Chapter 11. Laser Cavity Modes
Chapter 11.
Laser Cavity Modes
Chapter 11. Laser Cavity Modes
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 of
a 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.
Chapter 11. Laser Cavity Modes
Outline
Table of Contents
1
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator
Fabry-Perot Cavity Modes
Longitudinal Laser Cavity Modes and Mode Number
Requirements for the Development of Longitudinal Laser Modes
2
Transverse Laser Cavity Modes
Fresnel-Kirchhoff Diffraction Integral Formula
Transverse Modes in a Cavity with Plane-Parallel Mirrors
Transverse Modes in a Cavity with Curved Mirrors
Transverse Mode Frequencies
Single-Polarization Modes
3
Properties of Laser Modes
Spatial Dependence
Frequency Dependence
Mode Competition
Spectral Hole Burning
Spatial Hole Burning
Chapter 11. Laser Cavity Modes
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 cavity
radiation, 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
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator WTS §11.2
Consider a beam of light with amplitude E0 interacting with a single reflecting
surface, 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 θ. The reflected
amplitude is E0 r and the transmitted amplitude is E0 t (r and t are 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 amplitude E0 t propagates to the
second mirror, where again part of it is reflected and part is transmitted. The
reflected portion (E0 tr ) 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 to E0 r , E0 t 2 r , E0 t 2 r 3 , E0 t 2 r 5 , etc., while the amplitudes transmitted forward
from the second mirror are equal to E0 t 2 , E0 t 2 r 2 , E0 t 2 r 4 , etc. We will return to this
point shortly.
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fig. 3: The extra path length of a ray reflected from two
surfaces - see WTS Fig. 11-3
Fig. 2: Multiple reflections from two reflective surfaces see WTS Fig. 11-2
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator cont’
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 length
between the mirrors. From Fig. 3, we see that this path length is equal to 2d cos θ.
The phase difference between successive reflected or transmitted coefficients is
therefore φ = kz = 2kd cos θ = 4πd cos θ/λ.
We can now write down the total transmitted field amplitude by summing up the
individual terms. The transmitted field is
or,
Et = E0 t 2 + E0 t 2 r 2 e i φ = E0 t 2 r 4 e i 2φ + · · ·
Et = E0 t 2 (1 + r 2 e i φ + r 4 e i 2φ + · · · ) = E0 t 2
∞
X
r 2n e inφ .
(11.1)
(11.2)
n=0
The sum represents a convergent geometric series, which can be expressed as
∞
X
1
r 2n e inφ =
.
(11.3)
1
−
r 2 e iφ
n=0
Therefore, the total transmitted field and intensity can be written as
E0 t 2
|t |4
|t |4
2
2
=
Et =
I
=
E
E
I
,
=
2 .
t
0
t
0
2
1 − r 2 e iφ
1 − r 2 e i φ 1 − r 2 e iφ (11.4)
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator cont’
WTS §11.2
Of course, it is possible that there is a phase change φr /2 upon reflection for each
amplitude component. At a dielectric interface, the phase change is either 0
(internal) or π (external), while at a metal interface, the phase change can take any
value. We incorporate this phase change by writing
r = |r | e i φr /2 .
(11.5)
2
2
Defining the intensity reflection and transmission as R = |r | and T = |t | , we can
write It as
T2
It = I0
(11.6)
, where Φ = φ + φr .
2
|1 − Re i Φ |
We can then rewrite the denominator of this equation as
2
1 − Re i Φ (11.7)
= (1 − Re i Φ )(1 − Re −i Φ ) = 1 − Re i Φ − Re −i Φ + R 2
=
=
1 − 2R cos Φ + R 2
#
"
4R
2 Φ
(1 − R )2 1 +
.
sin
2
(1 − R )2
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator cont’
WTS §11.2
If we then set
4R
,
(1 − R )2
(11.8)
1
T2
.
(1 − R )2 1 + F ′ sin2 (Φ/2)
(11.9)
F′ =
then we can write
It = I0
The expression 1/(1 + F ′ sin2 (Φ/2)) is referred to as the Airy function (not to be
confused with the other Airy function or the other other Airy function). F ′ is called the
coefficient of finesse.
If we assume that there is no absorption, then R = 1 − T , and we have simply
1
It
.
