Laser Modes

7/26/2019 Laser Modes

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Chapter 11. Laser Cavity Modes

Chapter 11.Laser Cavity Modes


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Chapters 3 through 10 dealt with various aspects of the gain medium. In chapter 7,

we briefly mentioned that mirrors are used at the ends of the laser amplifier in order

to increase the effective length of the amplifier. At the time, we did not connect the

mirrors to the concept of cavity radiation, although the latter point was discussed in

chapter 6 in relation to thermal equilibrium and blackbody radiation.

In this chapter, we shall consider the properties associated with the optical cavity ofa laser that has mirrors on either end of the gain medium; these properties are

significant in determining the output characteristics of the laser beam. We will begin

by discussing the Fabry-Perot optical cavity, which leads to the concept of

longitudinal modes. Then, we will analyze a cavity with mirrors of finite size at the

ends of the amplifier, along with the associated diffraction losses. This will lead to

the development of transverse modes in the laser cavity. The effects of the

longitudinal and transverse modes on the laser properties will be briefly discussed.


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Outline

Table of Contents

1 Longitudinal Laser Cavity ModesFabry-Perot ResonatorFabry-Perot Cavity ModesLongitudinal Laser Cavity Modes and Mode NumberRequirements for the Development of Longitudinal Laser Modes

2 Transverse Laser Cavity ModesFresnel-Kirchhoff Diffraction Integral FormulaTransverse Modes in a Cavity with Plane-Parallel MirrorsTransverse Modes in a Cavity with Curved MirrorsTransverse Mode FrequenciesSingle-Polarization Modes

3 Properties of Laser ModesSpatial DependenceFrequency DependenceMode CompetitionSpectral Hole BurningSpatial Hole Burning


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Longitudinal Laser Cavity Modes

Longitudinal Laser Cavity Modes WTS 11.2

When we enclose an amplifying medium with mirrors, they place boundary

conditions on the EM field of the laser beam. In chapter 6, we studied cavityradiation, and it was mentioned that the electric field must be zero at the reflecting

surfaces of the mirrors.

To begin this chapter, we will analyze the case where a beam of light is incident

upon a two-mirrored cavity - known as a Fabry-Perot resonator- when there are no

optical elements or gain media between the mirrors. We will then consider the effect

of placing an amplifying medium between the mirrors.

Fig. 1: Transmitted and reflected rays when an EM wave arrives at a reflecting surface - see WTS Fig. 11-1


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Fabry-Perot Resonator WTS 11.2

Consider a beam of light with amplitude E0 interacting with a single reflectingsurface, as shown in Fig. 1. We assume initially that the index of refraction is the

same on both sides of the surface. The angle of incidence is . The reflectedamplitude isE0rand the transmitted amplitude isE0t (randtare the amplitude

reflection and transmission coefficients, which lie between 0 and 1. As we are

considering only stable, time-independent waves, we will suppress the time

dependence of the field throughout this development.

Next, we add a second mirror, parallel to the first and separated by a distance d.

This is shown in Fig. 2.The initially transmitted amplitudeE0tpropagates to the

second mirror, where again part of it is reflected and part is transmitted. The

reflected portion (E0tr) propagates back to the first mirror, where it is once more

partially reflected and partially transmitted, and so on.From the figure, we see that the amplitudes reflected backward from the first mirror

are equal toE0r,E0t2r,E0t

2r3,E0t2r5, etc., while the amplitudes transmitted forward

from the second mirror are equal to E0t2,E0t

2r2,E0t2r4, etc. We will return to this

point shortly.


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Fig. 2: Multiple reflections from two reflective surfaces -see WTS Fig. 11-2

Fig. 3: The extra path length of a ray reflected from twosurfaces - see WTS Fig. 11-3

C C


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Fabry-Perot Resonatorcont WTS 11.2

The various field components that are reflected or transmitted from the Fabry-Perot

resonator have a relative phase that is determined by the round-trip path lengthbetween the mirrors. From Fig. 3, we see that this path length is equal to 2dcos.The phase difference between successive reflected or transmitted coefficients is

therefore = kz= 2kdcos = 4dcos/.

