Title
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Frequency modulation Fourier transform spectroscopy 
Julien Mandon, Guy Guelachvili, Nathalie Picqué 
Laboratoire de Photophysique Moléculaire, CNRS; Univ. Paris-Sud, Bâtiment 350, 91405 
Orsay, France 
Corresponding author: 
Dr. Nathalie Picqué, 
Laboratoire de Photophysique Moléculaire 
Unité Propre du CNRS, Université Paris Sud, Bâtiment 350 
91405 Orsay Cedex, France 
Phone number: +33 1 69156649 
Fax number: +33 1 69157530 
Email: nathalie.picque@ppm.u-psud.fr
Web: http://www.laser-fts.org  
Abstract: A new method, FM-FTS, combining Frequency Modulation heterodyne laser 
spectroscopy and Fourier Transform Spectroscopy is presented. It provides simultaneous 
sensitive measurement of absorption and dispersion profiles with broadband spectral coverage 
capabilities. Experimental demonstration is made on the overtone spectrum of C2H2 in the 1.5 
µm region. 
OCIS codes: 120.6200, 300.6300, 300.6380, 300.6360, 300.6310, 300.6390, 120.5060 
120.6200  Spectrometers and spectroscopic instrumentation, 300.6300 Spectroscopy, Fourier 
transforms, 300.6380  Spectroscopy, modulation, 300.6360  Spectroscopy, laser, 300.6310  
Spectroscopy, heterodyne, 300.6390 Spectroscopy, molecular, 120.5060  Phase modulation 
mailto:nathalie.picque@ppm.u-psud.fr
http://www.laser-fts.org/
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Improving sensitivity is presently one of the major concern of spectroscopists. This may 
be obtained both from the enhancement of the intrinsic signal, and from the reduction of the 
background noise. In this latter case, modulation has been one of the most effective approach. 
In particular, Frequency Modulation (FM) absorption spectroscopy [1] has reached detection 
sensitivity near to the fundamental quantum noise limit, by shifting the frequency modulation 
of the measurements to a frequency range where the 1/f noise becomes negligible. Moreover, 
FM spectroscopy benefits from high-speed detection and simultaneous measurement of 
absorption and dispersion signals. Since Bjorklund’s first demonstrations [1,2] of the 
efficiency of FM spectroscopy with a single-mode continuous-wave dye laser, the technique 
has been widely used as a tunable laser spectroscopic method in fields such as laser 
stabilization [3], two-photon spectroscopy [4], optical heterodyne saturation spectroscopy [5], 
trace gas detection [6]. In most schemes, the laser wavelength is scanned across the 
atomic/molecular resonance to retrieve the line shape. More rarely, the modulation frequency 
is tuned. However in both cases, the measurements are limited to narrow spectral ranges. 
This letter reports the first results in FM broadband spectroscopy. This work is  
motivated by our ongoing effort of implementing a new spectroscopic approach 
simultaneously delivering sensitivity, resolution, accuracy, broad spectral coverage and rapid 
acquisition. The basic idea, named FM-FTS, is to associate the advantages of FM 
spectroscopy and high-resolution Fourier transform spectroscopy (FTS). FTS is able to record 
at once extended ranges, with no spectral restriction. In particular it gives easy access to the 
infrared domain. In this letter, a new way of modulating the interferogram is implemented. 
The key concept is that a radio frequency (RF) modulation is performed. The beat signal at 
the output port of the Fourier transform spectrometer is modulated at constant RF, which is 
about 104 times greater than the audio frequency generally delivered by the interferometer 
optical conversion. Together with the advantage, over classical FTS, of measurements 
performed at much higher frequency, our approach benefits from the synchronous detection 
ability and from the simultaneous acquisition of both the absorption and the dispersion of the 
recorded profiles. 
The experimental principle is presented in Fig. 1. The light emitted by the broadband 
source is first passing through the interferometer. The output beam is then phase-modulated 
by an electro-optic modulator (EOM) before entering the absorption cell and falling on the 
fast detector. The synchronous detection of the detector signal is realized by the lock-in 
amplifier at the EOM driver reference frequency fm. Recorded data are finally stored on the 
computer disk with their corresponding path difference position ∆. Their Fourier transform is 
the spectrum. 
