. [Siegman, Chapter 18].

From the preceding discussion, it is clear that w(z) is a measure of the transversesize of the beam, which of course, varies as the beam propagates. Figure (2.4) shows

the variation of spot size as a function of the propagation distance z measured fromthe beam waist which is dened to be the plane where the spot size has itsminimum value w0. In writing the expression for beam spot size(2.11b) have chosenthis plane to be the z = 0-plane. An examination of Eqs. (2.11a)-(2.11f)shows thata gaussian beam is uniquely determined by the location of its waist and spot sizewo.

A gaussian beam spreads in a nonlinear fashion during its propagation. Near thewaist the spread is slow so that the beam remains collimated. Far from the waistthe beam spreads linearly with distance from the waist. The characteristic distancethe beams travels from the waist before the spot size increase to 2w

o(or the beam


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w o z

w ( z )Confocal parameter b =2 z R

Rayleigh range z R waist z = 0

2w o

Rayleigh range z R

2w o

FIGURE 2.4

Variation of spot size with propagation near beam waist.

spot area doubles) is

zR = kw20 / 2 = nw20

. (2.17)

This distance zR is called the Rayleigh range. Notice that there are two pointslocated on the opposite sides of the beam waist where the spot size has the value 2wo. The distance between these 2wo spot size points is the confocal parameter

b = 2 zR =2n w20

. (2.18)

Confocal parameter b is a measure of the distance over which a beam may beconsidered to have uniform cross-section near its waist. It plays an important rolein the theory of laser resonators which will be discussed shortly.

In the far eld z zR the spot size grows linearly with z as [see Fig. (2.20)]

w(z) w0z

zR. (2.19)

The far eld divergence angle of the beam may be dened by the ratio of the fareld spot size to the distance

= w(z)z

= w0zR

= n w0

. (2.20)

For paraxial approximation to be good we require < 1/ which translates tominimum spot size w0 > /n . From the dependence of the confocal parameter [Eq.(2.18)] and the beam divergence angle [Eq. (2.20)] on w0 we see that a beams withsmaller waist spot size will remain collimated over shorter distances and will spreadmore rapidly in the far zone.

The origin of gaussian beam divergence is di ff raction, which arises whenever awave is conned to a nite transverse size. In fact, the far eld beam divergence


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66 Laser Physics

w o z 0

w ( z )

Beam waist z =0

= / n w

FIGURE 2.5

Far eld divergence of a gaussian beam.

angle is of the same order as the angle associated with the Fraunhofer (far eld)diff raction of a plane wave by a circular aperture of radius a wo

D = 0 .61

na. (2.21)

A Gaussian laser beam thus has the smallest possible divergence allowed by Maxwellsequations. Since di ff ractive phenomena cannot be described by ray optics, wave op-tics must always be used when dealing with gaussian beams.

The parameter R(z) [Eq.(2.11e)] is the radius of curvature of the very nearly

spherical phase fronts at z. This can be seen by writing the Gaussian beamE o(, z)ei (kz t ) [Eq.(2.11a)] in the limit z >> z R

E (r , t ) =Ae i (z)

1 + z2/z 2Re

2 /w 2 (z) eik (z+

22R ( z ) ) e it

iAzR e 2 /w 2 (z) 1

zeik (z+

2

2z ) e it , (2.22)

where we have used the approximation

R(z) = z 1 + z2R /z

2

z , z zR . (2.23)Let us compare it with a spherical wave emitted by a point source on the z-axis atz = 0. Near the zaxis this wave has the form

E s (r , t ) = E oei(kr t )

r= E o

eik 2 + z2

2 + z2e it

E oeik (z+

2

2z )

ze it , 2 z2 . (2.24)


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68 Laser Physics

to the confocal parameter R = 2 zR = b. The center of curvature for the wavefrontat z = zR is located at z = zR and the center of curvature of the wavefront atz = zR is located at z = zR as seen in Fig. (2.7). The curved wavefronts atz = zR have special signicance in stable resonator theory. If these wavefrontsare replaced by two mirrors with matching radii of curvature, we will form a stableresonator. This resonator will have mirrors of radius of curvature R and spacing Lwith R = b = 2 zR = L. Since the focal length of a mirror of radius R is f = R/ 2,the focal points of these two mirrors will coincide at the center of the resonator.The two mirrors then form a symmetric confocal resonator, thus giving rise to theconfocal parameter b = 2 zR .