We can now write down the total transmitted field amplitude by summing up the

individual terms. The transmitted field is

Et= E0t2 + E0t

2r2ei = E0t2r4ei2 + (11.1)

or,

Et= E0t2(1 + r2ei + r4ei2 + ) = E0t2

n=0

r2nein. (11.2)

The sum represents a convergent geometric series, which can be expressed as

n=0

r2nein = 1

1 r2ei . (11.3)

Therefore, the total transmitted field and intensity can be written as

Et =

E0t2

1 r2ei , I

t= E2

t = E2

0 |t

|4

1 r2ei2 = I0 |t

|4

1 r2ei2 . (11.4)

Ch t 11 L C it M d


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Of course, it is possible that there is a phase change r/2 upon reflection for eachamplitude component. At a dielectric interface, the phase change is either 0

(internal) or (external), while at a metal interface, the phase change can take anyvalue. We incorporate this phase change by writing

r= |r| eir/2. (11.5)Defining the intensity reflection and transmission asR= |r|2 andT= |t|2, we canwriteItas

It= I0T2

|1 Rei|2, where = + r. (11.6)

We can then rewrite the denominator of this equation as

1 Rei2 = (1 Rei)(1 Rei) = 1 Rei Rei + R2 (11.7)= 1 2Rcos + R2

= (1 R)21 +

4R

(1 R)2 sin2

2

.

Chapter 11 Laser Cavity Modes


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If we then set

F = 4R(1 R)2 , (11.8)

then we can write

It= I0T2

(1 R)21

1 + Fsin2(/2). (11.9)

The expression 1/(1 + F sin2(/2)) is referred to as theAiry function(not to beconfused with the other Airy function or the other other Airy function). F is called thecoefficient of finesse.

If we assume that there is no absorption, thenR= 1T, and we have simplyIt

I0 =

1

1 + Fsin2(/2) . (11.10)

This is plotted (as a function of /2) in Fig. 4,for three different values ofR. We seethat the function has a periodic series of maxima (with a value of unity) for

sin(/2) = 0, or /2 = n,n= 0,1, 2, . The minima occur at /2 = (2n+ 1)/2.The minimum values depend onF (and thus onR), but can be very small forreasonably large values ofR.



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Fig. 4: Transmitted intensity from a Fabry-Perot resonator vs. phase change - see WTS Fig. 11-4



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We will refer to the values of that maximizeIt/I0 as max, and therefore

max= 2n= 4

dcos + r. (11.11)

We can obtain the FWHM of the Airy function for large values of R(R>0.6 or so) byapproximating sin(/2) as /2. The value of at which the Airy function reduces to

half of its maximum value will be referred to as . This is obtained by setting1

1 + F(/2)2 =

1

2, (11.12)

which leads to

=

2

F . (11.13)The FWHM is simply twice this value:

FWHM = 2 = 4

F. (11.14)



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More important than the FWHM is the ratio of the separation between peaks to the

FWHM. Because the peaks are separated by = 2, we have

F=

FWHM =

2

4/

F=

F

2 =

R

1 R. (11.15)

Fis referred to as thefinesseof the cavity. In the case that the two mirrors have

different reflectivities,F=

(R1R2)1/4

1 (R1R2)1/2 . (11.16)We are generally more interested in the width and separation of the peaks in terms

of frequency (rather than phase). On p. 377 it is derived that

sep= c2d

, (11.17)

where is the refractive index of the medium between the mirrors. It follows that

FWHM=sep

F

= c(1 R)

2dR. (11.18)



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Fabry-Perot Resonatorcont WTS 11.2The sharpness of a cavitys frequency transmission peaks is described by the

quality factoror Q-factor, which is simply the ratio of their center frequency 0 totheir width:

Q= 0

FWHM=

2d

R0

c(1 R) , (11.19)

or, when the mirror reflectivities differ,

Q=2d(R1R2)

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c[1 (R1R2)1/2] . (11.20)

For most laser cavities, a high Qis desired. This helps to ensure that the spectral

width of the output is narrow - even narrower than the emission linewidth of the gain

medium. We will come back to this point later in the chapter.



