Download Absorption of low-loss optical materials measured at 1064 nm by a

Absorption of low-loss optical materials measured
at 1064 nm by a position-modulated collinear
photothermal detection technique
Vincent Loriette and Claude Boccara
A collinear photothermal detection bench is described that makes use of a position-modulated heating
source instead of the classic power-modulated source. This new modulation scheme increases by almost
a factor 2 the sensitivity of a standard mirage bench. This bench is then used to measure the absorption
coefficient of OH-free synthetic fused silica at 1064 nm in the parts per 106 range, which, combined with
spectrophotometric measurements, confirms that the dominant absorption source is the OH content.
© 2003 Optical Society of America
OCIS codes: 160.6030, 160.4760, 120.3940.
1. Introduction
Large interferometric detectors of gravitational
waves are among the first instruments for which optical absorption inside substrates may become a limitation.1,2 As those instruments are highly sensitive
to thermal deformation induced by absorption inside
components that are crossed by the laser beam, the
absorption levels of the various substrates should be
lower than 2 parts in 106 共ppm兲 per centimeter at the
wavelength of Nd:YAG, from which the laser used in
these detectors is made. Previous studies showed
that the best material for the first generation of gravitational wave detectors is synthetic fused silica.3
For many years4 large efforts have been made successfully to decrease the absorption of the best silica
below 1 ppm兾cm. At the Nd:YAG wavelength the
main cause of absorption in high-purity fused silica is
the presence of hydroxyl groups. The absorption
band at 1380 nm is sufficiently strong 关␣OH共1380 nm兲
⬇ 63 dB兾km ppm⫺1兴 to have a dominant effect at
1064 nm. Today the absorption level of OH-free
fused silica has decreased below 0.5 ppm兾cm at 1064
The authors are with the École Supérieure de Physique et
Chimie Industrielles de la Ville de Paris, Laboratoire d’Optique
Physique, Centre National de la Recherche Scientifique, Unité
Propre de Recherche 5, 10 rue Vauquelin, 75005 Paris, France. V.
Loriette’s e-mail address is [email protected].
Received 21 May 2002; revised manuscript received 1 October
2002.
0003-6935兾03兾040649-08$15.00兾0
© 2003 Optical Society of America
nm and continues to decrease. Detection of such a
low absorption level is difficult because the thickness
of the silica substrates shaped to become optical components is usually not larger than a few centimeters,
so the total absorption integrated over the sample
thickness never exceeds a few ppm. This level is of
the same order of magnitude as the scattering losses
of superpolished surfaces and also of the same order
of magnitude as the 0.7-dB兾km Rayleigh scattering
losses of fused silica at 1064 nm. To detect sub-ppm
absorption, a highly sensitive photothermal deflection instrument must be used. In this paper we describe the instrument that is used for measurements
of bulk absorption below the 1-ppm level and evaluate the level of OH contamination. This instrument
is an improved and optimized version of a mirage
bench.5 First we describe this bench, focusing our
attention on the original part of the bench. We also
recall the exact expression for the deviation of a probe
beam in collinear photothermal deflection geometry;
the derivation of this equation is presented in Appendix A. We then describe the current state of the art
in low-loss silica manufacturing by showing results of
absorption and OH contamination measurements
performed on various fused-silica samples.
2. Collinear Mirage Bench
A schematic of the setup is presented in Fig. 1. The
geometry of this setup is similar to that of the original
mirage scheme,5 except for the heat source modulation. The sample is heated by a P0 ⫽ 24-W TEM00
Nd:YAG laser modulated pump beam. The pump
beam is focused inside the sample with a two-mirror
1 February 2003 兾 Vol. 42, No. 4 兾 APPLIED OPTICS
649
of the lateral displacement is chosen to maximize the
mirage signal, and we obtain it by optimizing angular
displacement ␪ of the tilted mirror. Its optimal
value is given by the formula
␪ opt ⫽ 2
Fig. 1. Schematic of the collinear mirage setup. To modulate the
pump beam, either a mechanical chopper is inserted between the
two mirrors of the periscope or the second periscope mirror is tilted
by use of a piezo-motorized mirror holder. The 24-W Nd:YAG
pump beam is focused inside the sample by an f ⫽ 200 mm lens
共L1兲. The 1-mW He–Ne probe beam is focused by an f ⫽ 80 mm
lens 共L2兲. An interference filter 共F兲 and a mirror protect the twoquadrant detector 共D兲 from spurious light.