R = b

R

R = b

z = z R

Confocal parameter b z = z

R

z = 0

S

S

FIGURE 2.7

Wavefronts and ray trajectories in the waist region for a wave moving from leftto right. Far from the beam waist the wavefronts are part of spheres with centerof curvature at the center of the waist. The hyperbolas are rays indicating thedirection of energy ow in a gaussian beam.

The direction of energy ow in a gaussian beam is indicated in Figure (2.7) for agaussian beam traveling from left to right. Energy is transported along rays whichare the directed curves

x2 + y2 = 0 1 + z2/z 2R , (2.27)where 0 is the ray coordinate at the waist. The Poynting vector - S which describesthe ow of energy (energy ux density, W/m 2) is tangential to the rays. To the leftof the waist, Poynting vector has a small radial component pointing toward the axiscorresponding to a converging (focusing) beam. To the right of the waist it has asmall radial component pointing away from the axis corresponding to a diverging(defocusing) beam. Energy ow across the waist is from left to right as in a planewave moving in the z-direction.


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70 Laser Physics

References

1. H. Kogelnik and T. Li, Laser Beams and Resonators , Proc. of IEEE 54 ,1312-1329 (1966).

2. A. Siegman, Lasers (University Science Books, Mill Valley, CA 1986), Chapter20.

In using this relation we will nd it convenient to rewrite Eq. (2.11d) as

q (z) = z izR = distance from the waist i Rayleigh range of the beam . (2.29)

Example 1: Gaussian beam propagation in a homogeneous medium starting at its waist.

Let us take the beam waist location to be the z = 0 plane. At the waist, thewavefronts are planar, so that the complex beam parameter is pure imaginary givenby

q i = izR = i w20

. (2.30)

The matrix for free propagation over a distance z in a homogeneous medium is

A BC D =

1 z0 1 . (2.31)

Using this matrix, we nd that the beam parameter at a distance z from the waistwill be given by

q (z) = q i A + Bq i C + D

= q i + z

or1

q (z)

1R(z)

+ i2

kw2(z)=

1q i + z

=1

izR + z=

izR + zz2R + z2

. (2.32)

Equating the real and imaginary parts from the two sides, we obtain the famil-iar expressions [ 12 kw

2 = 12 kw20(w/w 0)2 = zR (w/w 0)2] for the wavefront radius of

curvature and spot size

R(z) = z2

+ z2R

z= z + z

2R

z, (2.33a)

w(z) = w0 z2R + z2z2R = w0 1 + z2

z2R. (2.33b)

Example 2: Passage of a gaussian beam through a lens.The ray transfer matrix for the lens is

A B

C D=

1 0 1

f 1. (2.34)


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Let q 1 be the incident beam parameter just before the lens. Then the output beamparameter q 2 (just after the lens) is given by

q 2 =q 1 A + Bq 1 C + D

=q 1

(q 1/f ) + 1,

or1q 2

= 1f

+1q 1

,

or 1R2(z)

+ i 2kw22

= 1f

+ 1R1(z)

+ i 2kw21

. (2.35)

Equating the real and imaginary parts on the two sides gives

1R2

=1

R1

1f

, (2.36a)

w2 = w1 . (2.36b)

Thus, a lens changes the curvature of the phase front but leaves the spot sizeuna ff ected. A related problem is the focusing of a gaussian beam by a mirror of focal length f = R/ 2.

2.2.1 Gaussian Beam Focusing

Consider a gaussian beam incident from left on a lens of focal length f . Let theincident beam waist be located a distance d1 from the lens and let the spot radiusthere be w01 . After passing through the lens the beam has a new waist at d2 andspot size w02 at the new waist. We are interested in nding d2 and w02 .