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p y


Fabry-Perot Cavity Modes WTS 11.2

Notice that the transmission peaks are equally-spaced in frequency; from eq.

(11.17), we see that they occur at frequencies

maxn = nc

2d (11.21)

(actually, the equal spacing is only valid over small frequency ranges or in the case

that the cavity is filled with gas or vacuum; otherwise, the material dispersion()causes a drift in the separation).

The corresponding peaks in terms of wavelengthoccur at

maxn = 2d

n . (11.22)

While these are not equally-spaced on a scale, they appear to be so for very largevalues ofn.

We can rewrite this equation as

d= n

maxn

2

. (11.23)

This indicates that the peaks occur when an integer number of half-wavelengths fit

into the cavity lengthd; these form standing waves with zero electric field at the

mirrors. Each of these standing waves is called amodeof the cavity.



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p y


Fabry-Perot Cavity Modes WTS 11.2

In order for the resonance condition to be applicable over a large mirror surface, the

mirror quality - the variation in dat different transverse positions within the cavity -

must be less than roughly/10.

Assuming perfect surface quality, it is instructive to consider the ratio of the intensity

insidea F-P cavity to that transmitted through it. For mirror reflectivityR, theintensity reflected from the mirror is R/T= R/(1 R) times greater than that whichis transmitted. The intensity ratio is therefore

IinIt

= 1 + R

1

R. (11.24)

ForR= 99%, this ratio is 199. A Fabry-Perot cavity can therefore serve as anenergy storage device for the cavity modes.



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Longitudinal Laser Cavity Modes and Mode Number WTS 11.2

Now, consider what happens when a gain medium is inserted within a F-P cavity.

When the gain medium is initially pumped, spontaneous emission is emitted in alldirections, across the entire gain bandwidth. However, the photons that are directed

axially (toward the mirrors) are reflected such that they return through the gain

medium. These stimulate emission on their next pass through the amplifier; the

stimulated emission is also directed axially. Eventually, a highly directional beam

evolves in the axial direction; it approximates a plane wave with very low divergence.

Not all wavelengths within the gain bandwidth can build up to a high intensity,however. Only those which satisfy the cavitys boundary conditions are significantly

enhanced. These are termedlongitudinal modesof the laser cavity; they occur at

wavelengths within the gain bandwidth that are an integer multiple of twice the cavity

length.

The frequencies of the longitudinal laser modes are given simply by = nc/2d inthe case that the mirrors are placed immediately at the ends of the gain medium (sothat the refractive index is constant throughout the cavity). For some lasers, the gain

medium only represents a fraction of the cavity. Here, if the gain medium and cavity

have lengthL andd, respectively, with refractive index L andC, then

=

nc

2

1

C(d L) + L L (11.25)



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Requirements for the Development of Longitudinal LaserModes WTS 11.2

Longitudinal modes may develop within any frequency region in which:

the gain within the laser amplifier at that frequency exceeds the losses (chapter

7), and

there exists an integral value of nsuch that the the frequency in question

satisfies the appropriate equation from the previous slide.

In broadband gain media (as with dye lasers or many solid-state lasers), there may

be thousands of longitudinal modes, especially if the cavity lengthdis large. On the

other hand, gas lasers may support very few longitudinal modes - or even just a

single one - due to their narrow gain bandwidth. Figure5shows how two distinct

modes can both satisfy the cavity resonance condition.



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Fig. 5: Diagram of two longitudinal laser modes - see WTS Fig. 11-6



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Requirements for the Development of Longitudinal LaserModes WTS 11.2

It is often the case that not all of the potential longitudinal modes will actually appearin the laser output. In the case of homogeneous broadening, the mode at the

highest value of (gain minus loss) will develop first after the laser is turned on. As

that mode develops, it removes the population from the upper laser level. The result

is thatthe entire gain spectrum will be reduced, because all of the upper level

population contributes equally at any wavelength over the emission spectrum for

homogeneous broadening (section 4.3). Homogeneously broadened laserstherefore often have only one longitudinal mode.