In more details, the electric field E at the output of the interferometer may be written as: 
0 ( )( , ) 1 exp exp( )d c.c.     (1)
c c ct i i tc
ω ω ω
⎡ ⎤∆⎛ ⎞
∆ = + − +⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
where E0 is the electric field amplitude of the source at ωc optical pulsation, c is the velocity 
of light and c.c the conjugate complex of the preceding expression in Eq. 1. The EOM effect 
on the beam is assumed to have a low modulation index M. As a consequence, each carrier 
wave of pulsation ωc, has two weak sidebands located at ± ωm = ± 2π fm. Equation (1) 
becomes:  
( ) ( ) }
0 ( )( , ) 1 exp exp
M exp M exp d c.c.     (2)
c m c m c
t i i t
i t i t
ω ω ω ω ω
⎡ ⎤∆⎛ ⎞
∆ = + −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
⎡ ⎤ ⎡ ⎤+ + − − +⎣ ⎦ ⎣ ⎦
When interacting with the gas, the carrier and the sidebands experience attenuation and phase-
shift due to absorption and dispersion. Following the notations introduced in [1], this 
interaction may be written as exp(-δ(ω)- i φ(ω)) where δ is the amplitude attenuation and φ is 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
the phase shift. The following convention is adopted: δn and φn denotes for n = 0, ±1 the 
respective components at ωc and ωc± ωm. Then Eq. 2 may be written: 
( ) ( ){
( ) ( ) ( ) ( ) }
1 1 1 1
( , ) 1 exp exp exp
M exp exp M exp exp d c.c.     (3)
c m c m c
t i i i t
i i t i i t
ω δ φ ω
δ φ ω ω δ φ ω ω ω+ + − −
⎡ ⎤∆⎛ ⎞
∆ = + − − −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
⎡ ⎤ ⎡ ⎤+ − − + − − − − +⎣ ⎦ ⎣ ⎦
The intensity I detected by the fast photodetector is proportional to : 
*I( , ) ( , ) ( , ).     (4)t t t∆ ∝ ∆ ∆E E  
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 1 1
0 1 0 1 0 1 0 1
0 1 1 0 0 1 0 1
I( , ) exp 2 exp 2 exp 2 1 cos
2M cos exp cos exp cos 1 cos
2M sin exp sin exp sin 1 cos
δ δ δ ω
ω δ δ φ φ δ δ φ φ ω
ω δ δ φ φ δ δ φ φ ω
+ + − −
− − + +
⎛ ⎡ ⎤∆⎛ ⎞⎡ ⎤∆ ∝ − + − + − +⎜ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦⎝
⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − − − − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
∆⎛ ⎞⎡ ⎤+ − − − − − − − + ⎜⎣ ⎦ ⎝ ⎠
( ) ( ) ( )
( ) ( ) ( )
1 1 1 1
1 1 1 1
2M cos 2 exp sin 1 cos
2M sin 2 exp cos 1 cos d .                                        (5)
m c c
ω δ δ φ φ ω
ω δ δ φ φ ω ω
+ − − +
+ − − +
⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
⎞⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − + ⎟⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎠
After synchronous detection at fm frequency and with the assumption that |δ0-δj |<<1 and |φ0-
φj|<<1 (with j = ±1), the in-phase Icos(∆) and the in-quadrature Isin(∆) parts of the electric 
signal are given by  
( )( )cos 0 1 1I ( ) M 1 cos exp 2 d .     (6)c cc
ω δ δ δ ω− +
⎡ ⎤∆⎛ ⎞
∆ ∝ + − −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
( )( )sin 0 1 1 0I ( ) M 1 cos exp 2 2 d .     (7)c cc
ω δ φ φ φ ω+ −
⎡ ⎤∆⎛ ⎞
∆ ∝ + − + −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
Summarizing, two interferograms are simultaneously measured, allowing to obtain 
broadband FM spectra. The in-phase interferogram provides spectrally resolved information 
on the difference of absorption experienced by each group of two sidebands. The in-
quadrature interferogram gives the difference between the average of the dispersions 
experienced by the sidebands and the dispersion undergone by each carrier. 
 For this first experimental demonstration, a narrow-band emission source covering 0.25 
cm-1 (7.5 GHz) has been implemented as a test source. It is made of a fiber-coupled 
distributed feedback laser diode emitting around 1530 nm with an output power of a few mW. 
