David E. Cooper and T. F. Gallagher. In this paper, we describe experimental and theoretical investigations of two variations of frequency modu- lation (FM) ...
Double frequency modulation spectroscopy: frequency with low-bandwidth detectors
high modulation
David E. Cooper and T. F. Gallagher
In this paper, we describe experimental and theoretical investigations of two variations of frequency modulation (FM) spectroscopy that use two electrooptic modulators. In the first variation, both modulators are frequency modulators (FM-FM), and, in the second, one is a frequency modulator and one is an amplitude modulator (FM-AM). The essential advantage of FM-FM and FM-AM spectroscopy is that sensitive lowbandwidth detectors, such as photomultiplier tubes, can be used to detect signals generated by the absorption of sidebands displaced from the carrier by frequencies far above the detector cutoff frequency. These two variations are complementary
in the sense that, in situations where optical power is at a premium, the
FM-FM scheme is most appropriate, and in situations where modulator drive power is at a premium, the FM-AM scheme is most appropriate. Using either of these variations, we have detected the absorption of 700-MHz sidebands with photomultiplier
1.
Introduction
Frequency modulation (FM) spectroscopy' is a sensitive optical spectroscopic technique for measuring absorption and dispersion in practically any optical medium. One of the most promising areas for the application of FM spectroscopy is in the detection of atmospheric trace gases and hazardous materials. Absorptions as small as 10-4 have been easily detected with
FM spectroscopy using either single-mode1 or multimode2 cw lasers. It appears that an optimized (shotnoise-limited) visible wavelength FM system should be capable of detecting an absorption as small as 10-6 with a 1-sec integration time. The reason the technique is so sensitive becomes apparent if we quickly review the principles of FM spectroscopy. A laser beam of frequency WL is phasemodulated at frequency Q,which' is typically, but not necessarily, far greater than the linewidth AWL of the laser, i.e., Q >> AWL.
and
AWL
Typical values are Q = 500 MHz
= 1 MHz. In the limit of low-modulation
index, the laser beam acquires sidebands at
WL
iQ, and
when the modulated laser beam impinges on a square law detector, such as a photodiode, each sideband beats
David Cooper is with SRI International, Electro-Optics Systems Laboratory, 333 Ravenswood Avenue, Menlo Park, California 94025,
and T. F. Gallagher is with University of Virginia, Physics Department, Charlottesville, Virginia 22901. Received 29 October 1985. 0003-6935/85/091327-08$02.00/0. © 1985 Optical Society of America.
tubes whose cutoff frequencies lie below 100 MHz.
with the carrier to produce a component of the photocurrent at Q. However, the two beat signals are 1800 out of phase and, therefore, cancel. If prior to photodetection the modulated beam traverses a medium whose complex index of refraction differs for the two sidebands, this sideband cancellationis incomplete, and a photocurrent at Q is produced.