The ray transfer matrix for beam passage from the rst waist to the second waistis

A BC D =

1 d20 1

1 0 1

f 11 d10 1 =

1 d2f d1 + d2 d1 d2

f 1

f 1 d1f

(2.37)

The complex beam parameter q 2 at the second waist is then given by

q 2 =Aq 1 + BCq 1 + D

=(1 d2/f )q 1 + ( d1 + d2 d1d2/f )

q 1/f + (1 d1/f ), (2.38)

where q 1 is the beam parameter at the rst waist. At the two waists, the complexbeam parameters are pure imaginary,

q 1 = izR 1 i nw201/ , q 2 = izR 2 i nw

202/ . (2.39)

Using these in the transformation equation (2.38) we obtain

izR 2 = izR 1 A + B izR 1 C + D

=( izR 1 A + B )( izR 1 A + B )

(zR 1 C )2 + D 2

= izR 1 (AD BC ) + ( z2R 1 AC + BD )

(zR 1

C )2 + D 2(2.40)


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72 Laser Physics

Equating the real an imaginary parts of the expression on the right hand side tothe corresponding terms on the left we nd

R e[q 2] 0 =z2R 1 AC + BD(zR 1 C )2 + D 2

, (2.41a)

I m[q 2] zR 2 = zR 1 (AD BC )

(zR 1 C )2 + D 2. (2.41b)

Using the fact AD BC = 1 and z0i = nw20i / we nd from Eq. (2.41b) that thenew waist spot size is given by

w202 =w201

(zR 1 /f )2 + (1 d1/f )2=

f nw01

2 11 + ( f /z R 1 )2(1 d1/f )2

. (2.42)

From Eq. (2.41a) we nd, since the denominator is not zero, z2R 1 AC + BD = 0,which leads us to

z2R 1 1 d2f

1f + d1 + d2

d1d2f 1

d1f = 0 . (2.43)

On simplifying and solving this equation for d2 we obtain

d2 =z2R 1 /f d1(1 d1/f )

(zR 1 /f )2 + (1 d1/f )2= f 1

(1 d1/f )(zR 1 /f )2 + (1 d1/f )2

. (2.44)

A plot of exit waist position d2/f as a function of the incident waist position d1/f is shown in Figure (2.8). To see the variation of the exit waist spot size with d1,we nd it is convenient to plot the Rayleigh range zR 2 /f = w202/ f which is ameasure of the spot size in units of f as a function of d1/f . This is shown in Fig.(2.9).

Let us compare these results with the predictions of geometrical optics. If weconsider the outgoing beam waist as the image of the incident beam waist, thengeometrical optics gives the location of new beam waist d2 and spot size wo2 to be

d2 =fd 1

d1 f = f 1

11 d1/f

(2.45)

w2o2 = w201

d2

d1

2

=w201

(1 d1/f )2 (2.46)

The prediction of geometrical optics for the second beam waist location d2 is shownby the dashed curve in Fig.(2.8). We see that gaussian beam and geometrical opticspredictions agree when |d1/f | 1 and zR 1 /f 1, that is, when the lens is locatedin the far zone of the incident beam waist. The disagreement between the twopredictions is complete as d1 f . In this case, geometrical optics predicts d2 and w202 , whereas gaussian beam results are

d2 f and w202 = f

nw01

2. (2.47)


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=0.4

1

2.5

FIGURE 2.8

Beam waist location d2/f after a gaussian beam passes through a positive lens of focal length f as a function of the incident beam waist location d1/f for diff erentvalues of the ratio zR 1/f .

A noteworthy features of Fig. 2.8 is that the distance d2 for the second waist fromthe lens has a maximum. The maximum occurs for d1/f 1.5. Similarly, thespot size for a gaussian beam after passing through has a nite maximum which isattained for d1/f 1.