The current of the laser diode is modulated at about 20 Hz by a ramp generator. At each path 
difference step, while the interferometer is recording one interferogram sample, the laser 
frequency excursion is equal to 7.5 GHz, corresponding to one period of the triangular ramp. 
Consequently, for the interferometer, the laser diode behaves as a continuous emission source 
emitting over 0.25 cm-1. The interferometer output light is phase-modulated at fm = 150 MHz 
by the EOM and passes through an 80-cm cell filled at 10 hPa with acetylene in natural 
abundance. The light is next focused on an InGaAs nanosecond infrared photodetector, which 
according to Eq.5 delivers a signal proportional to the intensity of the beam containing a beat 
signal at the RF modulation frequency. The amplified detector signal is mixed with the 
reference signal at fm, down to d.c., using a commercial high frequency dual-phase lock-in 
amplifier. The reference may be phase-shifted with respect the signal used to drive the EOM. 
The two channels detected in-phase and in-quadrature are measured simultaneously.  
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Figure 2 shows a typical in-phase interferogram of C2H2. Its shape is characteristic of an 
interferogram of first-derivative type line-shapes. The 3 cm period amplitude modulation is 
due to the beat between the two strongest acetylene lines in the explored spectral domain. 
Figure 3 shows the two narrow-band spectra, Fourier transform of the in-phase (absorption) 
and in-quadrature (dispersion) interferograms. The spectral domain extension is limited by the 
tuning capabilities of the diode laser, which was used as a test source. This does not restrict 
the generality of the present demonstration. The lines belong to the ν1+ν3 and ν1+ν3+ν51-ν51 
overtone bands of 12C2H2. The unapodised spectral instrumental resolution: 12.5 10-3 cm-1 
(0.375 GHz) is narrower than the Doppler width of the lines. Signal to noise ratio is of the 
order of 1200. The total recording time of the order of 15 minutes is due to the need of 
adapting the interferometer recording mode procedure to the rather low laser diode frequency 
excursion period.  
The present validation of FM-FTS with a narrow band light source made the experience 
much simpler. Indeed, in wideband FTS, processing the signal of the interferogram needs 
special dynamic range solutions. Thanks to the only 0.25 cm-1-wide spectrum analysed in this 
experiment, a sophisticate RF detection chain, presently under development, was not 
necessary. The design of our Connes-type interferometer allows a balanced detection of the 
signals recorded at the two output ports. This will be helpful to remove the part of the 
interferogram which is not modulated by path difference and to consequently improve the 
dynamic range of the measurements. Similar solutions have already been successfully 
practiced for time-resolved FTS [7]. In FM-FTS, they are formally even easier to implement 
since the signal may be band-pass filtered around the modulation radio-frequency.  
In the present experimental set-up, the light should sequentially reach the equipment 
parts as shown in Fig.1. Briefly, to have a broadband equivalent of FM tunable laser 
spectroscopy, the sidebands generated by the EOM must not be resolved by the spectrometer. 
Also, since each carrier and its sidebands have to experience different attenuation and phase-
shift, the EOM must be placed before the cell containing the gas of interest. This matter will 
be discussed in more detail elsewhere.  
This first FM-FTS experiment demonstrates the feasibility of coupling broadband laser 
sources, Fourier spectrometers and RF detection. This opens new perspectives in high 
sensitivity multiplex spectroscopy. FM-FTS may be coupled to a large variety of high 
brightness sources. This includes broadband cw lasers, supercontinua sources, mode-locked 
lasers as demonstrated recently [8], and Amplified Spontaneous Emission sources. Frequency 
nonlinear conversion may also be used when no laser source is available in the spectral range 
of interest.  
FM-FTS may be practiced with any kind of Fourier transform spectrometers, including 
commercially available instruments, at the expense of reasonable modifications in the signal 
detection scheme. The approach is also suitable at low spectral resolution. In such case, 
modulation frequencies lying in the GHz domain may be used. Moreover,  FM-FTS induces 
new practices in Fourier transform spectroscopy. The modulation frequency is very high. The 
optical fringes generated by the interferometer can then be scanned at a much higher 
frequency than what is usually practiced nowadays. Path difference variation of the order of 1 
m/s, is easily affordable. It corresponds to acquisition times expressed in second when 
presently the most efficient existing high resolution interferometers need 1 to 10 hours to 
record interferograms.  Additionally, due to the low étendue of the analysed laser beams in 
our method, miniaturized instruments may be implemented. In addition to the radio-frequency 
detection scheme, sensitivity may be further enhanced by using an external optical resonator, 
thus increasing the effective absorption length. 
