Laser Physics Chapter 3

7/28/2019 Laser Physics Chapter 3

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3

Laser Resonators

References

1. A. Siegman, Lasers (University Science Boks, Mill Valley, 1986), Chapters 19and 11.

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

3. A. Yariv, Quantum Electronics (John Wiley and Sons, Inc., Hoboken, NJ,1989), Chapter 7.

Having seen examples of paraxial beams, we now address the question how theymight be generated. Gaussian beams are generated by lasers incorporating stableopen resonators of the type we have discussed. In fact we will show that Gaussianbeams with suitably chosen beam parameters can be used to find the modes of openresonators.

3.1 Gaussian Beam Modes of an Optical Cavity

Let us consider the reflection of a Gaussian beam incident from left (complex beamparameter qi) on a mirror of radius of curvature R. The complex beam parameterqo of the outgoing (reflected) beam will then be given by

qo =qi A + B

qi C+ D

. (3.1)

The ray transfer matrix for the mirror isA BC D

=

1 0

2/R 1

(3.2)

where R is the radius of curvature of the mirror (+ for concave and for convexmirrors). Using this in Eq. (3.2) and 1/q = 1/R + i2/kw2, we find that the spotsize and radius of curvature of the outgoing beam are

wo = wi (3.3)1

Ro=

1

Ri 2

R. (3.4)

87


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88 Laser Physics

Thus the spot size remains unchanged but, in general, the radius of curvature of thephase front changes according to Eq. (3.4). For a flat mirror (R ) the radiusof curvature of the reflected wave also remains unchanged Ro = Ri. The reflectedwave moving to the left is of the same type (converging or diverging) as the incidentwave.

Now suppose the mirror radius of curvature R matches the radius of curvature Ri(+ for diverging and for converging) of the incident wavefront so that Ri = R andthe mirror surface coincides with the incident wavefront. Then radius of curvatureof the reflected wave

Ro = Ri , (3.5)

has the same magnitude as that of the incident wavefront but opposite sign. Thismeans the reflected wave traveling to the left will be a converging wave if the incidentwave is a diverging wave and vice versa. In traveling to the left the reflected beamwill retrace the path of the incident wave. If a second mirror is now placed in thepath of the reflected wave traveling to the left such that the radius of curvature ofthe mirror matches the radius of curvature of the wavefront at the location of thesecond mirror, the beam will be reflected to the right retracing its path. Thus whentwo mirrors such that their reflecting surfaces coincide with the incident wavefronts

of a Gaussian beam the beam is trapped between the mirrors bouncing back andforth between the mirrors. The conditions for Gaussian beam trapping are easilywritten down.

R2R1z2

z=0

z1

L

FIGURE 3.1

Relative to the beam waist (z = 0), the mirrors in a two-mirror cavity are locatedat z = z1 and z = z2.

Suppose the mirrors with radii of curvatures R1 and R2 are placed at position z1and z2, respectively, relative to the beam waist [see Fig.(3.1)]. Then the trappingof a Gaussian beam requires the curvature of the mirror to match the wavefront


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Laser Resonators 89

curvature at the location of the end mirrors 1

R(z1) z1 +z2Rz1

= R1, (3.6a)

R(z2) z2 +z2R

z2

= R2, (3.6b)

where zR =12kw

2o = nw

20/ is the Rayleigh range for the beam. The negative sign

in the first equation arises because the Gaussian wavefront radius of curvature ispositive for a diverging beam and negative for converging beam, traveling to theright, whereas the radius of curvature of a mirror is taken as positive for a concavemirror and negative for a convex mirror. For a Gaussian beam of minimum spotsize w0, Eqs. (3.6a) and (3.6b) determine the radii of curvature and the positionsof the mirrors relative to the waist that will trap the beam.

Often, the opposite problem, where the radii of curvatures R1 and R2 of the

mirrors and their separation L is given and beam waist size w0 and its locationrelative to M1 (or M2) are to be determined, is also of interest. Let us denote thelocations of the mirrors M1 and M2 by, respectively, z2 and z1 relative to the beamwaist. Then we have to solve Eqs. (3.6a) and (3.6b) with

z2 z1 = L (3.6c)

Rewriting Eqs. (3.6a)-(3.6c)

z2Rz1 =

(R1 + z1) (3.7a)z2Rz2

= R2 z2 (3.7b)

z2 = z1 + L (3.7c)

Taking the ratio of the first two equations and eliminating z2 from the result withthe help of the last equation we obtain

z2

z1

= R1 + z1

R2 z2or z2(R2 z2) = (R1 + z1)z1

or (z1 + L)(R2 z1 L) = (R1 + z1)z1

or z1(R1 + R2 2L) = L2R2L

or z1 = L(R2 L)

R1 + R2 2L(3.8)

Similarly, z2 is given by

z2 =L(R1 L)

R1 + R2 2L, (3.9)

1Note that these condition hold only at the end mirrors ( mirrors that force the reflected beam toretrace the path of the incident beam traveling in opposite direction.)