For an inhomogeneously broadened laser, all of the modes that meet the two

requirements on the previous slide will be present, provided that the natural linewidth

is narrower than the separation between modes. The presence of many longitudinal

modes usually leads to the phenomenon of spectral hole burning, which will bedescribed later in this chapter. Figure6describes the longitudinal modes of an

inhomogeneously broadened laser. The top portion of the figure shows the gain as a

function of frequency, as well as the cavity losses (assumed to be

frequency-independent). The middle portion shows the Fabry-Perot resonances of

the cavity. In the bottom portion, these effects are combined, and the possible

modes only occur where there is net gain.



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Fig. 6:Resulting laser cavity modes when a gain bandwidth of a laser amplifier is combined with resonances of a two-mirror

laser cavity - see WTS Fig. 11-7



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Transverse Laser Cavity Modes

Transverse Laser Cavity Modes WTS 11.3In the previous section, we analyzed the effects of two parallel reflecting surfaces of

infinite extent. This led to discrete longitudinal modes of the laser, each resembling a

plane wave. Such a description is not physically accurate, though. In a physical laser

cavity, the mirrors must be of finite extent, so plane wave solutions of the cavity are

not possible. The finite lateral size of the beam will cause it to diffract, leading tolosses within the laser cavity that have not been considered up to now.

Here, we make two modifications to our previous analysis. First, we assume that the

laser mirrors are of finite extent and of circular shape. Also, we will assume that the

source of light originates from the laser amplifier between the mirrors, rather than

from a plane wave incident from outside the cavity. We will begin by assuming that

the mirrors are flat, and then compare these results with those that assume slightlycurved mirrors; it will be seen that the diffraction losses are much lower for certain

mirror curvatures.



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Fresnel-Kirchhoff Diffraction Integral Formula WTS 11.3

We will not present a rigorous development of the Fresnel-Kirchoff diffraction integral

here. You may have encountered it in PC237 (or in PC495A). We simply present theresult that, for a source point that is positioned symmetrically with respect to an

aperture,

UP= ik4

A

UAeikr

r [cos(n, r) + 1] dA . (11.26)

This equation represents the field at a point Pto the right of an aperture A due to a

point sourceSof amplitudeU0 to the left of the aperture, as shown in Fig. 7.Thefactor (n, r) is the angle that the vector r makes with the normal to the apertureplane,n.

Fig. 7: Symbols used in the Fresnel-Kirchhoff diffraction integral formula - see WTS Fig. 11-8



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Transverse Modes in a Cavity with Plane-Parallel Mirrors WTS 11.3

Consider the case of a laser cavity consisting of two parallel, circular mirrors,

separated by a distanced, as shown in Fig. 8. We will evaluate a distribution of light

beginning at various points on the primed mirror, radiating toward the unprimed

mirror, and then reflecting back to a point on the primed mirror.

By the symmetry of the cavity, for a steady-state mode to develop, the amplitude

distribution of the light on the two mirrors must be identical. We therefore consider asource point functionU(x, y) at point (x, y) on the unprimed mirror, which is the sumof the contributions of radiation from all points leaving the primed mirror that arrive at

(x, y). This source pointU(x, y) then radiates back to the primed mirror to arrive atvarious points (x, y) with an amplitude functionU(x, y), after having traveled adistancer, where (from the figure):

r=

d2 + (x x)2 + (y y)2, (11.27)and is defined as the angle between dandr.