periscope and a 5-D lens and creates a modulated and
nonuniform temperature increase that is proportional to absorption coefficient ␣ of the sample. The
pump beam’s waist is w ⫽ 100 ␮m, which gives a
confocal parameter of zr ⫽ 59 mm. Usually one
modulates the beam power, for example, with a mechanical or an electro-optic chopper,2 but it can be
shown 共see Appendix A兲 that modulating the pump
beam’s position inside the sample while maintaining
its power constant generates a larger signal but with
a different signature, as can be seen from Fig. 2. We
obtain the position modulation of the pump beam by
tilting one of the periscope mirrors with a piezo actuator. The rotation of the beam axis in front of
the focusing lens gives a lateral displacement x0 of
the heat-source inside the sample. The amplitude
0.597w
,
f
(1)
which is independent of the modulation frequency.
In our setup its value is ␪opt ⫽ 0.6 mrad, which permits a sensitivity gain of a factor of 1.84 共see Appendix A兲 compared with that of the chopped beam while
the amount of spurious light in the setup as well as
the level of vibration is decreased. One can understand intuitively that the signal is larger by nearly a
factor of 2 just because all the available power from
the pump laser is used to heat the sample, as is not
the case when power modulation is used. In the
latter case half of the power is lost, either reflected by
the mechanical chopper or trapped in a polarizer.
When sub-ppm absorption samples are tested, multiplication by 1.84 of the signal amplitude is a significant improvement because, as can be seen from Fig.
3, the signal-to-noise ratio is less than 10. A low
power He–Ne probe laser, focused by a 12.5-D lens,
crosses the heated region and is periodically deflected
by the thermal gradient. The probe beam’s waist is
40 ␮m in the sample. The probe laser’s deviation ␾
is recorded by a two-quadrant photodetector followed
by a lock-in amplifier. At modulation frequencies
that satisfy the relation
␻ ⬎⬎ D兾w 2,
(2)
where D is the thermal diffusivity of the sample 共D ⬇
0.84 ⫻ 10⫺6 m2 s⫺1 for fused silica兲 and w is the pump
beam’s radius, the propagation distance of the heat
wave inside the sample during one modulation pe-
Fig. 2. 共a兲 Theoretical and 共b兲 experimental mirage signal amplitude 共solid curves兲 chopped-pump beam and 共dashed curves兲 positionmodulated pump beam.
650
APPLIED OPTICS 兾 Vol. 42, No. 4 兾 1 February 2003
Fig. 3. Typical mirage signal recorded with 共a兲 a position-modulated and 共b兲 a chopped pump beam. The sample absorption is ␣ ⫽ 0.6
ppm兾cm. The absorption is calculated by comparison of these data to those for the same measurement performed with a reference sample.
riod, ␮ ⫽ 共2D兾␻兲1兾2, is negligible compared with the
pump beam’s waist, and the heat source is confined
inside the volume defined by the pump beam. The
deviation inside the sample is then given by the approximate formula, which is rigorously valid at high
frequency,
␾⫽
⫺1 dn ␣P 0 1 D
n dT ␬ sin ␤ ␻
1
冑2␲
g共 x兲,
(3)
where x is the distance between the probe beam and
the pump beam, ␻ ⫽ 2␲f, ␤ is the angle between the
two beams, ␬ is the thermal conductivity, and n is the
index of refraction of the sample. For a chopped
beam,
g共 x兲 ⫽
冉 冊
⫺8ix
⫺2x 2
;
3 exp
w
w2
(4)
two silica standards have absorption levels of approximately 300 and 100 ppm兾cm. The use of different
types of material, water and then fused silica, for our
standards makes the calibration procedure sensitive
to uncertainties in measured geometrical parameters, such as angles of incidence and beam waists,
and adds the uncertainties of optical and thermal
parameter values, such as thermal conductivities and
diffusivities. The sum of those added uncertainty
sources leads us to estimate the measurement accuracy to a conservative value of 20%. The bench sensitivity is limited mainly by technical noise and
degraded by a continuous background signal. The
background signal is generated by spurious light
from the pump beam that strikes the detector. As
an order of magnitude, for a given value of the required sensitivity the maximum amount of unwanted
light on the detector, ␦P, is given by the approximate
expression
and for a position-modulated beam,
冉 冊 冉 冊兺 冋 冉 冊
冉 冊册冋 冉 冊 冉 冊
冉 冊册
8
⫺2x 2
⫺x 02
g共 x兲 ⫽ i 3 exp
exp
w
w2
w2
⫺ I p⫹1
2
0
2
x
w
⫹ x 0 I 2p⫹2
x 0 I 2p
4x 0 x
w2
⬁
p⫽0
x 02
共⫺1兲 p I p 2
w
4x 0 x
4x 0 x
⫺ 2xI 2p⫹1
2
w
w2
.