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90 Laser Physics

To calculate the waist size we substitute the expression (3.12) for z1 in (3.9) [or Eq.(3.13) in Eq. (3.10)] and obtain

z2R = (R1 + z1)z1 =

R1 L(R2 L)

R1 + R2 2L

L(R2 L)

(R1 + R2 2L)

= L(R2 L)(R21 + R1R2 2LR1 LR2 + L

2

)(R1 + R2 2L)2

=L(R2 L)(R1 L)(R1 + R2 L)

(R1 + R2 2L)2

Using the result z2R = nw20/ we find that the waist size is given by

w0 =

n

L(R2 L)(R1 L)(R1 + R2 L)

(R1 + R2 2L)2

1/4, (3.10)

The equations for z1, z2 and w0 can be put in more compact forms by introducingthe g-factors defined by

gi = 1L

Ri, i = 1, 2. (3.11)

Then we can write

R1 =L

1 g1R2 =

L

1 g2. (3.12)

Using these in the equation for z1 we find

z1 = L2(1/(1 g2) 1)

L/(1 g1) + L/(1 g2) 2L

= L g2(1 g1)g1 + g2 2g1g2

Similar procedure for z2 and w0 can be carried out. We then obtain

z1 = g2(1 g1)

g1 + g2 2g1g2L, (3.13)

z2 = g1(1 g2)g1 + g2 2g1g2 L (3.14)

w0 =

L

n

g1g2(1 g1g2)

(g1 + g2 2g1g2)2

1/4(3.15)

Using the propagation law for the spot size of a Gaussian beam, we find the spotsizes at the mirrors are given by

w1 = w01 +z21z2

R

= L

n g2

g1(1g1g2)

1/4

(3.16)

w2 = w0

1 +

z22z2R

=

L

n

g1

g2(1 g1g2)

1/4(3.17)


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Laser Resonators 91

An inspection of Eqs. (3.15), (3.16) and (3.17) for spot sizes shows that for physical(real and positive) beam spot sizes g1 and g2 must have the same sign so that theproduct g1g2 is positive and confined to the range

0 < g1g2 < 1 (3.18)

If this condition is not satisfied stable Gaussian beam solutions do not exist forthe given mirror configuration. Resonators with R1, R2, and L satisfying condition(3.18), admit Gaussian beam modes and are said to be stable. The condition forstability of resonators for admitting Gaussian beam mode solutions coincides withthe condition for trapping paraxial rays.

!4

!3

!2

!1

0

1

2

3

4

!4 !3 !2 !1 0 1 2 3 4

g1

g2

symmetric confocal

(g1=g2=0)

spherical

planarP

C

S

convex-concave

convex-convexconcave-convex

concave-concave

1

1

!1

!1

g1=g2=1

g1=g2= !1

FIGURE 3.2

Stable two-mirror resonators lie in the region bounded by the two branches of the

rectangular hyperbola g1g2 = 1 and the coordinate axes g1 = 0 and g2 = 0 in theg1 g2 plane.


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92 Laser Physics

3.1.1 Gaussian Beam Modes of an Arbitrary Cavity

For an arbitrary cavity with a round trip matrix

A BC D

starting from some reference

plane, we can express complex beam parameter q of the trapped Gaussian beam interms of cavity parameters. For a stable resonator, the complex beam parameter q

for a confined Gaussian beam must be reproduced after one round trip, that is,

q =Aq+ B

Cq+ D q2C q(AD) + B = 0 . (3.19)

Solving this equation gives

q =AD

2C

1

2C

(AD)2 4BC . (3.20)

Using the result ADBC det M = 1 we can write the quantity under the radical

sign as (AD)2 + 4(AD 1) = (A + D)2 4. The result for q then becomes

q =AD

2C i

1

2C

4 (A + D)2 . (3.21)

Now by definition, complex beam parameter q is given by

q(z) = z izR = (distance from the waist) i (Rayleigh range) , (3.22)

where zR = w20/n and we have assumed that beam waist is located at z = 0.

Equating the real and imaginary parts from the two sides of Eq. (3.21) we have the

result

z =AD

2C,

w2o =

n

4 (A + D)2

2|C|,

(3.23)

where for the beam spot size to be real w20 must be positive and real. This isensured by the constraint 1 < 12(A + D) < 1, already stated in the discussion ofthe periodic focusing system. For a two mirror cavity (two mirrors with radii ofcurvature R1 and R2 separated by a distance L), with the reference plane in frontof mirror M1 and proceeding toward M2, the round trip matrix is

A BC D

=

2g2 1 2Lg2

2

L(2g1g2 g1 g2) 4g1g2 2g2 1

, gi 1

L

Ri. (1.96*)

From this matrix we find

1

2(AD) = 2g2(1 g1) ,

1

2(A + D) = 2g1g2 1 .

Using these in the expressions derived for z and wo we find

z = g2(1 g1)

g1 + g2 2g1g2L , wo =

L

n

g1g2(1 g1g2)

(g1 + g2 2g1g2)2

1/4.


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Laser Resonators 93

As expected, the expression for z coincides with z1 the location of mirror M1 relativeto the waist and wo is the spot size at the waist.

If the reference plane happens to be the plane of beam waist, we must have thedistance from the waist z = 0. The expression for z in part (a) then shows thatthe corresponding matrix must have A = D. Another way of thinking about it is

in terms of 1/q 1R + i 2kw2 . At beam waist the radius of curvature R , whichimplies A = D. With A = D, the spot size at the waist is given by

w2o =

n

1A2|C|

=

n

1D2|C|

, 1 < A, D < 1 . (3.24)

3.2 Mode Frequency Spectrum

Paraxial beams separable in Cartesian coordinates are known as Hermite-Gaussbeams

Em(r, t) = Aw0

w(z)H

2x

w(z)

Hm

2y

w(z)

e

2/w2(z)it sin[m(z)] (3.25)

where

w(z) = w0

1 + z2/z2R, (3.26a)

R(z) = z + zR/z, (3.26b)zR = nw

20/, (3.26c)

m(z) = kz + k2/2R(z) m(z) (3.26d)

m(z) = ( + m + 1) tan1(z/zR) . (3.26e)