A question of practical importance when discussing applications such as lasertraps, cutting, drilling, and laser fusion is how small focal spots are possible toboost power density. For a given focal length f , we can reduce the size of w02 bymaking zR 1 /f large and, since zR 1 = nw 201/ , this means we need to make w01as large as possible. But w01 cannot be larger than the lens aperture if signicantbeam power loss is to be avoided and may need to be even smaller if we allow forbeam spreading from the rst waist to the lens. One way to address this is tocollimate the incident beam with a confocal parameter many focal lengths long andplace the lens in the near zone so that the input beam spot size does not changesignicantly from the rst beam waist to the lens. Under these conditions w01 islimited by the lens aperture. If we want the lens to transmit 99% of the incidentpower, then incident beam spot radius w01 must satisfy the condition

1

2

w01 =

1

2D , (2.48)


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74 Laser Physics

=0.4

FIGURE 2.9

Gaussian beam spot size in terms of Rayleigh range after focusing by the lens of focal length f for zR 1 /f = 0 .4, 1, 2.5. Geometrical optics predictions are shown by

grey curves for zR 1 /f = 0 .4 and 2.5. The dashed curve is the gaussian beam resultfor zR 1 /f = 1.

where D is the lens aperture (diameter). Under these conditions ( zR 1 /f 1 andthe constraint w01 = D ), Eq. (2.42) leads to the following expression for the focalspot radius (spot size at the second waist)

w02 = f

n w01=

nf D

=

n (f # ) , (2.49)

where f # is the f -number of the lens. A small f # implies a fast lens (high lightgathering capability) and a large f # a slow lens (low light gathering capability).The best lenses have f # 1, while most have f # > 1. Thus the smallest spotradius of focal spot is about the size of a wavelength.

The location d2 of the focal spot (second waist) from Eq. (2.44), under the sameconditions, is given by

d2

f 1

f 2

z2R 1

1 d1

f = 1

z2R 2

f 2 1

d1

f 1 , (2.50)


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where last step follows because zR 2 , the Rayleigh range for the focused beam, isusually much less than f . This means the second waist is very nearly in the focalplane of the lens.

The peak power density at the second waist is

I 02 =2P

w202=

2P ( /n )2

2 , (2.51)

where 2 = 22 is the solid angle into which the second beam waist radiates [Fig.1.4] and 2 is the divergence angle for the focused beam. Similarly, the peak intensityat the rst waist is

I 01 =2P

w201=

2P ( /n )2

1 , (2.52)

where 1 = 21 is the solid angle into which the rst waist radiates. The quantityB = I 01/ 1 = 2P ( /n )2 = I 02/ 2 is called the brightness (power emitted per unitarea per unit solid angle: W/m 2sr) of the source. The brightness of a source isan invariant in the sense that linear optics elements (mirrors, lenses etc.) do notchange it.

By expressing w02 in terms of w01 , the result for the peak intensity in the secondwaist can also be written as

I 02 =2P

f 2 1. (2.52*)

From this expression we see that the power density that can be obtained by focusinga beam of given power is inversely proportional to the solid angle divergence of thebeam being focused. High degree of directionality (smallness of ) of laser beams

thus is of crucial importance for obtaining high power densities in the focal spot.By contrast, a thermal source (ordinary lamp) emits in all directions (2 steradian).If it delivers a power P over an ideal lens aperture, leading to the focal spot powerdensity [See Fig. 1.4]

I P f 2

12

. (2.53)

Example: Consider a He:Ne laser with P = 1 mW, =633 nm, and spot size w01=1mm. Then its divergence angle ( n = 1), solid angle and intensity are

= wo

= 0.633 10

6 10 3

0.2 10 3 rad

1 = 2 = (0.2 10 3)2 1.3 10 7 sr

I 01 =2P

w20=

2 10 3 (10 1)2

W/cm 2 = 64 mW/cm 2

If this laser is focused by a lens of f = 2 .5 cm (the human eye), the peak intensitywill be

I 02 =2P

f 2 1=

2 10 3

6.25 1.3 10

7 2.5 103 W/cm 2 . (2.54)


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76 Laser Physics

Thus, direct viewing of even a lower-power laser beam can result in severe retinaldamage. Thermal lamps would have to emit hundreds of thousands of watts tomatch the intensities achievable by focusing even modest power lasers.

The large intensities achievable by lasers are a direct consequence of their lowdivergence. While care must be exercised in dealing with laser beams, their usein repairing detached retinas and other surgical procedures has become practicallyroutine.