=
′
I0
1 + F sin2 (Φ/2)
(11.10)
This is plotted (as a function of Φ/2) in Fig. 4, for three different values of R . We see
that the function has a periodic series of maxima (with a value of unity) for
sin(Φ/2) = 0, or Φ/2 = nπ, n = 0, 1, 2, · · · . The minima occur at Φ/2 = (2n + 1)π/2.
The minimum values depend on F ′ (and thus on R ), but can be very small for
reasonably large values of R .
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fig. 4:
Transmitted intensity from a Fabry-Perot resonator vs. phase change - see WTS Fig. 11-4
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator cont’
WTS §11.2
We will refer to the values of Φ that maximize It /I0 as Φmax , and therefore
Φmax = 2πn =
4π
d cos θ + φr .
λ
(11.11)
We can obtain the FWHM of the Airy function for large values of R (R > 0.6 or so) by
approximating sin(Φ/2) as Φ/2. The value of Φ at which the Airy function reduces to
half of its maximum value will be referred to as Φ′ . This is obtained by setting
1
1+
F ′ (Φ′ /2)2
=
1
,
2
(11.12)
which leads to
2
Φ′ = √ .
F′
The FWHM is simply twice this value:
4
FWHM = 2Φ′ = √ .
F′
(11.13)
(11.14)
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator cont’
WTS §11.2
More important than the FWHM is the ratio of the separation between peaks to the
FWHM. Because the peaks are separated by ∆Φ = 2π, we have
√
√
π F′
2π
π R
∆Φ
=
=
F=
.
(11.15)
√ =
FWHM
2
1−R
4/ F ′
F is referred to as the finesse of the cavity. In the case that the two mirrors have
different reflectivities,
π(R1 R2 )1/4
F=
.
(11.16)
1 − (R1 R2 )1/2
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
∆νsep =
c
,
2ηd
(11.17)
where η is the refractive index of the medium between the mirrors. It follows that
∆νFWHM =
∆νsep
c (1 − R )
=
√ .
F
2πηd R
(11.18)
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fabry-Perot Resonator cont’
WTS §11.2
The “sharpness” of a cavity’s frequency transmission peaks is described by the
quality factor or Q-factor, which is simply the ratio of their center frequency ν0 to
their width:
√
2πηd R ν0
ν0
Q=
(11.19)
=
,
∆νFWHM
c (1 − R )
or, when the mirror reflectivities differ,
Q=
2πηd (R1 R2 )1/4 ν0
.
c [1 − (R1 R2 )1/2 ]
(11.20)
For most laser cavities, a high Q is 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.
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
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
nc
νnmax =
(11.21)
2ηd
(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 η(ν)
causes a drift in the separation).
The corresponding peaks in terms of wavelength occur at
2d
λnmax =
.
(11.22)
n
While these are not equally-spaced on a λ scale, they appear to be so for very large
values of n.
We can rewrite this equation as
d=n
!
λmax
n
.
2
(11.23)
This indicates that the peaks occur when an integer number of half-wavelengths fit
into the cavity length d ; these form standing waves with zero electric field at the
mirrors. Each of these standing waves is called a mode of the cavity.
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
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 d at different transverse positions within the cavity must be less than roughly λ/10.
Assuming perfect surface quality, it is instructive to consider the ratio of the intensity
inside a F-P cavity to that transmitted through it. For mirror reflectivity R , the
intensity reflected from the mirror is R /T = R /(1 − R ) times greater than that which
is transmitted. The intensity ratio is therefore
Iin
1+R
=
.
It
1−R
(11.24)
For R = 99%, this ratio is 199. A Fabry-Perot cavity can therefore serve as an
energy storage device for the cavity modes.
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
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 all
directions, 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 cavity’s boundary conditions are significantly
enhanced. These are termed longitudinal modes of 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 ν = nc /2ηd in
the case that the mirrors are placed immediately at the ends of the gain medium (so
that 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 length L and d , respectively, with refractive index ηL and ηC , then
ν=
nc
1
2 ηC (d − L ) + ηL L
(11.25)
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Requirements for the Development of Longitudinal Laser
Modes 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 n such 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 length d is 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. Figure 5 shows how two distinct
modes can both satisfy the cavity resonance condition.
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Fig. 5:
Diagram of two longitudinal laser modes - see WTS Fig. 11-6
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
Requirements for the Development of Longitudinal Laser
Modes WTS §11.2
It is often the case that not all of the potential longitudinal modes will actually appear
in 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 that the 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 lasers
therefore 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 be
described later in this chapter. Figure 6 describes 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.
