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Frequency-modulation spectroscopy with a multimode dye laser R. K. Pattnaik, J. M. Supplee,
and E. A. Whittaker
A multimode
laser in frequency-modulation
spectroscopy
is broader
gives a signal that
than the
absorption line by an amount that is determined by how sharp the edges of the laser profile are, not by the total bandwidth of the laser. We have experimentally demonstrated that one can therefore obtain frequency-modulation signals that are narrower than the total bandwidth of the laser.
Introduction
In conventional absorption spectroscopy the absorption signal is given by the convolution of the laser line shape with the absorption profile. This convolution determines the width of the resulting signal. On the other hand, frequency-modulation (FM) spectroscopy yields a signal that is given by the derivative of the laser line shape convolved with the absorption profile. Therefore in both methods the signal is broader than the absorption profile. But in the FM case the amount of broadening is determined basically by how sharp the edge of the laser profile is, not by the total laser linewidth. For example, a laser with a relatively sharp edge, perhaps where the modes abruptly pass lasing threshold, can yield an FM signal that is only slightly broader than the absorption line. This can happen even if the total laser linewidth is much greater than the absorber width. We observed this effect in our computer modeling results,' and we report here an experimental verification.
ulation index M, the spectrum of this modulated beam consists of a strong line at c and weak sidebands at wc± Win. If this beam then passesthrough an absorbing sample and finally strikes a fast photodetector, the photocurrent signal is 2 ,3 cE0 2
[1 + (-1 - 81)Mcos .t
gF=c08exp(-280)
+ +-j - 2- 0)M sin wI),t].
+ (
(1)
Here the leading fraction simply gives the power at the carrier frequency w,, and 6 and + are respectively the absorption and the phase shift of the sample. The subscript is 0, + 1, or -1, respectively, depending on whether the carrier, the upper-sideband, or the lower-sideband frequency applies. Using phasesensitive heterodyne detection, one can extract the absorption-quadrature signal2 3 CE
2
I = M 80 ( -1-°)
(2)
Background and Theory
FM spectroscopy is a well-established
technique
for
enhancing the detection sensitivity of laser absorption spectroscopy.2-4 This technique usually involves a tunable laser that operates on a single axial
mode of frequency w,. The optical beam is then phase modulated at frequency w(. For a weak mod-
The authors are with the Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030. J. M. Supplee is also with the Department of Physics, Drew University, Madison, New Jersey 07940. Received 31 May 1991.
0003-6935/92/183429-04$05.00/0.
This again assumes weak absorption, with exp(-280)
1.
We are specifically interested in the absorptionquadrature signal [Eq. (2)] but for the generalized case of a multimode (multiple longitudinal modes) laser.' Because the multimode source can be considered a superposition of modes, one can simply sum the absorption-quadrature signals [Eq. (2)] of each mode. One thereby obtains the total absorptionquadrature signal as a function of the tunable laser center frequency,' I(Co) = M
cE n
2 2
8&rr(8,
1 -
n, )
(3)
© 1992 Optical Society of America.
20 June 1992 / Vol. 31, No. 18 / APPLIED OPTICS
3429
Here n simply numbers the modes; for example, 8,-1 is the sample absorption at the frequency of the lower sideband of the nth mode. For a small modulation frequency, the subtraction shown in Eq. (3) is equivalent to taking a derivative of the absorption 8. However, for closely spaced modes (spacing less than or approximately equivalent to the absorber half-width) the sum can be represented as an integral. Using both of these assumptions' changes Eq. (3) to I(wd)
l~twjc 2M_wed8(w) dw -
Wrt
modes
2MwomAft Wrt
Jmodes
Em ) dP(w
8(w)
-')
d(ww
P(w) = POexp(
)2]
2 2(W
)I
(5)
then approximation (4) can be integrated to yield 2(27r)12M(omPoA °)rt
UcrF' 2
W(u' + u 2 ) 31 2
dw
d
(4)
Here P (= cE 2 /8r) is the power of the various active laser modes, and wrt (round-trip frequency) is the frequency spacing between laser modes. The two integrals in approximation (4) are equal by integration by parts. Because the argument of P is w - w, the derivative in the last term of approximation (4) can be changed to d/dw, with only a sign change. The d/dw, can then be factored out in front of the integral, which shows that the FM signal [approximation (4)] is proportional to the derivative of the conventional absorption signal. This means that many of the following comments about signal linewidth also apply to the derivative of the conventional absorption signal. With FM spectroscopy the differentiation is simply intrinsic. Perhaps more important is the fact that FM spectroscopy offers significant sensitivity advantages. The second integral form in approximation (4) is a convenient way to picture the case of a laser envelope P that has fairly sharp edges, perhaps where the modes suddenly fall below lasing threshold. In that case' the sharp edges cause dP/dw to include a Dirac delta function at each edge. These delta functions can be the primary contribution to the integral in