Note that instead of traveling Gaussian waves we now have standing Gaussianwaves because the gain medium will generate Gaussian beams traveling in oppositedirections which will overlap to form standing waves. This imposes an additionalrequirement - for the field to be single valued we require that the round trip phase

shift be an integer multiple of 2 or single pass phase shift m(z2) m(z1) bean integer multiple of,

m(z2) m(z1) k(z2 z1) ( + m + 1)

tan1z2zR tan1

z1zR

+ k2

1

R2+

1

R1

= p (3.27)

where we have used R(z1) = R1 and R(z2) = R2. The term involving k2 is

negligible compared to kL as is a few spot sizes in magnitude and beam spot size

is small compared with both R and L,

1

kL

k2(R1 + R2)

R1R2

w2

RL


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94 Laser Physics

The tan1 terms can be combined to give

tan1z2zR tan1 z1

zR= tan1

(z2 z1)zRz2R + z1z2

= tan1LzR

z2R + z1z2. (3.29)

Using the expressions for z1, z2, and zR given in Eqs. (3.13)-(3.15) in this equation

we obtain

tan1z2zR tan1 z1

zR= tan1

1 g1g2

g1g2= cos1

g1g2 (3.30)

Using the results of Eqs. (3.28) and (3.30) in Eq. (3.27) the mode condition becomes

kL ( + m + 1) cos1g1g2 = p (3.31)This equation determines k which now depends on three indices

kmp =

Lp + ( + m + 1)

cos1

g1g2

. (3.32)

From the relation kmp = n2mp/c we find the resonator eigen frequencies aregiven by

mp =c

2nL

p + ( + m + 1)

cos1

g1g2

(3.33)

From this we see that each axial mode frequency 00p has a number of transversemode (indices mp with p fixed) frequencies associated with it. Consecutive axialand transverse modes are separated, respectively, by

ax 00p+1 00p =c

2nL, (3.34)

tr (+1)mp mp = (m+1)p mp =c

2nL

cos1

g1g2

. (3.35)

Resonator mode functions can then be written as (mp = 2mp)

Emp = Aw0

w(z)H

2x

w(z)

Hm

2y

w(z)

e

2

w2(z)impt

sin[mp(z)], (3.36)

wheremp(z) =

kmpz ( + m + 1)m(z) + k2/2R(z)

(3.37)

3.2.1 Symmetric Resonators

An important class of stable resonators are the symmetric resonators for whichR1 = R2 R and z1 = z2 = L/2 so that g1 = g2 g = 1 L/R. Theseresonators lie along the line g1 = g2 in the g1 - g2 plane [see Fig(1.25)]. By symmetrythe beam waist occurs at the center of the resonator z = 0 and the minimum spot

size there is given by

w0 =

L

2n

1 + g

1 g

1/4=

2n[L(2R L)]1/4 . (3.38)


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Laser Resonators 95

Spot sizes at the two mirrors are equal and given by

w1 = w2 =

L

n

1

1 g2

1/4=

n

LR2

2R L

1/4. (3.39)

We will discuss three special cases of symmetric stable resonators.

Long Radius Near Planar Resonators

These resonators are characterized by R1 R2 R L so that g1 g2 =1 L/R 1. Then beam spot sizes are

wo =

n

RL

2

1/4(1 L/2R)1/4

n

RL

2

1/4, (3.40)

w1 w2 =

n RL

2 1/4 1

(1

L/2R)1/4 w0 1 +

L

4R (3.41)Hence in near planar resonators beam waist is large and varies little over the lengthof the resonator (w1 = w2 w0). This is a useful feature for extracting energymore efficiently from the gain medium in a laser. Large spot size means that laserbeams produce by such cavities will have low divergence. These advantages areaccompanied by the difficulty in maintaining the alignment of mirrors. Since theseresonators are at the edge of instability, small perturbations in mirror orientationcan easily misalign resonator axis. Mirrors with large radius of curvature are difficultto manufacture. For example a D = 2.50 cm diameter mirror with R = 20 m has a

sag at the center of only D

2

/8R 4 m. Nevertheless resonators utilizing mirrorswith radius of curvature R 10 m are commonly used in gas lasers.Frequency spectrum of these resonators can also be discussed. Since g1 = g2 =

1 L/R 1 the angle = cos1g1g2 = cos1(1 L/R) is a small angle closeto zero. Using the small angle approximation cos 1 2/2 we find cos 1 2/2 = g1g2 = (1 L/R) so that the angle is given by

=

2L/R.

Then the mode frequency spectrum for near planar resonators becomes

mp =c

2nL

p + ( + m + 1)

2L/R

(3.42)

The axial and transverse mode separations are given by

ax =c

2nL(3.43)

tr =c

2nL

1

2L

R

ax (3.44)

Hence piled up on the high frequency side of each axial mode is a cluster of transversemodes as shown in Fig. 3.3. Note that for a fixed value of axial index p, all transversemodes with + m = const are degenerate. For example, modes 01p and 10p aredegenerate. Similarly 02p, 11p and 20p are degenerate.


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96 Laser Physics

p p+1p!1

10

01

11

20

02

21

12

30

03

00 "#ax

"#tr

p p+1p!1

p+1p!1 p #

#

p+1/2

12

34

0Near planar

Confocal

Near

Concentric

#

p!1/2

12

4

0

3

FIGURE 3.3

Frequency spectrum of symmetric two-mirror resonators with Hermite-Gaussmodes. The dashed lines show the trajectories of mode frequencies as the resonatorevolves from plane parallel to a concentric resonator.

Symmetric Confocal Resonator

Another important symmetric resonator that lies, literally, at the heart of stableresonators is the symmetric confocal resonator. This resonator is characterizedby g1 = 0 = g2 (point C) corresponding to R1 = R2 R = L. Since for aconcave spherical mirror the focal point lies a distance R/2 in front of it, it isclear that for L = R the focal points of the two mirrors coincide. Hence the nameconfocal. Note that there are other confocal resonators indicated by the dashedcurve L = 12(R1 + R2). All confocal resonators except the symmetric resonator areunstable.