With FM-FTS, both the absorption and the dispersion associated with each spectral 
features are measured simultaneously. Despite its recognized interest for lineshape parameters 
retrieval, traditional dispersion spectroscopy has been poorly developed, only at low spectral 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
resolution, mostly due to its experimental complexity. FM-FTS should represent an easy 
manner of getting this information over extended spectral domains, which may induce new 
interest to the experimental investigation of dispersion profiles. 
References 
[1] G.C. Bjorklund, Frequency-modulation spectroscopy: a new method for measuring weak 
absorptions and dispersions, Optics Letters 5, 15-17 (1980). 
[2] G.C. Bjorklund, M.D. Levenson, W. Lenth, C. Ortiz, Frequency-modulation (FM) 
spectroscopy. Theory of  lineshapes and signal-to-noise analysis, Applied Physics B 32, 145-
152 (1983). 
[3] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley and H. 
Ward, Laser phase and frequency stabilization using an optical resonator, Applied Physics B 
31, 97-105 (1983).  
[4] W. Zapka, M. D. Levenson, F. M. Schellenberg, A. C. Tam, G. C. Bjorklund, Continuous-
wave Doppler-free two-photon frequency-modulation spectroscopy in Rb vapor Optics 
Letters 8, 27-29 (1983) 
[5] J.L. Hall, L. Hollberg, T. Baer, H.G. Robinson, Optical heterodyne saturation 
spectroscopy, Applied Physics Letters 39, 680-682 (1981). 
[6] P. Maddaloni, P. Malara, G. Gagliardi, P. De Natale, Two-tone frequency modulation 
spectroscopy for ambient-air trace gas detection using a portable difference-frequency source 
around 3 µm, Applied-Physics-B-Lasers-and-Optics B85, 219-22 (2006). 
[7] N. Picqué, G. Guelachvili, High-information time-resolved Fourier transform 
spectroscopy at work, Applied Optics 39, 3984-3990 (2000). 
[8] J. Mandon, G. Guelachvili, N. Picqué, Frequency Comb Spectrometry with Frequency 
Modulation, in preparation, 2007. 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Figure captions 
Fig. 1. 
Schematic of the experimental setup. 
Fig. 2. 
Absorption interferogram using in-phase RF detection with FM-FTS. Maximum path 
difference is 40 cm corresponding to 12.5 10-3 cm-1 unapodized resolution. 
Fig. 3. 
FM-FTS dispersion and absorption spectra of the acetylene molecule at 1528.6 nm. The 
middle plot represents the line relative intensities taken from the HITRAN database. 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Broadband 
source
FTS EOM Sample Cell
Driver
Lock-In
Detector
Fig. 1. Schematic of the experimental setup. 
0 10 20 30 40
Path difference (cm)
FM-FTS Interferogram (absorption channel)
Fig. 2. Absorption interferogram using in-phase RF detection with FM-FTS. Maximum path 
difference is 40 cm corresponding to 12.5 10-3 cm-1 unapodized resolution. 
6541.50 6541.75 6542.00
Dispersion
Absorption
(2) R
ν1+ν3+ν51-ν51
ν1+ν3
(cm-1)
Fig. 3. FM-FTS dispersion and absorption spectra of the acetylene molecule at 1528.6 nm. 
The middle plot represents the line relative intensities taken from the HITRAN database. 
	Frequency modulation Fourier transform spectroscopy

  A new method, FM-FTS, combining Frequency Modulation heterodyne laser
spectroscopy and Fourier Transform Spectroscopy is presented. It provides
simultaneous sensitive measurement of absorption and dispersion profiles with
broadband spectral coverage capabilities. Experimental demonstration is made on
the overtone spectrum of C2H2 in the 1.5 $\mu$m region.