Chapter 11. Laser Cavity Modes
Longitudinal Laser Cavity Modes
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
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Laser Cavity Modes WTS §11.3
In 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 to
losses 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 slightly
curved mirrors; it will be seen that the diffraction losses are much lower for certain
mirror curvatures.
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
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 the
result that, for a source point that is positioned symmetrically with respect to an
aperture,
"
ik
e ikr
[cos(n, r) + 1] dA .
UP = −
UA
(11.26)
4π
r
A
This equation represents the field at a point P to the right of an aperture A due to a
point source S of amplitude U0 to the left of the aperture, as shown in Fig. 7. The
factor (n, r) is the angle that the vector r makes with the normal to the aperture
plane, n.
Fig. 7:
Symbols used in the Fresnel-Kirchhoff diffraction integral formula - see WTS Fig. 11-8
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
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 distance d , 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 a
source point function U (x , y ) at point (x , y ) on the unprimed mirror, which is the sum
of the contributions of radiation from all points leaving the primed mirror that arrive at
(x , y ). This source point U (x , y ) then radiates back to the primed mirror to arrive at
various points (x ′ , y ′ ) with an amplitude function U ′ (x ′ , y ′ ), after having traveled a
distance r , where (from the figure):
p
r = d 2 + (x ′ − x )2 + (y ′ − y )2 ,
(11.27)
and θ is defined as the angle between d and r .
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Fig. 8:
Two parallel circular mirrors considered as apertures when applying the Fresnel-Kirchhoff integral formula to a laser
cavity - see WTS Fig. 11-9
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Modes in a Cavity with Plane-Parallel Mirrors cont’
WTS §11.3
To determine the field distribution U ′ (x ′ , y ′ ) that results from U (x , y ), which itself
results from U ′ (x ′ , y ′ ), it is helpful to “unfold” the cavity as shown in Fig. 9. This is
possible 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 distribution U ′ (x ′ , y ′ ) is given by the Fresnel-Kirchhoff integral:
"
ik
e ikr
U ′ (x ′ , y ′ ) = −
U (x , y )
(cos θ + 1)dxdy .
4π
r
A
Fig. 9:
Equivalent aperture description of a two-mirror reflective laser cavity - see WTS Fig. 11-10
(11.28)
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Modes in a Cavity with Plane-Parallel Mirrors cont’
WTS §11.3
We seek solutions for the case in which the light has bounced back and force
between 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 γ. This factor represents diffraction
losses 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 that U ′ and U are proportional for every
point (x ′ , y ′ ) and (x , y ) on the two mirrors. This can be expressed by writing the F-K
integral as (note the misprint in the text)
"
U ′ (x ′ , y ′ ) = γU (x , y ) =
U (x , y )K (x , y , x ′ , y ′ )dxdy ,
(11.29)
A
where
ik
e ikr
.
(11.30)
(cos θ + 1)
4π
r
This is an integral equation in U ; K is the kernel of the equation and γ is the
eigenvalue. There are an infinite number of solutions Un and γn to this equation
(n = 1, 2, 3, · · · ). They are referred to as the transverse modes of the resonator.
K (x , y , x ′ , y ′ ) = −
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Modes in a Cavity with Plane-Parallel Mirrors cont’
WTS §11.3
It is important to note that γn is a complex number: γn = γn e i φn . γn represents the
change in amplitude after a round trip, and φn represents a possible phase shift. The
energy loss per transit is therefore
2
(11.31)
energy loss / round trip = 1 − γn .
Solutions of the integral equation can be obtained by making a simple approximation
for its kernel:
′
′
(11.32)
K (x , y , x ′ , y ′ ) = Ce −ik1 (xx +yy ) ,
where C and k1 are constants. A justification for this approximation is beyond the
scope of this course (although it’s central to the topic of Fourier optics). The integral
equation becomes
"
′
′
γU (x , y ) = C
U (x , y )e −ik1 (xx +yy ) dxdy
(11.33)
A
This equation tells us that U (x , y ) is its own Fourier transform.
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Modes in a Cavity with Plane-Parallel Mirrors cont’
WTS §11.3
The simplest function that is its own Fourier transform is the Gaussian function,
2
2
2
2
2
(11.34)
U (x , y ) = e −ρ /w = e −(x +y )/w ,
where ρ is the radial distance to any point (x , y ) from the center of the mirror. w is a
scaling constant that represents the value of ρ at which the field is reduced to a
fraction 1/e of its peak value (intensity is reduced to a fraction 1/e 2 ).