For the symmetric confocal resonator the beam waist lies at the center of the

resonator. Spot sizes at the waist and mirrors are given by

w0 =

L

2n, (3.45)

w1 =

L

n=

2w0. (3.46)

Thus in propagating from the beam waist to an end mirror the spot radius increasesby

2. Hence in the confocal resonator mirror separation is exactly equal to the

confocal parameter of the beam. On the average (averaged over the entire resonator)the confocal resonator has the smallest spot size of all stable resonator. This featureis useful if high power density is desired over the entire resonator length. Small modevolume associated with small beam size, however, may not be the most efficient for


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Laser Resonators 97

energy extraction in a laser if the gain volume does not match the mode volume.Small mode volume of the fundamental mode may encourage transverse modes tooscillate and this may degrade the spatial profile of the beam. Matching gain profileto mode volume is a fundamental consideration in the design of laser resonators.

This resonator is insensitive to small perturbations of mirror position and orien-

tation because any tilt of either mirror still leaves the center of curvature locatedat the other mirror. This merely displaces the resonator axis by a small amount.This insensitivity combined with high degree of degeneracy of mode spectrum makessymmetric confocal resonators ideal for scanning spectrum analyzer applications.

Frequency spectrum of the confocal resonator has high degree of degeneracy. Sinceg = 0 it follows that = cos1(0) = /2 and the mode spectrum becomes

mp =c

4nL(2p + m +p + 1). (3.47)

This spectrum gives axial and transverse modes separated by

ax =c

2nL, (3.48)

tr =c

4nL. (3.49)

Thus the spectrum has a simple structure and successive modes in a confocal (axialor transverse) are separated by

c =c

4nL. (3.50)

The spectrum has high degree of degeneracy. For example, modes 00p, 20(p 1), 11(p 1), 01(p 1), 40(p 2), 31(p 2), 22(p 2), 13(p 2), 04(p 2), etc aredegenerate.

Concentric or spherical resonators

These resonators lie near the boundary of the stability region in the third quadrantof the g1 g2 plane. They are characterized by L 2R. Writing L = 2R(1 )where 0 < L, R we obtain

g1 = g2 = 1L

R= 1 2(R )

R= 1 + 2

R(3.51)

Using this we obtain for the beam spot sizes

w0 =

L

2n

1 + g

1 g

1/4

L

2n

2/R

2

1/4=

L

2n

R

1/4(3.52)

w1 = Ln

11 g2

1/4

L

n 1

2 (2/R)1/4

= L2n

R1/4

(3.53)

In this case the minimum spot size w0 at the center of the resonator becomesvery small whereas the spot size at the mirrors z2 = z1 R becomes very large


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98 Laser Physics

w1 = w2 w0. The central spot has very high power density and may be useful insome nonlinear optical applications but from the point of view of energy extractionfrom the gain medium in a laser this is not very useful. The beam divergence is alsovery large. Spherical resonators are also very sensitive to small mirror misalignmentswhich can produce large changes in the resonator axis. res

Frequency spectrum of a spherical resonator is also easily calculated. FromEq.(3.51) we note g1 = g2 = 1 + 2/R 1. The angle = cos1

g1g2 =

cos1 [1 + 2/R] cos1(1) is close to . Writing this angle as = weobtain

cos( ) = cos

1 2

2

=

1 + 2

R

(3.54)

This gives 2

/R and = 2R. The spectrum for a spherical resonator

can then be writen

mp =c

2nL

p +

( + m + 1)

2

R

=c

2nL

p + + m + 1 ( + m + 1) 2

R

(3.55)

Mode frequency separation is then given

ax =c

2nL , (3.56)

trans =c

2nL

1 2

R

= ax

1 2

R

(3.57)

Thus the transverse mode spacing is only a little less than the axial mode separation.This means the modes 01p = 10p approach 00(p+1) from the low frequency side,modes 02p = 11p = 02p approach 00(p+2) from the low frequency side. Thuseach axial mode carries a pile of mode on its low frequency side. The trajectories

of various transverse modes as a resonator changes from a near-planar throughconfocal to near-spherical is shown in Fig. 3.3.

3.3 Resonator Losses and Resonance Properties

In our discussion of resonators so far, we have ignored losses. Losses are both

necessary and unavoidable. For example, to couple energy into the cavity, or toextract energy from it, mirror reflectivity must be less than unity. In addition,modes also suffer losses due to diffraction, scattering and absorption. These lossesare specified for one round trip of the circulating field inside the cavity.


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Laser Resonators 99

3.3.1 Losses

3.3.1.1 Transmission losses

These losses are incurred by the field due to finite transmissivity of the mirrorsor absorption in their reflective coating. These losses are localized at the mirrors.

When a wave circulating inside the cavity encounters a mirror of reflectivity R, afraction R of its power is reflected back into the cavity and a fraction 1R = T+ Ais lost either due to mirror transmission ( a fraction T) or absorption (a fraction A)by the mirror coating. Mirror loss is usually dominated by the transmission loss.For this reason mirror losses are simply referred to as transmission losses.

In laser physics it is customary to characterize mirror transmission and absorptionlosses in terms of the loss factor, a dimensionless quantity, defined by

L lnR = ln(1 TA) . (3.58)

For small mirror transmission and absorption, the loss factor reduces to the frac-tional power loss suffered by the wave at the mirror

L = ln(1 TA) T+ A . (3.59)

For large mirror losses, L can exceed unity. For example for a mirror of 30%reflectivity, the loss factor is 1.20, whereas for a mirror of 95% reflectivity it is 0.05.In the latter case it is equal to the mirror transmittance T= 1R.