Title
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Frequency modulation Fourier transform spectroscopy 
Julien Mandon, Guy Guelachvili, Nathalie Picqué 
Laboratoire de Photophysique Moléculaire, CNRS; Univ. Paris-Sud, Bâtiment 350, 91405 
Orsay, France 
Corresponding author: 
Dr. Nathalie Picqué, 
Laboratoire de Photophysique Moléculaire 
Unité Propre du CNRS, Université Paris Sud, Bâtiment 350 
91405 Orsay Cedex, France 
Phone number: +33 1 69156649 
Fax number: +33 1 69157530 
Email: nathalie.picque@ppm.u-psud.fr
Web: http://www.laser-fts.org  
Abstract: A new method, FM-FTS, combining Frequency Modulation heterodyne laser 
spectroscopy and Fourier Transform Spectroscopy is presented. It provides simultaneous 
sensitive measurement of absorption and dispersion profiles with broadband spectral coverage 
capabilities. Experimental demonstration is made on the overtone spectrum of C2H2 in the 1.5 
µm region. 
OCIS codes: 120.6200, 300.6300, 300.6380, 300.6360, 300.6310, 300.6390, 120.5060 
120.6200  Spectrometers and spectroscopic instrumentation, 300.6300 Spectroscopy, Fourier 
transforms, 300.6380  Spectroscopy, modulation, 300.6360  Spectroscopy, laser, 300.6310  
Spectroscopy, heterodyne, 300.6390 Spectroscopy, molecular, 120.5060  Phase modulation 
mailto:nathalie.picque@ppm.u-psud.fr
http://www.laser-fts.org/
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Improving sensitivity is presently one of the major concern of spectroscopists. This may 
be obtained both from the enhancement of the intrinsic signal, and from the reduction of the 
background noise. In this latter case, modulation has been one of the most effective approach. 
In particular, Frequency Modulation (FM) absorption spectroscopy [1] has reached detection 
sensitivity near to the fundamental quantum noise limit, by shifting the frequency modulation 
of the measurements to a frequency range where the 1/f noise becomes negligible. Moreover, 
FM spectroscopy benefits from high-speed detection and simultaneous measurement of 
absorption and dispersion signals. Since Bjorklund’s first demonstrations [1,2] of the 
efficiency of FM spectroscopy with a single-mode continuous-wave dye laser, the technique 
has been widely used as a tunable laser spectroscopic method in fields such as laser 
stabilization [3], two-photon spectroscopy [4], optical heterodyne saturation spectroscopy [5], 
trace gas detection [6]. In most schemes, the laser wavelength is scanned across the 
atomic/molecular resonance to retrieve the line shape. More rarely, the modulation frequency 
is tuned. However in both cases, the measurements are limited to narrow spectral ranges. 
This letter reports the first results in FM broadband spectroscopy. This work is  
motivated by our ongoing effort of implementing a new spectroscopic approach 
simultaneously delivering sensitivity, resolution, accuracy, broad spectral coverage and rapid 
acquisition. The basic idea, named FM-FTS, is to associate the advantages of FM 
spectroscopy and high-resolution Fourier transform spectroscopy (FTS). FTS is able to record 
at once extended ranges, with no spectral restriction. In particular it gives easy access to the 
infrared domain. In this letter, a new way of modulating the interferogram is implemented. 
The key concept is that a radio frequency (RF) modulation is performed. The beat signal at 
the output port of the Fourier transform spectrometer is modulated at constant RF, which is 
about 104 times greater than the audio frequency generally delivered by the interferometer 
optical conversion. Together with the advantage, over classical FTS, of measurements 
performed at much higher frequency, our approach benefits from the synchronous detection 
ability and from the simultaneous acquisition of both the absorption and the dispersion of the 
recorded profiles. 
The experimental principle is presented in Fig. 1. The light emitted by the broadband 
source is first passing through the interferometer. The output beam is then phase-modulated 
by an electro-optic modulator (EOM) before entering the absorption cell and falling on the 
fast detector. The synchronous detection of the detector signal is realized by the lock-in 
amplifier at the EOM driver reference frequency fm. Recorded data are finally stored on the 
computer disk with their corresponding path difference position ∆. Their Fourier transform is 
the spectrum. 