One case of practical
interest is where a differential absorption between the two sidebands occurs. Since there is no intrinsic laser noise at Q for the case where Q >>AWL, the absorption
appears against a background that is in principle shotnoise-limited. Even in cases where Q - AWL,it has been demonstrated 2 that laser noise at Q poses no significant problems in cw FM experiments because this noise bears no fixed phase relationship to the modulator
drive frequency and, therefore, cancels when the photocurrent is heterodyned with the local oscillator in a rf mixer. In the monitoring of an absorption using FM spectroscopy, optimum sensitivity is obtained when the modulation frequency Q exceeds the absorption linewidth. For Doppler-broadened gases, this means Q 2 GHz, whereas for atmospheric pressure-broadened gases this requires Q- 10-20 GHz. In the visible wavelength region, neither condition poses very serious
problems, since LiTaO 3 modulators 3 and GaAs Schottky barrier photodiodes4 with sufficient bandwidth have been demonstrated. Although the highspeed GaAs photodiodes do not have the sensitivity of most conventional visible wavelength detectors, this is usually not a serious problem, since one can easily put several milliwatts of power on the detector with laser light sources. However, an interesting and promising 1 May 1985 / Vol. 24, No. 9 / APPLIED OPTICS
1327
extension of FM spectroscopy is to the 8-12-,gm atmospheric window region, where numerous molecular species have strong and characteristic absorption features, and where carbon dioxide and semiconductor diode laser sources are available. For work in this region, CdTe modulators with 16-GHz bandwidths have been constructed.5 However, similarly fast and sensitive detectors are unavailable. Sensitive HgCdTe detectors are limited to frequencies below -2 GHz,6 and ultrafast 8-10-GHz pyroelectric detectors suffer from
ET(t) = Eo
pq=-+
The components of the laser power spectrum that are of interest in FM spectroscopy are those at the difference frequency nQ, -
Ps(t) =
87r
+ mQ 2
spectral region. We begin with a theoretical description
of double frequency-modulated (FM-FM) light and amplitude-modulated FM light (FM-AM) in Sec. II. Although the case of double frequency-modulated light has been treated theoretically in the literature,7 the motivation for that treatment was quite different from
given by
-
T(t)ER(t) Jp(Ml)Jq(M2)Jp'(M1)Jqd(M2)
= cEo3
This bandwidth limitation of sensitive photodetectors
FM approaches that circumvent the detector bandwidth limitation and present experimental results we have obtained using these approaches in the visible
(2)
X expi(WL+ PQ + qQ2)t.
very low sensitivity. is a severe problem in the extension of FM spectroscopy to nonvisible spectral regions and in the visible spectral region under conditions where optical power levels are constrained to be low. We discuss here two alternative
Jp(Ml)Jq(M2)T(WL+PQj + qQ2)
8wrp,qp',q' X TpqTtq, expi[(p - p')Ql + (q - q')Q2]t-
(3)
We will, for reasons that will become apparent, restrict this general equation to the special case of Q, = 2W + a, Q2= c, and (p -p') = ±k, (q - q') = 1. Under these
conditions, the component terms of Eq. (3) that have frequencies ka occur whenever
= 2k and are given
explicitly by P,(k at) =
°E Jp(Mi)Jq(M2)
87r p,q
X [Jp-k(M1)Jq+1(M2)TpqT;_kq+expi(kQ%- 19 2)t + Jp+k(Ml)Jq-L(M2)TpqT;+kq-1 (4) X exp - i(kQ, - Q2)t].
the one stated here, and in addition the formalism needs
to be extended to correspond with the actual experimental cases discussed in Sec. III.
We will further restrict Eq. (4) to the case where k = 1. The idea here is that W is an arbitrary frequency, chosen
11. Theoretical Description
to be coincident with a particular absorption feature of interest, and a is a much smaller frequency, chosen to lie within the passband of the photodetector in use and also much smaller than the linewidth of the absorption feature. Under these conditions Eq. (4) reduces to
We consider here a description of the electric field and power spectrum of laser light at frequency WL that is directed first through a frequency modulator operating at Ql, then through either a second frequency modulator or an amplitude modulator operating at Q2, and finally through an optical medium that interacts differently with the different frequency components of the double modulated light. The beam is subsequently monitored with a photodetector that generates a photocurrent proportional to the square of the electric field. A.
FM-FM Light
We first consider the case where both modulators are
frequency modulators, and for convenience we use the notation of Ref. 7. The electric field for this case can be written as
Ps(Ut) = CE2Jp(M1)Jq(M2) 87r pq X [Jp-1(Ml)Jq+2(M2)TpqT_lq+
2
exp(iat)
+ Jp+l(M)Jq-2(M2)TpqT;+,q2 exp(-iat)]
(5)
We note that if all the transmission factors Tij are set equal to unity in this expression, a situation corresponding to no absorption or dispersion, then with the help of Neumann's addition theorem for Bessel functions, 8 it is easily shown that PS (at) = 0, as expected.