(5)
Ip are the modified Bessel functions, and the amplitude of the pump beam’s lateral displacement in the
sample is x0. The derivation of Eqs. 共3兲–共5兲 is given
in Appendix A.
3. Calibration and Performances
As can be seen from Eqs. 共3兲–共5兲, the mirage signal
depends on many optical, thermal, and geometrical
parameters, so an absolute measurement is hardly
achievable at a reasonable level of accuracy. We
need to compare our results with known fused-silica
absorption standards. Those standards are in turn
characterized by comparison of their absorption coefficients with pure water at low pump power. Our
␦P ⬍⬍
V ␣ ␩ probe
P probe,
V dc ␩ pump
(6)
where Vdc is the dc voltage on the detector, V␣ is the
signal voltage that corresponds to the required sensitivity, ␩ are the quantum efficiencies of the detector
at the pump beam and the probe beam wavelengths,
and Pprobe is the probe beam’s power on the detector.
In our setup this level is ␦P ⬍ 10 nW for a 1-ppm兾cm
sensitivity. To limit the last-named effect the detector is protected by a high-reflectance mirror at 1064
nm with a transmittance of 10⫺4, followed by an interference filter. Technical noise is dominated by
beam wander induced by turbulence. Beam wander
is limited by use of a compact setup and choice of a
high modulation frequency. The higher the frequency, the lower the turbulence noise but the lower
the signal because of the 1兾␻ term in Eq. 共3兲. The
choice of modulation frequency is purely experimental: As long as Eq. 共2兲 is fulfilled for the samples as
well as for the standards, the only criterion for this
choice is the optimization of the signal-to-noise ratio.
We work at 130 Hz and obtain a 0.1-ppm兾cm equivalent noise at this frequency. Length l over which
1 February 2003 兾 Vol. 42, No. 4 兾 APPLIED OPTICS
651
the probe beam is significantly affected by the index
gradient is roughly equal to
l⬇
2w
.
sin ␤
(7)
So choosing a small value for ␤ should increase the
signal. However, the interaction length has to be
smaller than the sample thickness; otherwise surface
absorption may mask the small volume absorption
signal. We use an angle ␤ ⫽ 10° inside the sample,
which gives an interaction length l ⫽ 1.7 mm that is
suitable for testing centimeter-thick samples.
4. Results
We used the collinear mirage technique to evaluate
the absorption coefficients of various grades of Suprasil fused-silica samples. Figure 3 shows typical
measurements recorded on one of our best samples,
which had an absorption coefficient of 0.6 ppm兾cm at
1064 nm. The two measurements differ in the
choice of the modulation scheme. The 0.1-ppm兾cm
signal offset in the figure was induced by some light
coming from the pump laser. We calculated that
this offset was created by 2 nW of Nd:YAG radiation
striking one quadrant of the detector 共of a total of 12
W incident upon the sample兲.
To evaluate signal amplitude, we first measure our
100-ppm兾cm standard and fit the signal voltage over
the dc voltage ratio with a four-parameter function;
the parameters are signal offset ys, position of the
axis of symmetry xs, amplitude As, and distance between two maxima of the data ws. We then make
the same fit on our sample data, using the fact that
two of those four parameters, namely, x and w, have
the same value, so only a two-parameter fit is necessary for evaluating ys and sample signal amplitude
As. The sample absorption coefficient is then given
by ␣sample ⫽ ␣standardAsampleAs⫺1.