2.2.2 Hermite-Gauss beam solutions

So far we have discussed only the fundamental Gaussian beam solution. Thereare other solutions of the paraxial wave equation (2.9), which have more complexspatial structure. In general, the solutions of the paraxial wave equation (2.9) willbe labeled by two indices. The solutions separable in the Cartesian coordinatesystem are the Hermite-Gauss solutions given by

[E o]mn ( r ) = Aw0

wH m ( 2x/w )H n ( 2y/w )e i(m + n +1) + ik

2 / 2q. (2.55)

Here we have suppressed the zdependence of w(z), complex beam parameter q (z)and phase (z) for simplicity of writing. These quantities are independent of thebeam indices and are given by Eqs. (2.11b)-(2.11f). A Hermite-Gauss beam of indices m, n is sometimes denoted by HG mn .

H m (x) in Eq.(2.55) is a Hermite polynomial of degree m and argument x. Somelow order Hermite polynomials and recursion relations for computing the higherorder ones are listed below

H 0(x) = 1H 1(x) = 2 x

H 2(x) = 4 x2 2H 3(x) = 8 x3 12x

H m +1 (x) = 2 xH m (x) 2mH m 1(x)dH m (x)

dx= 2 mH m 1(x)

(2.56)

Hermite-Gauss beams maintain their form during propagation. Spot size w(z) setsthe length scale over which the beam prole changes signicantly in transversedirections, and zR sets the length scale over which beam prole changes signicantlyas the wave propagates. The intensity distribution for the beam with indices m, nis given by

I mn (x, y ) = I 0w20w2

H 2m ( 2x/w )H 2n ( 2y/w )e 2(x2 + y2 )/w 2 , (2.57)

where I 0 is given by

I 0 =

1

2 0n|A|2

c . (2.58)


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78 Laser Physics

astigmatic. If a elliptic beam is passed through a lens, beam waists after the lensdo not, in general, lie in the same plane. Fundamental elliptical beam has the form

E o( r ) = A w0xw0ywx (z)wy(z) eikx 2

2q x ( z ) +y 2

2qy ( z ) i (z)

1

q x (z)=

1

Rx (z)+ i

2

kw2x (z)1

q y(z)=

1Ry(z)

+ i2

kw2y(z)

w2x = w2ox 1 +

2(z zx )kw20x

2

w2y = w2oy 1 +

2(z zy)kw20y

2

Rx (z) = ( z Z x ) 1 + kw20x

2(z Z x )2

Ry(z) = ( z Z y) 1 +kw20y

2(z Z y)

2

(z) =12

tan 12(z Z x )

kw20x+

12

tan 12(z Z y)

kw20y

(2.62)

Beam waist occurs at z = Z x in the x z plane and at z = Z y for the y z plane.These two planes in general do not coincide. The beam in general has ellipticalprole. Semiconductor lasers emit this type of beams. Such beam can be convertedinto symmetric Hermite-Gauss beams by using prisms or cylindrical lenses. Forw0x = w0y = w0 (which also requires Z x = Z y = Z ), we recover the fundamentalcircularly symmetric gaussian beam with waist at Z .

2.2.4 Laguerre-Gauss Beams

Paraxial wave equation (2.9) admits beam solutions that reect other symmetries.For example, in the presence of circular cylindrical symmetry about the z-axis, Eq.(2.9) admits Laguerre-Gaussian beam solutions

[E o] p( r ) = A 2( p!) ( p + )! w0w 2w||

ei L || p22

w2e i(2 p+ +1) + i

k 22q . (2.63)

where L | p (x) is the associated Laguerre polynomial and w(z) and R(z) are inde-

pendent of the mode indices. Some low order associated Laguerre polynomials and


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recursion relations for computing the higher order polynomials are ( > 0)

L0(u) = 1

L1(u) = u + + 1

L2(u) =12

u2 2( + 2) u + ( + 1)( + 2)

( p + 1) L|| p+1 (u) = (2 p + + 1 u)L |

| p (u) ( p + )L

|| p 1(u)

udL || p (u)

du= pL || p (u) ( p + )L

|| p 1(u)

(2.64)

We see that the lowest order solution ( p = 0 = ) coincides with the fundamentalGaussian beam solution for the HG family. For p = 0 and = 1 we obtain theintensity

I 01() = I 0w20w2

22

w2e 2

2 /w 2 (2.65)

This has a dark center and is sometimes called the donut mode. Note that thedonut shaped intensity distribution sometimes seen in lasers is most often a mix-ture of HG 01 and HG 10 modes.