T L C i M d


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Fig. 8: Two parallel circular mirrors considered as apertures when applying the Fresnel-Kirchhoff integral formula to a lasercavity - see WTS Fig. 11-9


T L C it M d


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Transverse Modes in a Cavity with Plane-Parallel Mirrors contWTS 11.3

To determine the field distributionU(x, y) that results fromU(x, y), which itselfresults fromU(x, y), it is helpful to unfold the cavity as shown in Fig. 9. This ispossible since, for plane mirrors, sequential images through two mirrors appear as a

successive row of virtual images of the mirror apertures, each spaced by d.

The field distributionU(x, y) is given by the Fresnel-Kirchhoff integral:

U(x, y) = ik4

A

U(x, y) eikr

r (cos + 1)dxdy. (11.28)

Fig. 9: Equivalent aperture description of a two-mirror reflective laser cavity - see WTS Fig. 11-10




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We seek solutions for the case in which the light has bounced back and forcebetween the mirrors many times, so that it has reached a steady state transverse

profile. That is, the field has no further change in shape, although the overall

amplitude can decrease by a constant factor. This factor represents diffractionlosses around the edges of the circular mirrors, and is included in the factor a that

helps to determine the threshold gain in a laser (end of chapter 7).

Therefore, we need to find solutions such thatU andUare proportional for everypoint (x, y) and (x, y) on the two mirrors. This can be expressed by writing the F-Kintegral as (note the misprint in the text)

U(x, y) = U(x, y) =

A

U(x, y)K(x, y, x, y)dxdy, (11.29)

where

K(x, y, x, y) = ik4

(cos + 1)eikr

r . (11.30)

This is anintegral equationin U;K is thekernelof the equation and is theeigenvalue. There are an infinite number of solutionsUnandnto this equation(n= 1,2,3,

). They are referred to as thetransverse modesof the resonator.




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It is important to note thatnis a complex number: n=n ein. nrepresents the

change in amplitude after a round trip, and nrepresents a possible phase shift. Theenergy loss per transit is therefore

energy loss / round trip = 1 n

2. (11.31)

Solutions of the integral equation can be obtained by making a simple approximationfor its kernel:

K(x, y, x, y) = Ceik1(xx+yy), (11.32)

whereCandk1 are constants. A justification for this approximation is beyond the

scope of this course (although its central to the topic of Fourier optics). The integral

equation becomes

U(x, y) = C

A

U(x, y)eik1(xx+yy)dxdy (11.33)

This equation tells us that U(x, y) is its own Fourier transform.




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The simplest function that is its own Fourier transform is the Gaussian function,

U(x, y) = e2/w2 = e(x2+y2)/w2 , (11.34)

where is the radial distance to any point (x, y) from the center of the mirror. w is ascaling constant that represents the value of at which the field is reduced to afraction 1/eof its peak value (intensity is reduced to a fraction 1/e2).

There are in fact an infinite set of equations that are their own Fourier transforms.

They can be written as the products of Hermite polynomials and the Gaussianfunction:

Upq(x, y) = Hp

2x

w

Hq

2y

w

e(x2+y2)/w2 . (11.35)Here,pandqare integers that designate the order of the Hermite polynomials.

Each set of (p, q) represents a specific stable distribution of wave amplitude at oneof the mirrors; that is, a specific transverse mode of the open-walled cavity. TheHermite polynomials are defined by the function

Hm(u) = (1)meu2dm(eu

2)

dum , (11.36)

whereudenotes either 2x/wor 2y/w.




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The first few Hermite polynomials can be written

H0(u) = 1, H1(u) = 2u, H2(u) = 4u2 2, (11.37)

It should be noted that there is another defining equation for Hm(u) that differs veryslightly from eq. (11.36). The two equations lead to identical shapes, but different

scaling. Be aware of this if youre using another source for Hm(u).

Each of the transverse mode distributionsUpq(x, y) is designated as TEMpq, whereTEM stands for transverse electromagnetic. The lowest-order mode (TEM00) is

simply the Gaussian distributione(x2+y2)/w2 .