The main source of absorption at the Nd:YAG wavelength is the presence of OH ions. We verified the
correlation between the absorption coefficient and the
OH contamination, using 15 samples from different
batches with absorption ranging from 0.4 to 4 ppm兾cm.
We used Suprasil 311 and Suprasil 311SV samples.
Suprasil 311SV has a lower OH content than standard
Suprasil 311 silica and is optimized for use at 1064 nm.
For each sample we made four to eight measurements
at different positions and different depths. The mirage signal induced by volume absorption in a homogeneously absorbing sample does not depend on the
depth at which the measurement is performed. However, if spurious surface absorption is present, for example because of surface contamination, the
corresponding mirage signal’s amplitude and phase
depend on the distance between the two beams on the
sample surface, so the contribution of spurious surface
absorption to the total mirage signal depends on the
measurement position. The variable depth measurement allows us to verify that the signal induced by
volume absorption is not significantly affected by spurious surface absorption. We evaluated the OH con652
APPLIED OPTICS 兾 Vol. 42, No. 4 兾 1 February 2003
Fig. 4. Absorption coefficient versus OH content in various Suprasil 311 and Suprasil 311SV samples.
tamination by measuring the sample transmittance
and absorption coefficient ␣cm⫺1 at the fundamental
absorption band wavelength of 2.72 ␮m with a spectrophotometer 共Cary Model 5E兲. Provided that the
only source of absorption at this wavelength is OH, and
using a value of extinction in fused silica of ␣OH ⫽ 104
dB兾km ppm,6 we obtained the approximation
OHppm ⬇ 43 ⫻ 10 ⫺6 ␣ cm⫺1
(8)
at 2.72 ␮m. The result is presented in Fig. 4. Several kinds of information can be extracted from this
result: First, it allows us to evaluate the residual
共not induced by OH兲 absorption of our silica samples
to 0.2 ⫾ 0.2 ppm兾cm, which will give a minimum
value of the absorption factor at 1064 nm if only the
OH content is reduced. Then a second, very important, result can be drawn: Within a given grade of
fused silica all measurements are in close agreement,
although we performed our measurements within a
2-year period, with samples extracted from different
batches. This result indicates that a stable level of
performance has been obtained at a high level and is
of major importance for achievement of large sets of
optical components that are manufactured one after
another and need to share the same quality level.
5. Conclusion
The absorption coefficient of OH-free fused silica was
found to have dropped below 5 ⫻ 10⫺7 cm⫺1 at 1064
nm. The absorption level varies from one fusedsilica grade to the other, but within a given grade the
absorption coefficient is constant. Those two results
show that fused silica satisfies the needs of the first
generation of interferometric gravitational wave detectors.
Appendix A. Derivation of the Mirage Signal
In volume absorption a sinusoidally position- or
intensity-modulated pump beam is focused inside the
sample and the probe beam forms a small angle ␤
of the pump beam during its propagation inside the
sample. The source term can be written as
P ␻共 x, y, z兲 ⫽ ␣P 0
冉 冊 冋
册
2
⫺2y 2
⫺2
exp
exp
共x
2
2
␲w
w
w2
⫺ x 0 cos ␻t兲 2 .
(A7)
For a chopped pump beam the calculation is the
same, except that the source term in Eq. 共A7兲 should
be replaced by
P ␻共 x, y, z兲 ⫽ ␣
Fig. 5. Geometry of the beams inside the samples projected on the
xOy, xOz, and yOz planes. The average distance between the probe
beam and the oscillating pump beam is x, x0 is the amplitude of the
pump beam oscillations, and ␤ is the angle between the two beams.
i␻
1
T ⫹ P ␻ ⫽ 0,
D
␬
(A1)
where T is the time Fourier component of the temperature at modulation frequency ␻. We make use
of three-dimensional spatial Fourier transforms:
⫺3兾2
⌰ ⬅ 共2␲兲
⫺3兾2
T ⫽ 共2␲兲
兰
兰
P ␻共 x, y, z兲 ⫽ ␣P 0
冋
2
⫺2共 x 2 ⫹ y 2兲
exp
␲w 2
w2
冉 冊 冋
冋
册
冋
exp
⫺x 02
4x 0 x
cos共␻t兲
2 exp
w
w2
⫻ exp
⫺x 02
cos共2␻t兲 .
w2
册 冉
⬁
P ␻ exp i共 x␭ x ⫹ y␭ y ⫹ z␭ z兲dxdydz.