In more details, the electric field E at the output of the interferometer may be written as: 
0 ( )( , ) 1 exp exp( )d c.c.     (1)
c c ct i i tc
ω ω ω
⎡ ⎤∆⎛ ⎞
∆ = + − +⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
where E0 is the electric field amplitude of the source at ωc optical pulsation, c is the velocity 
of light and c.c the conjugate complex of the preceding expression in Eq. 1. The EOM effect 
on the beam is assumed to have a low modulation index M. As a consequence, each carrier 
wave of pulsation ωc, has two weak sidebands located at ± ωm = ± 2π fm. Equation (1) 
becomes:  
( ) ( ) }
0 ( )( , ) 1 exp exp
M exp M exp d c.c.     (2)
c m c m c
t i i t
i t i t
ω ω ω ω ω
⎡ ⎤∆⎛ ⎞
∆ = + −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
⎡ ⎤ ⎡ ⎤+ + − − +⎣ ⎦ ⎣ ⎦
When interacting with the gas, the carrier and the sidebands experience attenuation and phase-
shift due to absorption and dispersion. Following the notations introduced in [1], this 
interaction may be written as exp(-δ(ω)- i φ(ω)) where δ is the amplitude attenuation and φ is 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
the phase shift. The following convention is adopted: δn and φn denotes for n = 0, ±1 the 
respective components at ωc and ωc± ωm. Then Eq. 2 may be written: 
( ) ( ){
( ) ( ) ( ) ( ) }
1 1 1 1
( , ) 1 exp exp exp
M exp exp M exp exp d c.c.     (3)
c m c m c
t i i i t
i i t i i t
ω δ φ ω
δ φ ω ω δ φ ω ω ω+ + − −
⎡ ⎤∆⎛ ⎞
∆ = + − − −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
⎡ ⎤ ⎡ ⎤+ − − + − − − − +⎣ ⎦ ⎣ ⎦
The intensity I detected by the fast photodetector is proportional to : 
*I( , ) ( , ) ( , ).     (4)t t t∆ ∝ ∆ ∆E E  
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 1 1
0 1 0 1 0 1 0 1
0 1 1 0 0 1 0 1
I( , ) exp 2 exp 2 exp 2 1 cos
2M cos exp cos exp cos 1 cos
2M sin exp sin exp sin 1 cos
δ δ δ ω
ω δ δ φ φ δ δ φ φ ω
ω δ δ φ φ δ δ φ φ ω
+ + − −
− − + +
⎛ ⎡ ⎤∆⎛ ⎞⎡ ⎤∆ ∝ − + − + − +⎜ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦⎝
⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − − − − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
∆⎛ ⎞⎡ ⎤+ − − − − − − − + ⎜⎣ ⎦ ⎝ ⎠
( ) ( ) ( )
( ) ( ) ( )
1 1 1 1
1 1 1 1
2M cos 2 exp sin 1 cos
2M sin 2 exp cos 1 cos d .                                        (5)
m c c
ω δ δ φ φ ω
ω δ δ φ φ ω ω
+ − − +
+ − − +
⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
⎞⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − + ⎟⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎠
After synchronous detection at fm frequency and with the assumption that |δ0-δj |<<1 and |φ0-
φj|<<1 (with j = ±1), the in-phase Icos(∆) and the in-quadrature Isin(∆) parts of the electric 
signal are given by  
( )( )cos 0 1 1I ( ) M 1 cos exp 2 d .     (6)c cc
ω δ δ δ ω− +
⎡ ⎤∆⎛ ⎞
∆ ∝ + − −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
( )( )sin 0 1 1 0I ( ) M 1 cos exp 2 2 d .     (7)c cc
ω δ φ φ φ ω+ −
⎡ ⎤∆⎛ ⎞
∆ ∝ + − + −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
Summarizing, two interferograms are simultaneously measured, allowing to obtain 
broadband FM spectra. The in-phase interferogram provides spectrally resolved information 
on the difference of absorption experienced by each group of two sidebands. The in-
quadrature interferogram gives the difference between the average of the dispersions 
experienced by the sidebands and the dispersion undergone by each carrier. 
 For this first experimental demonstration, a narrow-band emission source covering 0.25 
cm-1 (7.5 GHz) has been implemented as a test source. It is made of a fiber-coupled 
distributed feedback laser diode emitting around 1530 nm with an output power of a few mW. 
The current of the laser diode is modulated at about 20 Hz by a ramp generator. At each path 
difference step, while the interferometer is recording one interferogram sample, the laser 
frequency excursion is equal to 7.5 GHz, corresponding to one period of the triangular ramp. 