Clearly, when absorption and dispersion are present, many possible ways of generating beat signals at a exist. To proceed further, one needs to specify which transmission factors differ from unity. We will assume
EFM-FM(t)= /2E
p,q=--
Jp(M1)Jq(M2) expi(L
+ PQi + qQ2)t
+ c.c.,
(1)
where Ml and M2 are the modulation indices of the first
that the modulation index of the first modulator is small, Ml < 1, so that all terms other than p = 0, +1 are negligible. In addition, we will consider the case of differential absorption and dispersion between sideband groups centered at +W. These sideband groups
and second modulators, respectively, and the J, (M)
are the triplets (W,W + a, co- ) and (-co, -co + , -co
factors are Bessel functions of integer order. The effect of the optical medium is represented by a set of complex frequency-dependent transmission factors, Tpq T(L + pQ, + qQ2), and the transmitted electric field is, therefore,ET(t) = 1/2ET(t) + c.c.with
-
1328
APPLIED OPTICS / Vol. 24, No. 9 / 1 May 1985
a), respectively.
Hence in this case the corresponding
nonunity transmission factors appearing in Eq. (5) are (To1 , T,,, T. 3 ) and (To-, T-1 , T,_ 3 ). With these approximations, the beat signal appearing at a is given by
PS(ot) =-Jo(M)JI(M(J1(M 8w
--
2 )J1(M2 )[(8X+0
+ 2w + '5o-r)
J0 (Ml)
(5-Z- + 26_X+ -. +,)]cosat + J1(M2)J(M 2)[K). + (P-) (' + -- )] +Jl(M2)J3(M2)[( WD.- + "D-.+.) (% + 4!_,)Ilsinrt).
t, (Ml) (a}1(
(6)
1:
J-1 (M1 >
In deriving this expression, we have written Tpq = exp
_
0
(6pq+ ipq) and assumed a weak interaction limit between the medium and the optical field. In writing the absorption and phase terms, we have replaced the numerical subscripts by their corresponding frequencies. The essential features of this result are similar to those for conventional FM spectroscopy. The inphase
J1 (M1)J1 (M2 )
-J1
(M, )J
-
t
-w-cr s-so = -w+c
3
4
r
b,
-co-e = = '
= -(-w =
b
where 6 and 4 are the constant background absorption and phase shift, respectively. Our signal then reduces to P 8(Tt)
=
8
2MJW(M 2 )A6 cosot,
(7)
where A(3= (+ - ). The inphase term in this expression is identical to that obtained in conventional FM spectroscopy with the exception of the factor 2J2(M 2 ). Note, however, that no quadrature signal arises from the anomalous dispersion. If the second modulator is driven with M2 1.8, then J1 (M 2 ) takes on its maximum value yielding 2J (1.8) = 0.67. It is
lower sidebands are separated from the laser frequency WL by 2 + a; and we have drawn the sideband amplitudes assuming M1 1. The effect of the second fre-
quency modulator on this FM spectrum is shown in Fig. 1(b). For convenience, we have taken M 2 2.4, which
corresponds to the first zero of Jo. Consequently, the second modulator shifts all the optical power from the first modulator to different frequencies; i.e., there are now no components at WL, and WOL1 (2 + ). The presence of these components, however, does not change
the heterodyne signal obtained by the absorption feature located at co,since no J(Ml)JO(M 2 ) or J(Ml)-
JO(M2 ) factor appears in Eq. (6). Figure 1 shows clearly
that beat signals at a-arise from the sideband products
Feature Jj(
0
(b)
_
_ I
i
I
_
J(MI
J-1(M2)
~~J-1 (M1 XJ1(M2)
l l J.(M) J-1 (2)
I
,a
-2.-.
-
.1
I1 1
00
c
2.+1a
RELATIVE FREQUENCY
Fig. 1. Optical power spectrum of (a) pure FM light at frequency 2w + and modulation index M1 , and (b) the light produced by directing this FM light through a second frequency modulator operating
at frequency w and modulation index M 2.