In a two mirror cavity the wave is reflected once from each of the mirrors so thatin one round trip it has a fraction R1R2 of its initial power remaining inside thecavity and a fraction 1R

1R

2lost due to mirror transmission and absorption. The

loss factor in this case is L = lnR1R2. Generalizing this to an N-mirror cavity,we find the loss factor due to mirror transmission and absorption is

LT = ln[R1R2 RN] =N

j=1

lnRj =N

j=1

Lj (3.60)

3.3.1.2 Diffraction losses

An important parameter for discussing diffraction losses in finite diameter optical

resonators is the resonator Fresnel number NF, defined by

NF =na2

L(3.61)

where 2a is the transverse width of the resonator end mirrors, L is the length of theresonator, is the wavelength of light and n is the refractive index of the mediumfilling the resonator. This number is the number of Fresnel zones across one endmirror as seen from the center of the opposite mirror. Let us recall that the endmirror spot size in a symmetric confocal resonator of length L is w21 = L/n andother stable resonators have similar spot sizes at the mirrors. Then the expression

for NF can be written as

NF =n

L

a

=

a2

w21

1

. (3.62)


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100 Laser Physics

Thus, NF is the ratio of the resonator mirror area to the area of the lowest orderconfocal mode area divided by . If we note further, that the radius of an nth orderHermite-Gauss mode is to a good approximation n

n w1. Then the largest

order of Hermite-Gauss or Laguerre-Gauss mode that will fit within the mirroraperture is

nmax = a2

w21= NF . (3.63)

The exact calculation of losses must be done using Huygens-Kirchoff diffractiontheory. These calculations give both the losses and additional phase shifts, whichare the exact versions of the Guoy phase shifts (z) given in the ideal Gaussianlimit by the (z) = tan1(z/z0) formula. They, thus, determine the exact spacingof transverse modes in the finite-aperture resonators. In practice, for nF 10, theGaussian approximation for the mode shape is excellent for the lowest few modes.

A number of empirical formulas for the one-way power loss per pass Ldi

ff in finite-aperture resonators have been developed by various researchers.

Confocal square mirrors

Ldiff =

8

2NF e4NF NF 12

1 16N2F

e82N2

F/9 NF 0

(3.64)

Confocal circular mirrors

Ldiff =

162NF e

4NF NF 11 (NF)2 NF 0

(3.65)

Planar strip mirrors

Ldiff = 0.12 N3/2F

NF 1 (3.66)

Planar circular mirrors

Ldiff = 0.33 N3/2F

NF 1 (3.67)

3.3.2 Absorption and scattering losses

These losses due to scattering and absorption in the medium are distributed through-out the cavity. They are also referred to as distributed or internal losses. Such lossesare specified in terms of an absorption coefficient , with units []=m1 or morepractically cm1. After one round-trip, a wave circulating inside the cavity will

have a fraction e2L

of its initial power remaining inside the cavity and a fraction1 e2L lost due to absorption. Here 2L is the round trip length of the cavity.Sometimes the factor e2L is treated as an effective reflectivity and internal lossesper pass are specified in terms of an effective transmittance Ti 1e2L. The loss


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Laser Resonators 101

factor for absorptive losses is then defined in a manner analogous to transmissionlosses

Li

lnRi = ln(1 Ti) losses specified as Ti

ln[e2L] = 2L losses specified as (3.68)

The cavity loss factor is the sum of internal (distributed absorptive and scatteringlosses) and external loss (mirror transmission, absorption, and diffraction losses)factors

L = Li + Le = lnRi lnR1R2 RN (3.69)

3.3.3 Cavity/Photon life time

As a consequence of these losses any field inside the cavity will lose energy as itpropagates inside the cavity. If there are no sources to replenish the lost energy, fieldamplitude and therefore the intensity will decay with a time constant determined

by the losses.

R1

L

R2

Intra-cavity

intensity

t=0

Output

intensity

t=0

Pump

intensity

t=0!"

Absorption #

Power

reflectivity

FIGURE 3.4

Losses (transmission) allow an external pump field to build up a steady intra-cavity

and transmitted field amplitudes. If the pump is removed, cavity losses cause theintra-cavity and transmitted field amplitudes to decay. For simplicity, the fieldtransmitted by only one mirror is shown.

To be concrete, let us consider a two mirror cavity of length L. Let E0 be thefield amplitude and I0 the corresponding intensity at time t = 0. Then one roundtrip distance is 2L and the time for one round trip is

R = 2L/(c/n) = 2nL/c (3.70)

After each roundtrip the intensity is reduced by the factor2 R1R2e2L. After m

2Here we are ignoring diffraction losses which are usually small for large aperture mirrors. They are


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102 Laser Physics

round trips the intensity will be

I(tm = mR) = I0R1R2e

2Lm = I0 exp[(2L lnR1R2)m]

= I0 exp[LcR

mR]

= I0 exp[tm/c] (3.71)

where

tm = mR = m2nL

c(3.72)

c =R

2L lnR1R2=

2nL

cLc(3.73)

If we take the intensity (3.71) equation to hold for all times t, not just at discreteinstants tm, we can write it as

I(t) = I0 exp(t/c) . (3.74)

The light intensity in this approximation decays with a time constant c given by Eq.(3.75). It is the time for the cavity intensity to decrease to 1/e 37% of its initialvalue. It is a measure of the time for which the cavity can store electromagneticenergy. Time c is called the lifetime of the cavity. It is naturally related to thecavity loss factor Lc.