(A4)
Using
冋
exp
i␻
q ⫽␭ ⫹␭ ⫹␭ ⫹
D
2
x
2
y
2
z
⫺q 2⌰ ⫹
1 ␻
⌸ ⫽ 0.
␬
1. Source Term in Fourier Space
The pump beam is position modulated along the x
direction, and its axis is collinear to the z axis. In
our experiment the absorption coefficient is very low,
about 1 ppm兾cm, so we may neglect the attenuation
p
册 冉 冊
⫺i
4x 0 x
w2
冊
冉 冊
⬁
兺
i pJ p i
p⫽1
x 02
w2
⫻ cos 共2p␻t兲,
(A10)
or by use of the modified Bessel functions 关Ref. 7, Eq.
共8.406.3兲兴 In关 Jn共iz兲 ⫽ inIn共z兲兴 and the fact that Jn共⫺z兲
⫽ 共⫺1兲nJn共z兲:
冋
册 冉 冊
4x 0 x
4x 0 x
2 cos 共␻t兲 ⫽ I 0
w
w2
⬁
⫹2
冋
exp
(A6)
p
x 02
⫺x 02
2 cos 共2␻t兲 ⫽ J 0 i
w
w2
(A5)
in the Fourier equation, we obtain
冊
冉
⫻ cos 共 p␻t兲,
⫹2
exp
2
兺iJ
p⫽1
(A3)
兰
册
(A9)
4x 0 x
4x 0 x
cos 共␻t兲 ⫽ J 0 ⫺i
w2
w2
The spatial Fourier transform of the source term is
⌸ ␻ ⫽ 共2␲兲 ⫺3兾2
册
The terms cos共␻t兲 and cos共2␻t兲 can be developed in
Bessel series 关Ref. 7, Eq. 共8.511.4兲兴:
(A2)
⌰ exp关⫺i共 x␭ x ⫹ y␭ y ⫹ z␭ z兲兴d␭ xd␭ yd␭ z.
(A8)
⫻ exp
⫹2
T exp i共 x␭ x ⫹ y␭ y ⫹ z␭ z兲dxdydz,
册
Equation 共A7兲 can be developed in
with the pump. A schematic of the beam geometry
is presented in Fig. 5. To solve the heat equation we
include the source term at modulation frequency P␻
in the time Fourier transform of the heat equation:
ⵜ 2T ⫺
冋
2
2
⫺2共 x 2 ⫹ y 2兲
exp
.
P0
␲
␲w 2
w2
兺I
册 冉 冊
p⫽1
p
冉 冊
4x 0 x
cos 共 p␻t兲,
w2
⫺x
x 02
cos 共2␻t兲 ⫽ I 0 2 ⫹ 2
w
w
2
0
2
⬁
兺 共⫺1兲 I
p
p
p⫽1
⫻ cos共2p␻t兲.
冉 冊
x 02
w2
(A11)
The Fourier ␻ source term is thus the sum of the
cos共␻t兲 terms that come from the product of the
cos共 p␻t兲 terms of the first sum with the cos共2q␻t兲
terms of the second sum, with p ⫽ 2q ⫹ 1 or p ⫽ 2q ⫺
1. The first terms are
1 February 2003 兾 Vol. 42, No. 4 兾 APPLIED OPTICS
653
2I 0
冉 冊冉 冊 冉 冊冉 冊
冉 冊冉 冊 冉 冊冉 冊
冉 冊冉 冊 冉 冊冉 冊
and q is given by Eq. 共A5兲 with ␭z ⫽ 0:
x 02
4x 0 x
x 02
4x 0 x
I
⫺
2I
,
1
1
2
2
2 I1
w
w
w
w2
兰
x 02
4x 0 x
x 02
4x 0 x
⫺2I 1 2 I 3
⫹
2I
,
2
2
2 I3
w
w
w
w2
x 02
4x 0 x
x 02
4x 0 x
⫹2I 2 2 I 5
⫺ 2I 3 2 I 5
....
