mode samples a different area on the mirrors, the
differences of their respective ringdown times can
be associated with a spatial reflectivity distribution
function, which provides a measure for the inhomo-
geneity of the mirror coatings. Conversely, one can
use the differences in optical loss associated with
the respective transverse cavity modes to create a
“reflectivity map” of the mirrors–keeping in mind
that the contributions from the two mirrors cannot
be easily separated. Although of large historical
importance [
3,19], the coating inhomogeneity of
high-reflectivity mirrors has rarely been investi-
gated, aside from several early studies [
4] including
the first application of phase-shift CRD to the char-
acterization of mirror coatings by Herbelin et al. [
3].
In this report, we examine the influence of coating
inhomogeneity on the CRD measurement. In par-
ticular, we use the ringdown times of different trans-
verse cavity modes to creat e a “reflectivity map” for
the mirrors. We show that one can synthesize a
reflectivity map from a sum of TEM
m;n
basis func-
tions or from the spatial intensity distributions asso-
ciated with each experimentally obtained transverse
cavity mode. This may be regarded as an analogy to
the synthesis of a waveform using a sum of weighted
Fourier components.
2. Theory
The transverse mode pattern of cavity modes in
a two-mirror cavity can be described either in a
Cartesian coordinate system or in a cylindrical coor-
dinate system. In the former case Hermite–Gaussian
transverse mode patterns and their linear combina-
tions give an adequate representation of the inten-
sity distribution inside a cavity. The normalized
Hermite–Gaussian functions of order m and n can
be described as [
20–22]
HG
m
x
2
πw
2
1∕4
1
2
m
m!
p
H
m
2
p
x
w
exp
−x
2
w
2
m 0; 1; 2; 3; …
HG
n
y
2
πw
2
1∕4
1
2
n
n!
p
H
n
2
p
y
w
exp
−y
2
w
2
n 0; 1; 2; 3; …; (1)
where H
m
x and H
n
y are Hermite polynomial
functions of order m and n, respectively, and w is
the beam waist of the lowest order mode. The inten-
sity distribution of each TEM
m;n
mode can be calcu-
lated from the Hermite–Gaussian basis functions
I
m;n
x; yI
0
m;n
HG
m
x
2
HG
n
y
2
: (2)
The normalization factors in Eq. (
1) assure that for
each mode
1
Z
∞
x−∞
HGx
2
dx
Z
∞
y−∞
HGy
2
dy: (3)
And therefore,
I
0
I
0
m;n
Z
∞
x;y−∞
I
m;n
x; ydxdy: (4)
In the analysis of the experimental data we scale the
spatially integrated intensity of each mode to the
same value, I
0
. Examples of calculated I
m;n
x; y
functions are given in Fig.
1.
We intend to obtain information about the mirror
coating’s homogeneity from a “reflectivity map.” The
transverse modes on each of the mirrors can be added
incoherently and in the limiting case of a very large
number of transverse modes the sum of all their
intensities covers the mirror surface smoothly with
a maximum on the cavity axis:
¯
Ix; y
P
N;M
n;m0
I
m;n
x; y
N 1M 1
: (5)
The size of the area that can be inte rrogated in this
manner depends on the size of the highest order
HG-mode, which is about 4w
2
N 1
1∕2
M 1
1∕2
[22]. It is well-known that Hermite–Gaussian func-
tions form a complete basis set [
22], and the interrog-
ation area as well as the spatial resolution can be
increased by including more modes in the analysis.
In a practical application the number of available
mode functions is finite. We used N M 6 for 49
TEM modes in the simulation and a total of 13 or 15
modes in the present experiments. As indicated in
Eq. (
4), we give each mode the same weight.
When the mirror surface exhibits defects in the
coating, the reflectivity of each mode is reduced.
Figure
2(A) shows the location of a simulated defect
as the product,
¯
Ix; y · Rx; y, which is equivalent to
n,0 n,1 n,2 n,3 n,4 n,5 n,6
0,m
1,m
2,m
3,m
4,m
5.m
6,m
Fig. 1. Simulated intensity distribution functions for a set of
Hermite–Gaussian transverse cavity modes.
2918 APPLIED OPTICS / Vol. 53, No. 13 / 1 May 2014