Io

Ioe!1

t=0 t="c

time

Ioe!t/"c

0.37Io

FIGURE 3.5In the absence of a pump, the initial intra-cavity field intensity decays exponentiallyin a lossy cavity.

For large mirror reflectivities R1, R2 1 and negligible absorption, we can writeR1 = 1 T1 and R2 = 1 T2 where the mirror transmittances satisfy T1 ,T2 1.The expression for the cavity life time then simplifies to

c =2nL

c(2L + T1 + T2)

2nL

cL

(3.75)

easily incorporated by including multiplicative factors such as (1 Ldiff) with mirror reflectivitiesin Eq.(3.71)


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where the total round trip loss L = 2L + T1 + T2 is the sum of intrinsic (due todiffraction, absorption, scattering etc.) loss per round trip 2L and transmissionlosses per round trip

L = LI + LT , LI = 2L , LT = T1 + T2 . (3.76)

These equations are easily generalized to cavities containing additional optical ele-ments by including losses introduced by these elements into the calculation of totalloss.An order of magnitude estimate Let us estimate c for a cavity consisting of twomirrors of reflectivities R1 = 0.98 = R2 separated by 90 cm in air. Since the mirrorreflectivities are large (and therefore the losses small) we can use Eq. (3.75) for cto obtain

c =2nL

c

1

2L + T1 + T2=

2 0.90

3.00 108 1

0.04

s = 150 ns (3.77)

This differs by less than 2% from the result ( c = 148 ns) obtained by using Eq.(3.75).

The cavity lifetime, although short by ordinary standards, is typically long com-pared to an optical cycle. For o = 600 nm an optical cycle is o = /c =600 109/3.00 108 = 2.00 1015 s. The ratio c/o is then the number ofcycles an optical field will execute in one cavity life time. For our example this ratio

c

o=

150 109

2.00 1015= 7.5

107 (3.78)

is a very large number. Thus if a certain amount of energy is deposited in the cavity,it will take many millions of optical cycles for the cavity to empty out. The ratioc/o is a measure of the quality of a resonator and is related to the Q-factor of thecavity

Q = 2 co

= 22nL

L(3.79)

A cavity that stores energy for a long time (many optical cycles) is a high-Q (highquality) cavity. From the dependence of Q on cavity losses we see that lower theloss higher the quality. Expressing c in terms of Q

c = Qo

2=

Q

0(3.80)

we write the time dependence of the intensity as

I(t) = I0e0t/Q . (3.81)

Noting that the intensity is given by I = (W/V)c/n where W is the stored and V

is the effective mode volume, we can write an equation for the energy stored in theresonator

dW

dt= 0

QW . (3.82)


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104 Laser Physics

With the help of this equation we find that the energy lost in one optical cycleo = 2/0 is W = o

dWdt

= 2WQ so that Q can be written asQ = 2

W

W. (3.83)

Thus Q is 2 times the ratio of energy stored in the resonator to the energy lost inone optical cycle. Another way of stating this is that Q is 2 times the number ofoptical cycles it will take to lose the energy stored at a constant rate ofW peroptical cycle. Note that Q can be quite high even for lossy cavities. For example,the Q-factor of 50 cm long cavity with 90% loss in one round trip at = 600 nm is

Q = 22nL

L= 2

2 0.500.90 600 109 = 1.1 10

7 (3.84)

Such large values of Q arise because although the loss per pass is large, the loss per

optical cycle is still small.

3.3.4 Cavity linewidth (FWHM of the spectrum)

From the decay of the intensity (I |E|2) it follows that the amplitude of cavitymode (frequency 0 will have the time dependence

E(t) = E0et/2ci0t . (3.85)

Taking the Fourier transform of the amplitude we find

E() = 0

dt E(t)eit =E0

i(0 ) + 1/2c. (3.86)

This leads to the cavity mode power spectrum

E()2 = E02

( 0)2 + (1/2c)2=

E02 c

1

1/4c(0 )2 + (1/4c)2

, (3.87)

which is plotted in Figure . In the limit c the spectrum has the character ofdelta function centered at 0. In this limit the mode spectrum is a sharp spike (line)at 0. For finite c, the power in a mode is distributed over a band of frequenciescentered on mode frequency 0. Using FWHM to define the width c of the powerspectrum we find that

c =1

2c=

c

2nL

L

2=

0

2Q=

0

Q

. (3.88)

Thus the cavity mode acquires a width o/Q known as the cavity line width.3 Large

values ofQ imply sharply defined mode frequencies relative to the optical frequency.

3Care must be exercised in interpreting the term linewidth used by different authors as it is defineddifferently.


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Axial modes

Transversemodes

!!"

!

#!c

!"

!

"

!

"

+#!c/2!

"

$#!c/2 !

Io

1/2 Io

Cavity losses

FIGURE 3.6

The decay of field amplitude caused by cavity losses imparts a width to each ideal(loss-less) cavity mode frequency.

Another quality factor F, called the finesse of the cavity, is used to characterizethe sharpness of cavity modes. Cavity finesse is defined to be the ratio of thelongitudinal mode spacing

ax= c/2nL to the cavity linewidth

c

F=ax

c= ax2c =

c

2nL2

2nL

cL=

2

L. (3.89)

Cavity finesse gives an indication of the sharpness of mode frequencies relative tothe frequency spacing of cavity modes. For 90 cm long cavity with 4% loss per passwe find

c =3.00 108

2 0.90 0.04

2Hz = 1.06 MHz 1.1MHz . (3.90)

The axial mode separation is

ax =c

2nL=

3.00 1082 0.90 MHz = 167 MHz (3.91)

In this case c is small compared to the axial mode spacing and corresponds to ahigh finesse F= 152.