2
w
w
w
w2
⫽ ␪共␭ x, ␭ y兲.
(A12)
We write these terms in a more compact form, using
the definition
⌫p
冉 冊 冉 冊 冉 冊
x 02
x 02
x 02
⫽
I
⫺
I
,
p
p⫹1
w2
w2
w2
as
⬁
2
兺 共⫺1兲 ⌫ I
p
p 2p⫹1
p⫽0
冉 冊
(A14)
⌺共␭ y兲 ⫽ F
兰
⫹⬁
⫽
1
2␲
,
exp ⫺
(A16)
␾⫽
1 dn
␾⫽
n dT
1 dn
␾⫽
n dT
兰
兰
兰
⫺⬁
⫹⬁
⫺⬁
⫹⬁
⫺⬁
⳵T
ds,
⳵x
再
共2␲兲
⫺3兾2
冎
兰
(A19)
⌺共0兲 ⫽
(A20)
654
1
冑2␲␣P0⌽共␭x兲⌺共␭y兲
␬q 2
(A21)
APPLIED OPTICS 兾 Vol. 42, No. 4 兾 1 February 2003
兰
⫹⬁
⫺⬁
1
冑2␲
␭x
␭ x2 ⫹ i␻兾D
(A25)
.
⫺1 dn ␣P 0 1 D
n dT ␬ sin ␤ ␻
⫻
(A26)
兰
⫹⬁
1
冑2␲
␭ x⌽共␭ x兲exp共⫺ix␭ x兲d␭ x.
(A27)
⫺⬁
For a position-modulated pump the ␭x integral is
兰
⬁
⌽共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x
⫺⬁
冉 冊 兺 冉冊
兰 冋 冑 冉 冊 冉 冊册
⫻
⫺x 02
...
w2
⬁
F
⫺⬁
but, as the source term does not depend on z, we can
write ⌰共␭x, ␭y, ␭z兲 as ⌰ ⫽ ␪共␭x, ␭y兲␦共␭z兲, where
␪共␭ x, ␭ y兲 ⫽
2␲
␦共␭ y兲,
sin ␤
At high frequency Eq. 共A25兲 reads as
⫽ 2 exp
⫺i␭ x␪ exp关⫺i共 x␭ x ⫹ y␭ y
⫹ z␭ z兲兴d␭ xd␭ yd␭ z ds,
sin ␤
⫺i dn ␣P 0 ⌺共0兲
n dT ␬ sin ␤
␾⫽
(A18)
⳵T共 x, s sin ␤兲
ds,
⳵x
F ⫺1共1兲 ⫽
Whatever the kind of modulation, we have
Deviation of the Probe Beam
⫹⬁
冑2␲
⫻ ⌽共␭ x兲exp共⫺ix␭ x兲d␭ x.
The deviation of the probe beam inside the substrate
is calculated from the eikonal equation8 integrated
over probe beam path s ⫽ y兾sin ␤:
1 dn
n dT
(A23)
so
where F denotes a Fourier transform.
␾⫽
冎
⌽共␭ x兲⌺共␭ y兲
exp关⫺i共 x␭ x
q2
(A24)
w 2␭ y2
,
8
(A17)
2.
␭x
(A15)
冉 冊兺 冉 冊
冋 冑 冉 冊 冉 冊册
冋 冑 冉 冊册 冑 冉 冊
2
⫺2y 2
exp
␲w 2
w2
⫺⬁
exp共⫺is sin ␤␭ y兲ds ⫽
x 02
共⫺1兲 p⌫ p 2
w
2
⫺2x 2
4x 0 x
exp
I 2p⫹1
2
2
␲w
w
w2
兰 再兰
⫹⬁
⫺⬁
where 公2␲ comes from the z Fourier transform of P ,
and
⫻F
⫺i dn ␣P 0 1
n dT ␬ 2␲
The integral over s is easy to perform as
4x 0 x
.
w2
␻
p⫽0
␾⫽
⫹ s sin ␤␭ y兲兴d␭ xd␭ y ds.