approximation (4); for that case the integration is trivial and reveals that the FM signal I contains two copies of the absorption profile . This signal is not broadened by the width of the laser line. This is because it is the tuning of the sharp edge of the laser envelope through the absorber that causes the FM signal.' In contrast to the sharp-edge case discussed above, a laser envelope that falls off gradually results in an FM signal that is broader than the absorption line. But it is noteworthy that this broadening is governed by how sharply the laser envelope rises, not by the total width of the laser. Some key features of a non-sharp-edge case can be illustrated by considering the case of both a Doppler-broadened (Gaussian) absorption line and a laser envelope that is also modeled as having a Gaussian shape. Then approximation (4) can be integrated in closed form. That is, if we model the absorption profile 8 and the laser 3430
( ) =A exp, -_( -
I(w) =
dw
P(w
envelope P by
APPLIED OPTICS / Vol. 31, No. 18 / 20 June 1992
X
(w
-
fl)exp[( 2 j
+ r2)-
(6)
Equation (6) illustrates general features that apply to other line shapes too, not just Gaussians. Because r and cr' are squared before adding, the larger of these easily dominates. Therefore if the laser is narrower than the absorption line by even a modest factor, it will contribute negligible broadening. If the two widths are the same, then combining r and r' in this manner broadens the signal by a factor of /2. However, in any of these cases there is also some narrowing of the signal peak, simply because FM detection results in an odd signal [as given by Eq. (6)], and the FWHM of each lobe of x exp(-x2) is 68% as wide as the FWHM of exp(-x 2 ). If the Gaussian model illustrated above does not hold because the actual laser envelope is irregular,
then the situation may improve. That is, if the
actual laser envelope has a steep edge on either side,
then the tuning of that steep edge through the
absorber results in an FM signal that is approximately only as wide as the true absorber width. Thus actual experimental cases could fall between the Gaussian and sharp-edge extremes discussed above. With this in mind we use the measured laser envelope in Eq. (3) to calculate the expected FM signal. This expected result is then compared with the measured FM signal. Experimental Results and Discussion
To test this model, we measured the sodium D2 line absorption spectrum ( = 589.0 nm) with an argonion-pumped Coherent 599-01 jet stream dye laser. The laser was fitted with a three-plate birefringent filter, which resulted in a nominally 20-GHz-wide laser, and we scanned this with a computer-controlled motorized micrometer (Oriel 18006). The scan was calibrated by using both the D-D 2 line separation and the interference fringes that arose from thin glass cell windows of known free spectral range. As the expected FM spectroscopy (FMS) line shapes are sensitive to the laser line shape, we obtained a spectrum of the laser that was transmitted through a triple-pass plane-parallel Fabry-Perot interferometer. The mirrors were set to a free spectral range of
22 at the D2 wave-
Since the finesse is
54 GHz.
length for these mirrors, we should have a 2.5-GHz resolution, which is adequate to permit estimation of the laser line shape. A smoothed Fabry-Perot scan is shown in Fig. 1. We estimate that the laser FWHM is typically 9.5 GHz, with a slight asymmetry
in line shape. FMS measurements were made by phase modulating the laser at 320 MHz with a LiTaO3 electro-optic modulator. Since this modulation frequency is much less than the total laser bandwidth, we are in a totally different regime from previous FM experiments with multimode lasers.5 - 7 Those previous experiments maintained much of the fundamental spirit of single-
0.5
0 0-
mode FM spectroscopy by the choice of a modulation
frequency greater than the total laser bandwidth, which put the sidebands outside of the laser power envelope. In our experiment the modulated beam was mechanically chopped at 2 kHz and then passed through an 8-cm-long sodium cell that was maintained at a temperature of 170'C. At this temperature the reduction in the resonantly tuned transmitted beam owing to sodium absorption was negligible. The beam was detected with an EG&G FND-100 photodetector. The photocurrent was phase-sensitively demodulated by using an rf mixer and lock-in amplifier combination. Spectra were obtained by scanning the laser wavelength with the motor micrometer and by collecting the lock-in output with a computer. Concurrent with the FMS data scan, the sodium fluorescence that was emitted at right angles to the laser was recorded with a photomultiplier and a photon counter. The fluorescence scan is shown in Fig. 2, and this scan provided an extra check on the laser line shape. The fact that the sodium line is convolved with the laser line shape causes little broadening because the sodium line is narrow; it has a FWHM of 2.8 GHz because of a hyperfine structure and Doppler broadening. The measurements shown in Figs. 1 and 2 are consistent; the line shape asymmetry is present again, which leads us to conclude that this is a genuine feature of the laser output
50
25 0 -25 (GHz) FREQUENCY
-50
Fig. 2. Fluorescence spectrum obtained simultaneously with the FM signal.