3.4 Resonance Properties of Laser Resonators

Optical resonators can also be used as spectrum analyzers, filters or as build upcavities. For these applications the resonator mirrors must be partially transmitting


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106 Laser Physics

R1 R2

Absorption !

z=0 z= L

Ei

ER

ET

Ec

E+

E"

" R1 R1R2

FIGURE 3.7

E+ and E are the right and left traveling part of the same circulating field Ec.

to permit coupling of external fields to the cavity.

Consider a resonator filled with a dielectric of refractive index n and absorptioncoefficient . A monochromatic field Ei is incident on mirror M1. We assume thatthe incident field is spatially mode matched to one of the cavity modes. Let ERand ET denote the reflected and transmitted fields and E+ and E the right andleft going fields inside the resonator. All fields have the dominant time dependenceeit and dominant space dependence eikzz/2. Then in the steady-state the fieldsmust satisfy the boundary conditions at z = 0 and z = L

E+(0) =T1Ei(0) +

R1E(0) (3.92a)

ER(0) = R1Ei(0) +

T1E(0) (3.92b)

E(L) =R2E+(L) =

R2E+(0)e

L/2+i(kL+) (3.92c)

ET(L) =T2E+(L) =

T2E+(0)e

L+i(kL+) (3.92d)

Here represents phase changes, other than kz, experienced by the wave dueto propagation. It may includes, for example, the Guoy phase shift and any othercontributions due to the transverse spatial structure of the wave.

Since E+ and E are part of the field circulating inside the resonator it followsthat

E(0) =R2E+(L)e

L/2+i =R2E+(0)e

L+i2 . (3.93)

where the phase is related to the wave frequency = 2 and the refractive indexn of the medium filling the resonator by

kL + =nL

c + . (3.94)

With the help of Eq.(3.93) we can solve Eqs.(3.92) for E+(0) Ec, ET(L) ET, and


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ER(0) ER to find

Ec =

T1

1R1R2eLei2Ei(0) (3.95a)

ET =

T1T2e

Lei

1R1R2eLei2Ei(0) , (3.95b)ER =

R1

R2e

L+i2

1R1R2eLei2Ei(0) (3.95c)

From these equations, we find that the ratios of the transmitted, reflected, andintracavity intensities to the intensity of the incident wave can be written as

Tc ITI0

=Tcmax

1 + F sin2 , (3.96a)

Rc IRI0

= Rcmin + F sin2

1 + F sin2

, (3.96b)

Ic IcI0

=Icmax

1 + F sin2 , (3.96c)

where we have introduced the following definitions

F =4R1R2e

L

(1R1R2eL)2, (3.97a)

Tcmax =T1T2 e

L

(1R1R2eL)2 , (3.97b)

Rcmin =(R1

R2e

L)2

(1R1R2eL)2, (3.97c)

Icmax =T1

(1R1R2eL)2. (3.97d)

Here Tc and Rc represent, respectively, the power transmission and reflection coeffi-cients of the cavity and Ic represents the factor by which the intracavity intensity is

enhanced compared to the incident intensity. The significance of various quantitiesintroduced in Eqs.(3.97) will become clear in the following paragraphs. Equations(3.96) indicate that Tc, Rc, and Ic are periodic functions of. They pass througha series of maxima and minima as varies.

A plot of cavity transmission and reflection as a function of is shown in Figs.(3.8) and (3.9). It attains a maximum value Tcmax given by Eq.(3.97b) for

nL

c+ = p , (3.98)

where p is an integer. The maxima are separated by

=

. We also note from Fig.(3.8) that even when each mirror by itself may have a small transmission, cavityas a whole may have much larger transmission. This is a dramatic consequence ofwave interference. This can be seen by considering Fig. (??) where the transmitted


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108 Laser Physics

!=2"#L/c

R= 0.1

0.5

0.9

#p

(#p$%#c/2)2"L /c

#p+1

Tc max

Tc max1

2 %#c

(#p+%#c/2)2"L /c

FIGURE 3.8

Transmission from a Fabry-Perot cavity as a function of .

field is written as a superposition of partial transmissions as the wave injected intothe cavity bounces back and forth between the mirrors. With this interpretationthe integer p is referred to as the order of a transmission maximum and serves asan index to label it.

Half way between two consecutive maxima, the transmission has a minimum givenby

Tcmin =Tcmax1 + F

. (3.99)

These minima are located at

nL

c+ =

p +

1

2

(3.100)

The condition for maximum transmission (3.98) implies that maximum transmissionis attained when the frequency of the incident field coincides with the frequenciesdetermined by

p =c

2nL

p

. (3.101)

These frequencies are precisely the cavity mode frequencies given by Eq.(??). Itfollows that the cavity transmission is a maximum whenever the incident field fre-quency is in resonance with a cavity mode frequency.

Assuming that the incident field is spatially mode matched to a particular trans-verse mode, the frequency interval between successive maxima will be

p+1 p =c

2nL ax , (3.102)


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!=2"#L/c

R= 0.9

0.5

0.1

p (p +1)

Rc max

Rc max

1 2

0

FIGURE 3.9

Reflection from a Fabry-Perot cavity as a function of.

which is equal to the axial mode separation for the cavity. In the context of res-onators as spectrum analyzers, this interval is known as the free spectral range

FSR =c

2nL. (3.103)

From Fig. (3.8), we note that the sharpness of transmission peaks increases withincreasing coefficient of finesse F, which is controlled by the mirror reflectivities R1,R2, and internal losses 2L, in addition to the quality of the resonators construction.When the transmission and internal losses T1 = 1 R1, T2 = 1 R2, and Li =1 e2L are small, we can use the much simpler expression for the coefficient offinesse

F4R1R2e

L

(1R1R2eL)2

8[2 (T1 + T2 + 2L)](T1 + T2 + 2L)2

=8[2 L]

L2

16

L2. (3.104)

The last approximation is good within a few percent as long as the total loss L =LT + LI 0.1.