The Hermite-Gaussian solutions of eq. (11.33) were obtained by solving the integral

equation in Cartesian (x, y) coordinates, which is why they have x ysymmetry. It isalso possible to solve the F-K integral in cylindrical coordinates, resulting in solutions

which have cylindrical symmetry. These form a set of Laguerre-Gaussianmodes,which are now designated by a pair of integers indicating theradialand azimuthal

order. The L-G solutions will not be written out here.

In a laser cavity with perfect cylindrical symmetry, it is the L-G modes which will be

present. The H-G modes require a small degree ofastigmatism in the cavity (in

order to force a preferred orientation of Cartesian axes). One common method of

providing this astigmatism will be mentioned later in this chapter.




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a s e se ase Ca ty odes

Fig. 10: Mode patterns for various transverse laser modes: pure modes in (a) circular symmetry and (b) Cartesiansymmetry - see WTS Fig. 11-14




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y

Transverse Modes in a Cavity with Curved Mirrors WTS 11.3

Our derivation to this point assumed that the mirrors defining the cavity were flat. It

is not difficult to see that this situation leads to a considerable amount of diffraction

loss, particularly if the ratio of cavity length to mirror diameter is large.

This diffraction loss can be reduced considerably simply by curving the mirrors so

that diffraction is balanced by a small degree of focusing. This point will beelaborated upon considerably in chapter 12.

Here (Fig.11), we provide a brief analysis of the difference in diffraction loss

between planar and curved mirrors. The fractional loss for the two lowest-order

modes per round-trip transit is shown as a function of Fresnel numberN= a2/d,whereais the mirror radius. The mirror curvature is such that the cavity is confocal

(chapter 12). Clearly, the curved mirrors result in a reduction in loss of several orders

of magnitude.




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Fig. 11: Fractional power loss per transit vs. Fresnel number for a laser cavity - see WTS Fig. 11-11


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Fig. 12: A simplified description of two distinct transverse laser modes, showing the larger effective path length for ahigher-order mode - see WTS Fig. 11-13




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Single-Polarization Modes WTS 11.3

Each laser cavity moden, , mis actually two modes, representing the twoorthogonal polarizations transverse to the cavity axis. In many cases, we wish to

have a laser output that exhibits high polarization purity. In this case, it is necessary

to include an element in the cavity that provides polarization-dependent loss.

One very efficient arrangement is to use a Brewster-angle window, as shown in

Fig.13.You will recall that, at the Brewster angle, the p polarization (that which lies in

a plane normal to the plane of the window and perpendicular to the direction ofpropagation) exhibits zero reflectivity. The orthogonal s polarization has a non-zero

reflectivity (about 15% for an air-glass interface) at this angle. Therefore, including a

Brewster-angle plane in the laser cavity will produce sufficient

polarization-dependent loss in the cavity to suppress the s polarization, and produce

a highly polarized output beam.

A side-effect of the Brewster-angle window is that it breaks the cylindrical symmetry

of the laser cavity by introducing a slight amount of astigmatism (provided that the

beam is either converging or diverging at the window position). This aids in

producing H-G modes, rather than L-G modes, as described earlier in this chapter.




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Fig. 13: (a) Reflected intensity vs. angle for light reflected from an air-glass interface. (b) Brewster angle window providing

very low reflection loss for light polarized in the plane of the figure - see WTS Fig. 11-15


Properties of Laser Modes


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Spatial Dependence of Laser Modes WTS 11.4For the remainder of this chapter, we will summarize various characteristics of laser

modes. The description will be purely qualitative, as a quantitative analysis is

beyond the scope of this course. In each case, keep in mind that it is possible to

have more than one transverse or longitudinal laser mode oscillating simultaneously

within the laser cavity.

Each mode, with its associated mode number (n, , m) in a two-mirror cavity,represents a distinct standing wave, with zero electric field at the mirrors. They all

have a distinct three-dimensional spatial distribution of laser intensity between the

mirrors that is at least slightly different from that for any other mode, as indicated in

Fig.10.

Although all lasing modes use the same gain medium, they are in fact accessingdifferent spatial regions of the gain medium. Some may experience more gain than

others.