⌸ ␻ ⫽ 冑2␲␣P 0⌽共␭ x兲⌺共␭ y兲␦共␭ z兲,
⬁
(A22)
By using Eqs. 共A6兲, 共A15兲, and 共A22兲 in Eq. 共A20兲 we
obtain
(A13)
The Fourier ␻ term is thus
⫺x 02
⌽共␭ x兲 ⫽ 2 exp
w2
⌰ exp关⫺i共 z␭ z兲兴d␭ z ⫽ ␪共␭ x, ␭ y兲 冑2␲ F ⫺1关␦共␭ z兲兴
⬁
共⫺1兲 p⌫ p
p⫽0
x 02
w2
2
⫺2x 2
4x 0 x
exp
I 2p⫹1
␲w 2
w2
w2
⫻ ␭ x exp共⫺i␭ x x兲d␭ x.
(A28)
The Fourier transform can be converted into a Fourier convolution product 关Ref. 7, Eq. 共17.21.3兲兴:
f ⴱ g共␭兲 ⫽
1
冑2␲
兰
⬁
⫺⬁
f 共␭ ⫺ ␰兲 g共␰兲d␰.
(A29)
By remarking that
F
冉 冊册
冋冑
2
⫺2x 2
exp
␲w 2
w2
⫽ ⌺共␭ x兲,
(A30)
⫽
2
⫺2x 2
exp
w
w2
⌽共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x
⫺⬁
冉 冊兺
⬁
⫺x 02
⫽ 2 exp
w2
⫻
兰
⬁
p⫽0
兰
冉 冊
x 02
共⫺1兲 ⌫ p 2
w
p
⬁
(A34)
共⌺ ⴱ ⍀ 2p⫹1兲共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x
⫽
共⌺ ⴱ ⍀ 2p⫹1兲共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x,
(A31)
where
冋 冉 冊册
4x 0 x
w2
兰
.
⫹
(A32)
⫽
兰
⬁
⫺⬁
冋冑
1
2␲
兰
⬁
⫽
1
冑2␲
兰
⬁
⌺共␭ x ⫺ ␣兲⍀ 2p⫹1共␣兲d␣
⫺⬁
⍀ 2p⫹1共␣兲
⫺⬁
冋兰
⬁
⫺⬁
⫽
兰
再兰
冑 兰
冋兰
冑2␲
⫻
⬁
册
1
2␲
⫻
⫹i
冎
册
⫺⬁
⌺共␰兲共␰ ⫹ ␣兲exp共⫺i␰x兲d␰ d␣,
⫺⬁
(A35)
冉 冊
4x 0 x
w2
(A36)
(A33)
⌺共␰兲exp共⫺i␰x兲d␰
冋 冉 冊 冉 冊册
d
2x 0
4x 0 x
4x 0 x
4x 0 x
⫽ 2 I 2p⫹2
⫹ I 2p
,
I 2p⫹1
2
2
dx
w
w
w
w2
(A37)
so integral of Eq. 共A33兲 reads as
兰
⬁
共⌺ ⴱ ⍀兲共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x ⫽ i
冋 冉 冊
⫻ x 0 I 2p
兰
,
⍀ 2p⫹1共␣兲exp共⫺i␣x兲d␣ ⫽ 冑2␲ F ⫺1共⍀ 2p⫹1兲
冉 冊
⌺共␰兲共␰ ⫹ ␣兲exp共⫺i␰x兲d␰
⬁
册冎
dI 2p⫹1共 z兲 1
⫽ 关I 2p⫹2共 z兲 ⫹ I 2p共 z兲兴
dz
2
⍀ 2p⫹1共␣兲exp共⫺i␣x兲
⫺⬁
⫺⬁
⍀ 2p⫹1共␣兲exp共⫺i␣x兲d␣
⫺⬁
and 关Ref. 7, Eq. 共8.486.1兲兴
⫺⬁
兰
⬁
⫽ 冑2␲I 2p⫹1
⌺共␰兲共␰ ⫹ ␣兲exp关⫺i共␰ ⫹ ␣兲 x兴d␰ d␣
⌺共␰兲␰ exp共⫺i␰x兲d␰ ⫹ ␣
⫺4ix
w2
⫺⬁
where we have used a change of variable ␰ ⫽ ␭x ⫺ ␣.