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 Gaussian
function:
√ 
√ 
 2x 
 2y  −(x 2 +y 2 )/w 2


 e
.
Upq (x , y ) = Hp 
(11.35)
 Hq 
w   w 
Here, p and q are integers that designate the order of the Hermite polynomials.
Each set of (p , q) represents a specific stable distribution of wave amplitude at one
of the mirrors; that is, a specific transverse mode of the open-walled cavity. The
Hermite polynomials are defined by the function
2
Hm (u) = (−1)m e u
where u denotes either
√
√
2x /w or 2y /w .
2
d m (e −u )
,
dum
(11.36)
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Modes in a Cavity with Plane-Parallel Mirrors cont’
WTS §11.3
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 very
slightly from eq. (11.36). The two equations lead to identical shapes, but different
scaling. Be aware of this if you’re using another source for Hm (u).
Each of the transverse mode distributions Upq (x , y ) is designated as TEMpq , where
TEM stands for “transverse electromagnetic.” The lowest-order mode (TEM00 ) is
2
2
2
simply the Gaussian distribution e −(x +y )/w .
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 − y symmetry. It is
also possible to solve the F-K integral in cylindrical coordinates, resulting in solutions
which have cylindrical symmetry. These form a set of Laguerre-Gaussian modes,
which are now designated by a pair of integers indicating the radial and 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 of astigmatism 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.
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Fig. 10:
Mode patterns for various transverse laser modes: pure modes in (a) circular symmetry and (b) Cartesian
symmetry - see WTS Fig. 11-14
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
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 be
elaborated 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 number N = a 2 /λd ,
where a is 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.
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Fig. 11:
Fractional power loss per transit vs. Fresnel number for a laser cavity - see WTS Fig. 11-11
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Transverse Mode Frequencies WTS §11.3
Earlier in this chapter, we described longitudinal modes as those which have the
same optical path within the cavity but with slightly different frequencies - depending
on the mode number n - such that the electric field was zero at the mirrors.
Different transverse modes are modes which have different values of p and/or q
(regardless of n). Transverse modes with the same n but different p and q will have
slightly different optical path lengths ηd , owing to slightly different angular
distributions of the amplitude functions within the cavity. Because they still must
satisfy the boundary equations on electric field, these transverse modes will each
have a slightly different frequency as well; their effective cavity lengths d are
different. This is illustrated in Fig. 12.
In general, the frequency difference between longitudinal modes is significantly
greater than that between transverse modes.
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Fig. 12:
A simplified description of two distinct transverse laser modes, showing the larger effective path length for a
higher-order mode - see WTS Fig. 11-13
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
Single-Polarization Modes WTS §11.3
Each laser cavity mode n, ℓ, m is actually two modes, representing the two
orthogonal 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 of
propagation) 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.
Chapter 11. Laser Cavity Modes
Transverse Laser Cavity Modes
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
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
Spatial Dependence of Laser Modes WTS §11.4
For 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, ℓ, 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 accessing
different spatial regions of the gain medium. Some may experience more gain than
others.
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
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 n will have
different frequencies, since they have different optical path lengths through the cavity
(Fig. 12). Typically, adjacent longitudinal modes (n and n + 1) have a greater
frequency difference (eq. 11.17) than do two transverse modes with the same
longitudinal mode number n but different values of p and q (eq. 11.35).
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
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 by
stimulating 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 operate
independently 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 same n can 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.
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
Spectral Hole Burning WTS §11.4
Laser modes that are able to reach saturation intensity can significantly affect the
gain 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 the
upper 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 in
Fig. 14. This is referred to as spectral hole burning or 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.
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
Fig. 14:
Laser gain distribution within a laser amplifier due to spectral hole burning - see WTS Fig. 11-16
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
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, as
shown in Fig. 15; the periodicity is λ/2η.
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 as spatial hole burning.
The usual result of spatial hole burning is simply a waste of energy (pump energy is
used to increase Nu everywhere, 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 d is slightly unstable
(even by thermal vibrations), the laser may flip back and forth among two or more
longitudinal 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.
Chapter 11. Laser Cavity Modes
Properties of Laser Modes
Fig. 15:
Laser gain distribution within a laser amplifier due to spatial hole burning - see WTS Fig. 11-17