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Frequency Dependence of Laser Modes WTS 11.4

Each mode has a slightly different frequency. This is clear for different longitudinal

modes (eq.11.25). However, even transverse modes with the same nwill have

different frequencies, since they have different optical path lengths through the cavity(Fig.12). Typically, adjacent longitudinal modes (nandn+ 1) have a greaterfrequency difference (eq.11.17) than do two transverse modes with the same

longitudinal mode numbernbut different values of pandq(eq.11.35).




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Mode Competition WTS 11.4

In the case of homogeneous broadening, the waves associated with different modes

within the same gain medium are all competing for the same upper laser level

species. Each mode is attempting to grow toward reaching its saturation intensity bystimulating more emission into its mode.

The mode at the center of the gain profile, where gain is the highest, will reach its

saturation intensity first. This causes the entire gain curve (within the volume of that

mode) to decrease, since every atom in the upper level is affected by that saturation,

according to chapter 8.Thus, it will be difficult for more than one mode to lase. That is, unless the weaker

mode can feed on a spatial region of gain that is distinct from that of the strong

mode. As such, it is common for homogeneously broadened lasers to lase on a

single longitudinal mode but on more than one transverse mode, since the latter

have distinctly different spatial regions.

With inhomogeneous broadening, different longitudinal modes can operateindependently as long as their natural linewidths do not overlap, as they do not

compete for the same upper laser level species. While distinct longitudinal modes

are sufficiently separated in frequency that many can lase, different transverse

modes with the samencan be close enough in frequency that they must compete

for the same upper laser level species. In this case, they must seek gain in different

spatial regions from that of the strongest (usually the TEM00) mode.




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Spectral Hole Burning WTS 11.4

Laser modes that are able to reach saturation intensity can significantly affect thegain within the laser amplifier. For a homogeneously broadened medium, it was

mentioned in chapter 8 that as a longitudinal laser mode develops, the stimulated

emission process will reduce the gain profile to a value at which it equals the losses

within the laser cavity (mirror transmission, absorption and scattering).

In contrast, for inhomogeneous Doppler-broadened media, the population in theupper laser level will be reduced only at the frequencies where the modes are

developing, since different populations within the upper laser level contribute to

different frequency components of the gain spectrum. Thus, if the natural emission

linewidth of the transition is much narrower than the Doppler width (which is the case

for must visible gas lasers), then the gain spectrum while the laser is operating will

have periodic dips according to the positions of the longitudinal modes, as shown inFig.14.This is referred to asspectral hole burningor frequency hole burning.

The holes have a width equal to the natural linewidth and are burned down to the

point where gain is reduced to the value of cavity losses.




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Fig. 14: Laser gain distribution within a laser amplifier due to spectral hole burning - see WTS Fig. 11-16




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Spatial Hole Burning WTS 11.4

When the standing-wave pattern of a single longitudinal mode develops within a

homogeneously broadened gain medium, the laser intensity pattern is periodic, asshown in Fig.15; the periodicity is/2.

As long as the cavity length is very stable and the amplifier gain remains constant,

this pattern is stable. At the null points where the electric field is zero, there is no

stimulated emission and thus no reduction in the gain. Midway between the null

points, the electric field is maximum, and the gain is reduced strongly. This is

referred to asspatial hole burning.

The usual result of spatial hole burning is simply a waste of energy (pump energy is

used to increaseNueverywhere, but at the field nulls, there is no intensity available

to stimulate the emission of coherent photons; eventually, the energy is lost through

spontaneous emission). However, if the laser cavity length dis slightly unstable

(even by thermal vibrations), the laser may flip back and forth among two or morelongitudinal modes. This leads to the phenomenon of mode partition noise.

Spatial hole burning can be eliminated entirely by using a ring cavity, in which the

electric field forms a traveling wave rather than a standing wave (thus eliminating the

field nulls). This type of cavity will be discussed further in chapter 13.




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Fig. 15: Laser gain distribution within a laser amplifier due to spatial hole burning - see WTS Fig. 11-17

Documents

Laser Modes