The integral over ␰ is easy to solve and gives
⬁
冋兰
册
⫺⬁
⫽
d
dx
兰
⬁
册
⍀ 2p⫹1共␣兲exp共⫺i␣x兲d␣
⍀ 2p⫹1共␣兲
⬁
⬁
兰
⬁
but we have
⫺⬁
⫽
冉 冊再
⫺2x 2
2
exp
w2
冑2␲ w
⌺共␭ x
⫺⬁
⬁
⍀ 2p⫹1共␣兲␣ exp共⫺i␣x兲d␣
⫺⬁
⫺ ␣兲␭ x exp共⫺i␭ x x兲d␭ x d␣
1
⬁
1
⫻
⫻ ␭ x exp共⫺i␭ x x兲d␭ x
⫽
⍀ 2p⫹1共␣兲exp共⫺i␣x兲d␣
⫺⬁
共⌺ ⴱ ⍀ 2p⫹1兲共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x
⫺⬁
兰
兰
⬁
⫺4ix
w2
⫺⬁
The integral on the right-hand side of Eq. 共A31兲 is
⬁
2
⫺2x 2
exp
w2
冑2␲ w
1
⫻
⍀ 2p⫹1共␭ x兲 ⫽ F I 2p⫹1
冉 冊冋
⫺⬁
⫺⬁
兰
冊
⫺4ix
⫹␣ .
w2
The integral of Eq. 共A33兲 thus reads as
⬁
⬁
冉 冊
⫺8ix
2
⫺2x 2
⫺2x 2
exp
⫹
␣
exp
w3
w2
w
w2
we have
兰
冉 冊
冉 冊冉
⫽
冉 冊
4
⫺2x 2
3 exp
w
w2
冉 冊
冉 冊册
4x 0 x
4x 0 x
⫺ 2xI 2p⫹1
w2
w2
⫹ x 0 I 2p⫹2
4x 0 x
w2
.
1 February 2003 兾 Vol. 42, No. 4 兾 APPLIED OPTICS
(A38)
655
The integral of Eq. 共A31兲 thus reads as
兰
⬁
for a chopped pump and to x ⫽ 0 for a positionmodulated pump. In the latter case the signal’s amplitude is maximum when x0 ⫽ 0.597w and is greater
by a factor 1.84 than for the chopped signal.
⌽共␭ x兲␭ x exp共⫺i␭ x x兲d␭ x
⫺⬁
⫽i
冉 冊 冉 冊
冉 冊冋 冉 冊
冉 冊 冉 冊册
⬁
⫻
The authors thank Heraeus France for the loan of
the various samples of Suprasil 311.
8
⫺2x 2
⫺x 02
exp
exp
w3
w2
w2
兺
共⫺1兲 p⌫ p
p⫽0
⫺ 2xI 2p⫹1
x 02
w2
x 0 I 2p
4x 0 x
w2
4x 0 x
4x 0 x
⫹ x 0 I 2p⫹2
2
w
w2
References
.
(A39)
For a chopped beam the deviation 关Eq. 共A27兲兴 is thus
␾⫽
⫺1 dn ␣P 0 1 D
n dT ␬ sin ␤ ␻
with
g共 x兲 ⫽ i
冑2␲
g共 x兲,
(A40)
冉 冊 冉 冊兺 冋 冉 冊
冉 冊册冋 冉 冊 冉 冊
冉 冊册
8
⫺2x 2
⫺x 02
exp
exp
w3
w2
w2
⫺ I p⫹1
1
2
0
2
x
w
⫹ x 0 I 2p⫹2
x 0 I 2p
4x 0 x
w2
.
⬁
共⫺1兲 p I p
p⫽0
x 02
w2
4x 0 x
4x 0 x
⫺ 2xI 2p⫹1
2
w
w2
(A41)
The maximum deviation occurs for a distance between probe beam and pump beam equal to x ⫽ w兾2
656
APPLIED OPTICS 兾 Vol. 42, No. 4 兾 1 February 2003
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