JO(M1 )Jl(M 2 ) J1 (M1 )JA(M2 ) and J(Ml)J(M 2 ) J1 (Ml)J3 (M2 ). These are exactly the factors appearing in Eq. (6). B.
FM-AM Light
Let us now consider the case in which the second modulator is an amplitude modulator, constructed by, placing a frequency modulator between linear polarizers
crossed at 1450, so that no light is transmitted if the second modulator is not driven. (For modulators with birefringent crystals, a bias voltage must be used.) In this case, the electric field can be written 9 EFM-AM(t)= /2E0
thus possible to obtain signals in FM-FM spectroscopy
comparable with those in conventional FM spectroscopy, provided the second modulator is driven at high modulation index. The origin of the heterodyne beat signals appearing at a in FM-FM spectroscopy becomes clear if we consider the effect of both modulators in the frequency domain, as illustrated in Fig. 1. The FM light from the first modulator is shown in Fig. 1(a); the upper and
Absorption
(M2 ) Jo(
(cosat) term arises from a differential between weighted
averages of the absorptions of the upper and lower triplet groups. The quadrature (sinat) term arises from differentials between the average phase shifts of upper and lower sideband pairs. In the limit where a-is small relative to the linewidth of the absorption feature of interest, and for the casewhere the absorption is probed by the upper sideband triplet, we may write
3
Lq
Jp(M1)Jq(M2 ) expi(OL+ PQ1 + qQ2)t
- Z Jp(Mj)expi(WL +P%)tJ + c.c.
(8)
This is simply the FM-FM electric field given in Eq. (1)
minus the pure FM electric field from the first modulator. As a result, in the FM-AM configuration even when M2 is small, the components at WL and WL P Q1 are almost completely suppressed, but components at frequencies WL I p I q 2 with q # 0 are unattenuated. Thus one obtains a frequency spectrum essentially identical with that obtained in the FM-FM configuration when M 2 is large. Hence these two config-
urations are complementary in that FM-FM is preferable when optical power levels are low and the second modulator can be driven at high modulation index, and FM-AM is preferable when optical power levels are high
and the second modulator is constrained to be driven at low modulation index. The effect of an optical medium on the FM-AM electric field is again represented by a set of complex frequency-dependent transmission factors T and the transmitted field can be written ET(t) = 2[ET(t) eT(t)] + c.c. with ET(t) given by Eq. (2) and T(t) given
by 1 May 1985 / Vol. 24, No. 9 / APPLIED OPTICS
1329
Lens
Lens
Lens
Lens
Polarizing Beam Splitter
Fig. 2. Experimental configuration used for FM-AM spectroscopy and for FM-FM spectroscopy when no dc bias voltage is applied
to the second modulator crystal and the polarizers are removed.
eT(t) =
Eo
Jp(Ml)T(WL + Pil) expi(wOL+ PQI)t-
(9)
The power spectrum of this FM-AM light at the various beat frequencies of interest is given by P, (t) ETET (EWT + ET) + jTeT It isstraightforward to show that with the same conditions used to treat the FM-FM case, only the ETE' term givesrise to beat signals at a. Hence P, (at) for FM-AM light is identical to that for
FM-FM light and given by Eq. (6). w
111. Experimental Results
We have investigated experimentally both FM-FM and FM-AM spectroscopy in the visible spectral region using two LiTaO3 electrooptic modulators driven at frequencies in the 500-1500-MHz range.
In both cases,
0~ -J
-) 0
signals were recovered using either photomultiplier tubes or semiconductor photodiodes. We obtained the highest SNRs in the FM-AM configuration by driving the amplitude modulator cw and detecting the signal, with a photodiode. A.