Let us calculate the total power of the beam. With I o = 12 0cn|A|2, we have

P =

0 d2

0 d I (, z)= I o 2( p!) ( + p)!

w0w

2 2

0 d 2w2

L || p (22/w 2) 2 e 22 /w 2

= I o4 ( p!)

( + p)!wow

2 w2

4

0 du u L || p (u)2

e u

= I o( p!)

( + p)!w2o

( + p)!( p!)

= I ow2o (2.66)

Hence we can write the intensity of LG p beam as

I p =2P

w2 p!

( + p)! 2

w

2

L || p (22/w 2)

2e 2

2 /w 2 (2.67)

There are also the so-called Bessel beam or non-di ff racting beam solutions [3]. Inpractice, symmetries other than the rectangular symmetry are di ffi cult to realize.The presence of Brewster surfaces and other asymmetric optical elements in laserresonators naturally leads to beams with Cartesian symmetry. For this reason only

the Hermite-Gauss solutions are usually considered.


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80 Laser Physics

2.3 Laser Beam Quality

We have seen that the fundamental gaussian beam spot size, which sets the trans-verse length scale for the beam, varies as

w2(z) = w2o 1 + (z

zo)2

zR

2

= w2o +

wo

2

(z zo)2 ,

where zo is the location of beam waist. We also note that the spot size for a gaussianbeam is related to the transverse variance 2x or standard deviation x of a TEM 00beam by wx = 2 x and wo = 2 ox in the x direction, and wy = 2 y and wo = 2 oyin the y direction. If we dene the spot sizes for an arbitrary, non-gaussian beamas W x = 2 x and W y = 2 y , then it is possible to show that in the paraxialapproximation, the axial variations of these spot sizes in free space is given by

W 2x (z) = W 2ox + M

4x

W ox

2 (z zox )2 , (2.68a)

W 2y (z) = W 2oy + M

4y

W oy

2(z zoy)2 , (2.68b)

where M 2x and M 2y are the so called beam quality factors in the x and y directions.To extract their physical meaning, we consider the far-eld limits of these equations,which yield

W x (z) = M 2x

W ox (z zox ) , (2.69a)

W y(z) = M 2y

W oy(z zoy) . (2.69b)

A comparison of these equations with the corresponding result for the ideal gaussianbeam shows that the far-eld divergence of the given beam M 2 W o is a factor

of M 2 larger compared to the far-eld divergence W o of a gaussian beam of thesame waist spot size W o.

Equations (2.68) imply that the free-space propagation of the transverse spotsizes W x and W y for any real laser beam is determined by a waist spot size W oand waist location zo, exactly like the parameters wx (z) and wy(z) for a gaussianbeam. However, the propagation of a real beam, in addition to being dependent onthe ration /W o, also depends on the beam quality factor M 2 in the appropriatetransverse direction.

The beam quality factor thus dened is always M 2 < 1. Note also that theRayleigh range for this beam in the x coordinate is given by Z Rx = W 2ox /M 2x , sothat the far-eld divergence of the beam increases both as the waist size W ox gets

smaller and also as the beam quality factor M 2x gets larger. The axial propagation


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of an arbitrary laser beam is thus fully characterized by the six parameters W ox , zoxand M 2x in the x transverse direction and W oy , zoy and M 2y in the y direction. Thequadratic propagation equation written above will in fact be valid for any choiceof perpendicular x and y axes in the transverse plane. However, the parametervalues W ox , W oy give the most signicant description of the beam if the x and ycorrespond to the principal axes of the beam, that is, the axes in which the crossmoment xy over the beam intensity prole is zero. The most general real beamcan then be characterized by its waist asymmetry ( W ox = W oy , its conventionalastigmatism ( zox = zoy , and its divergence asymmetry ( M 2x /W ox = M 2y /W oy).

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Laser Physics Chapter 2