spectrum. We also note that the asymmetry appears consistently on the low-frequency side of the laser spectral profile, that it is independent of scan direction when measured by the fluorescence, and that it is insensitive to internal laser-cavity alignment. The FMS absorption signal we obtained is shown in Fig. 3. The selection of the correct rf phase for extracting the absorption signal was aided by the fact that at low modulation frequency (320 MHz is substantially less than the absorption linewidth) the dispersive FMS signal is small.3 Hence the rf phase could be adjusted for the absorption quadrature by first nulling the signal and then by shifting 90°. The experimental results thus obtained are shown in Fig. 3. These results show good agreement with the theoretically expected results, which are shown by the solid curve in the same figure. These theoretically expected results were calculated by using the actual laser-power spectrum (Fig. 2) in Eq. (3). We comment more generally on the FM signal that one may expect for laser envelopes that are different from ours (Fig. 2). At one extreme a smooth and symmetric laser envelope may be modeled as a Gaussian. For that case, Eq. (6) shows that the laser A
0.1-
1-
0.0
,1 -m-
~r_0.5 rJ2
-0.2
-0.3-50
-25
0
25
50
FREQUENCY (GHz) Fig. 1. Laser-power spectrum, measured with a Fabry-Perot interferometer.
-50
-25
0
25
50
FREQUENCY Fig. 3. Theoretical expectation of the FM signal (solid curve) as calculated from Eq. (3), and experimental results (dotted curve). 20 June 1992 / Vol. 31, No. 18 / APPLIED OPTICS
3431
width and the absorber width are squared before adding, so with a laser significantly wider than the absorber, such as our 9.5-GHz laser versus our 2.8-GHz absorber, we expect the laser width to totally dominate. However, the fact that Eq. (6) gives an odd function then causes a reduction to 68%, as mentioned above. We would therefore estimate a FWHM of -9.5
GHz(0.68)
= 6.5 GHz for that
case. At the other extreme a laser envelope with an abrupt edge yields' an FM signal with only the absorber width, 2.8 GHz for this experiment. Thus we see that knowledge only of the laser width can give some insight on the expected FM signal width. Since our actual laser envelope had one steep edge, the measured FM signal had one narrow peak with a FWHM of 3.0 GHz (Fig. 3), which is close to the
expectation for the hard-edge case. In summary, we have experimentally demonstrated that the FM signal peak can be narrower than the laser bandwidth for a multimode laser that has a sharp-edged profile. This fact may extend the use of FM spectroscopy as a simple and sensitive gas-phase analysis technique. While we have not made an effort to establish sensitivity accurately, it should be noted that the signal-to-noise ratio for the FM signal of Fig. 3 appears to be no worse than the signal-tonoise ratio for the fluorescence signal of Fig. 2. As noted above, these spectra were obtained under conditions such that the attenuation of the overall laser envelope by the sodium atoms was insignificant. Of course, much greater absorption would have been measured for a single-mode laser, but the added expense and complexity of such a device make it less attractive as a general-purpose analytic tool.
3432
APPLIED OPTICS / Vol. 31, No. 18 / 20 June 1992
Finally, we note that our results may also have applications to non-laser-based modulation spectroscopies. In those techniques, the source line shape will again dictate resolution limits, so efforts to modify that shape may have significant benefits. This research was supported in part by National Science Foundation grants ECSE-8811955 and ECSE8911318. We thank the referees for suggesting significant improvements. References
1. J. M. Supplee and E. A. Whittaker, "Theoretical modeling of multimode laser frequency-modulation spectroscopy," J. Opt. Soc. Am. B 8, 719-725 (1991). 2. G. C. Bjorklund, "Frequency-modulation
spectroscopy:
a new
method for measuring weak absorptions and dispersions," Opt. Lett. 5, 15-17 (1980). 3. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983). 4. M. Gehrtz, G. C. Bjorklund, and E. A. Whittaker, "Quantum-
limited laser frequency-modulation spectroscopy," J. Opt. Soc. Am. B 2, 1510-1526 (1985). 5. T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, and
W. Lenth, "Frequency-modulation spectroscopy with a pulsed dye laser," Opt. Lett. 7, 28-30 (1982). 6. D. E. Cooper and T. F. Gallagher, "Frequency-modulation
spectroscopy with a multimode laser," Opt. Lett. 9, 451-453 (1984). 7. G. R. Janik, C. B. Carlisle, and T. F. Gallagher, "Two-tone frequency-modulation spectroscopy," J. Opt. Soc. Am. B 3, 1070-1074 (1986).