FM-AM Experiments
The schematic of our experimental configuration for FM-AM spectroscopy is given in Fig. 2. The laser is a Spectra-Physics 102 He-Ne laser operating at 632.8 nm, which has a power of 2 mW and a cavity mode spacing of 641 MHz. A single cavity mode with -1 mW of
power is selected from this laser by means of a linear
-1460
0
1460
RELATIVE FREQUENCY (MHz)
Fig. 3. Optical power spectrum of pure FM light with a 1460-MHz modulation frequency.
polarizer. The laser beam is gently focused through the
first electrooptic modulator, which is driven cw at frequency 2W+ a = 1460 MHz by a Hewlett Packard (HP) 8620 sweep oscillator and a solid-state power amplifier capable of 10-W output. The drive power is adjusted to put 15% of the optical power in each of the two side-
250-Hz pulse repetition frequency and 50-,usecpulse width. The rf power level is adjusted to give a modulation index of M2 - 1. Due to the natural birefringence of LiTaO 3 , a dc bias voltage of up to 200 V must be ap-
placed between two linear polarizers crossed at ±450.
plied to the crystal to ensure that the transmitted beam is extinguished when no rf is applied to the modulator. The resulting frequency-amplitude modulated beam next impinges on a piezoelectrically scannable Spectra-Physics 410 etalon of 30-GHz FSR and 600-MHz bandwidth. We observe the retroflection from the
This modulator is driven at frequency w = 700 MHz by
etalon, which decreases as the etalon is scanned through
an EPSCO PG5kB pulsed cavity oscillator with a
the laser frequency, mimicking a 25%absorption.
bands, corresponding to a modulation index M1 0.8. Figure 3 shows a typical power spectrum of this pure FM light. The optical beam is next focused gently through
1330
a second electrooptic
modulator,
which is
APPLIED OPTICS / Vol. 24, No. 9 / 1 May 1985
I
Figure 4 shows a FM-AM signal obtained with 65 nW
I
of optical power incident on the PMT biased at 900 V. This signal exhibits the asymmetric feature of a normal FM signal with the two peaks separated by 1400MHz, twice the drive frequency of the amplitude modulator. The signal is exactly what is obtained in a normal FM configuration using a 700-MHz frequency modulator and a 1-GHz photodetector, but it has been recovered in our experiment by use of an additional amplitude modulator and a 100-MHz PMT. Although our signal processing is far from optimum, we estimate from the observed SNR (100) that an absorption as small as 2.5 X 10-3 could be measured in this configuration. This
CO -J
V/)
U-
-700 700 RELATIVE FREQUENCY (MHz) Fig. 4. FM-AM signal at 60 MHz resulting from absorption of sidebands at 700 MHz obtained with 65 nW of optical power incident on an RCA 931 PMT biased at 900 V.
is an impressive sensitivity given the small amount of optical power incident on the detector. We also performed FM-AM experiments using a semiconductor photodiode and a cw rf driver for the amplitude modulator. For these experiments, the frequency modulator is driven at 1060MHz by the HP 8620 sweep oscillator, and the amplitude modulator is driven at 500 MHz by a General Radio 1209B oscillator and Boonton 230A amplifier. The rf drive to the amplitude modulator is again adjusted to give a modulation index M2 - 1. The detector used is a HP 4220 PIN photodiode having a bandwidth in excess of 1 GHz. The IF signal at 60 MHz is again detected using the ZFM-3 mixer. However, to improve the SNR, we mechanically chop the laser beam at 100 Hz and detect the mixer output with a lockin amplifier referenced to the chopping frequency. FM spectra are recorded by driving an X- Y recorder with the lockin output as the frequency of the etalon is scanned. Figure 5 shows the FM-AM signal obtained with -0.5 mW of average power incident on the HP 4220 photo-
The retroflected beam is directed through two neutral-density filters having a combined transmittance of
diode. As before, the signal exhibits the asymmetry characteristic of a normal FM signal with the two peaks
3% onto a RCA 931 photomultiplier
separated by 1000 MHz, twice the drive frequency of the
tube with 600-900
V applied across the dynode chain. Under these conditions, the PMT bandwidth is