A measure of the sharpness of the transmission maxima is their FWHM (fullwidth at half maximum). Let us consider a transmission maximum at = p anddenote its FWHM by c. Then the frequency p +c/2 where the transmissionfalls to half of its maximum value at p is determined by Tc(p +

12c) =

12Tc(p).

With the help of Eq. (3.113a) this leads to

Tcmax

1 + F sin2(p + cnL/c)=

1

2Tcmax (3.105)


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110 Laser Physics

orF sin2(cnL/c) = 1 (3.106)

where we have used the result sin2(p + ) = sin2 . Solving this we find that cis determined by

sin(

c

nL/c) =

1

F (3.107)Here the negative root has been ignored because, by definition, c is positive.

If the coefficient of finesse F is large, the argument of the sine function is small.We can then use the small angle approximation sin and obtain

c =2

F

c

2nLFSRF

(3.108)

where FSR = c/2nL is the free spectral range and F is the finesse of the resonatorgiven by

F= F2

= (R1R2)1/4eL/2

1 R1R2eL

(3.109)

It is clear that the width c of a transmission peak is a small fraction of theseparation between successive transmission peaks (free spectral range) when theresonator finesse F is large. When a resonator is used as a spectrum analyzer, itsability to resolve two closely spaced spectral frequency in the input signal is limitedby the width c. Large finesse F is a measure of the resolving power of a resonator.By writing the finesse as the ratio

F=

FSRc =

2cR (3.110)

we see that F is the number of transmission peaks that can fit in one free-spectralrange. Also using the relation c = 1/2c and FSR = 1/R we see that thefinesse is 2 times the number of round trips that the wave makes in one cavity lifetime. For small losses, the expression for the finesse simplifies to

F=2

L=

2

LT + LI(3.111)

Using this, this expression for the finesse the width can be written as

c =cL

2nL

LR

(3.112)

Thus c is the fractional power loss per roundtrip.In terms ofF, we can rewrite the transmission and reflection coefficients and the

build up factor as

Tc =Tcmax

1 + (2F/)2 sin2 , (3.113a)

Rc =

Rcmin + (2F/)2 sin2

1 + (2F/)2 sin2 , (3.113b)

Ic =Icmax

1 + (2F/)2 sin2 , (3.113c)


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In the neighborhood of a transmission peak at p = pFSR we can write

= p + ( p) . (3.114)

Then, for a high-finesse cavity near a transmission peak

Tc =Tcmax

1 + (2F/)2 sin2[( p)/FSR ]

Tcmax1 + (2F/)2[( p)/FSR ]2

=Tcmax

1 + [2( p)/c]2

=Tcmax(c/2)

2

( p)2 + (c/2)2=Tcmaxc

2

1

c/2

( p)2 + (c/2)2

Tcmaxc2

S() (3.115)

Here the function S() is the line shape function. This form is particularly usefulwhen a high-finesse resonator is used as a filter.

We have derived three different expressions for c

c =

c2nL

1 1

R1R2eL

(R1R2)1/4eL/2 c2nLF

c2nL

12 (2L lnR1R2) Q

c2nL

12 (2L + T1 + T2)

(3.116)

In practice, the first and second expressions for c lead to the same answerwithin a few percent for R1 and R2 as large as 0.5. Furthermore, both the firstand second expressions give the same result as the third expression for R1, R2, and1 e2L 0.1.

Case I: R1 R2 = 0.98, 2L 0

c =

c2nL

1 0.02020305 = c2nL 6.4308 10

3

c2nL

1 0.020202071 = c2nL 6.4307 10

3

c2nL

1 0.02 = c2nL 6.3662 10

3

Case II: R1 R2 = 0.5, 2L 0

c =

c2nL

1 0.7071068 = c2nL 0.225

c2nL 1 0.69314718 = c2nL 0.221

c2nL

1 0.5 = c2nL 0.159


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112 Laser Physics

For L = 1.00 m in air, R2 = R2 0.98 and L = 0, we obtain the cavity linewidthto be

c =c

2L

1

2(T1 + T2) =

3.00 1082 1.00 2 0.04 = 9.55 10

5 Hz 1.0MHz

The finesse is found to beF=

c2L

=2

L= 157

3.4.1 Cavity Buildup

The intracavity field build up is also a periodic function of and has a maximumon transmission resonance

Icmax =T1

(1R1R2eL)2

(3.117)

For a high-finesse cavity R1 ,R2 1 and 2L 1

Icmax =4T1

(T1 + T2 + 2L)2(3.118)

IfT1 T2 + 2L, which is a condition for zero reflection from the cavity we have amaximum buildup factor

Icmax =1

T1(3.119)

IfT2 + 2L

T1 we have the largest buildup factor

Icmax =4

T1(3.120)

In practice, even modest buildup factors, like 10, give a 100-fold increase in nonlinearoptical experiments, such as second harmonic generation.

3.5 Problems

1. Roundtrip matrix and beam parameters:

(a) Given the round trip matrix

A BC D

for an optical cavity starting from some

reference plane, show that the complex beam parameter q of the trappedGaussian beam in that reference plane is given by

q =AD

2C i

4 (A + D)2

2C.

If the reference plane is a waist of the trapped Gaussian beam, determine theconditions on the matrix elements and the spot size at the waist in terms of

Laser Physics Chapter 3

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