Consequently, for the interferometer, the laser diode behaves as a continuous emission source 
emitting over 0.25 cm-1. The interferometer output light is phase-modulated at fm = 150 MHz 
by the EOM and passes through an 80-cm cell filled at 10 hPa with acetylene in natural 
abundance. The light is next focused on an InGaAs nanosecond infrared photodetector, which 
according to Eq.5 delivers a signal proportional to the intensity of the beam containing a beat 
signal at the RF modulation frequency. The amplified detector signal is mixed with the 
reference signal at fm, down to d.c., using a commercial high frequency dual-phase lock-in 
amplifier. The reference may be phase-shifted with respect the signal used to drive the EOM. 
The two channels detected in-phase and in-quadrature are measured simultaneously.  
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Figure 2 shows a typical in-phase interferogram of C2H2. Its shape is characteristic of an 
interferogram of first-derivative type line-shapes. The 3 cm period amplitude modulation is 
due to the beat between the two strongest acetylene lines in the explored spectral domain. 
Figure 3 shows the two narrow-band spectra, Fourier transform of the in-phase (absorption) 
and in-quadrature (dispersion) interferograms. The spectral domain extension is limited by the 
tuning capabilities of the diode laser, which was used as a test source. This does not restrict 
the generality of the present demonstration. The lines belong to the ν1+ν3 and ν1+ν3+ν51-ν51 
overtone bands of 12C2H2. The unapodised spectral instrumental resolution: 12.5 10-3 cm-1 
(0.375 GHz) is narrower than the Doppler width of the lines. Signal to noise ratio is of the 
order of 1200. The total recording time of the order of 15 minutes is due to the need of 
adapting the interferometer recording mode procedure to the rather low laser diode frequency 
excursion period.  
The present validation of FM-FTS with a narrow band light source made the experience 
much simpler. Indeed, in wideband FTS, processing the signal of the interferogram needs 
special dynamic range solutions. Thanks to the only 0.25 cm-1-wide spectrum analysed in this 
experiment, a sophisticate RF detection chain, presently under development, was not 
necessary. The design of our Connes-type interferometer allows a balanced detection of the 
signals recorded at the two output ports. This will be helpful to remove the part of the 
interferogram which is not modulated by path difference and to consequently improve the 
dynamic range of the measurements. Similar solutions have already been successfully 
practiced for time-resolved FTS [7]. In FM-FTS, they are formally even easier to implement 
since the signal may be band-pass filtered around the modulation radio-frequency.  
In the present experimental set-up, the light should sequentially reach the equipment 
parts as shown in Fig.1. Briefly, to have a broadband equivalent of FM tunable laser 
spectroscopy, the sidebands generated by the EOM must not be resolved by the spectrometer. 
Also, since each carrier and its sidebands have to experience different attenuation and phase-
shift, the EOM must be placed before the cell containing the gas of interest. This matter will 
be discussed in more detail elsewhere.  
This first FM-FTS experiment demonstrates the feasibility of coupling broadband laser 
sources, Fourier spectrometers and RF detection. This opens new perspectives in high 
sensitivity multiplex spectroscopy. FM-FTS may be coupled to a large variety of high 
brightness sources. This includes broadband cw lasers, supercontinua sources, mode-locked 
lasers as demonstrated recently [8], and Amplified Spontaneous Emission sources. Frequency 
nonlinear conversion may also be used when no laser source is available in the spectral range 
of interest.  
FM-FTS may be practiced with any kind of Fourier transform spectrometers, including 
commercially available instruments, at the expense of reasonable modifications in the signal 
detection scheme. The approach is also suitable at low spectral resolution. In such case, 
modulation frequencies lying in the GHz domain may be used. Moreover,  FM-FTS induces 
new practices in Fourier transform spectroscopy. The modulation frequency is very high. The 
optical fringes generated by the interferometer can then be scanned at a much higher 
frequency than what is usually practiced nowadays. Path difference variation of the order of 1 
m/s, is easily affordable. It corresponds to acquisition times expressed in second when 
presently the most efficient existing high resolution interferometers need 1 to 10 hours to 
record interferograms.  Additionally, due to the low étendue of the analysed laser beams in 
our method, miniaturized instruments may be implemented. In addition to the radio-frequency 
detection scheme, sensitivity may be further enhanced by using an external optical resonator, 
thus increasing the effective absorption length. 
With FM-FTS, both the absorption and the dispersion associated with each spectral 
features are measured simultaneously. Despite its recognized interest for lineshape parameters 
retrieval, traditional dispersion spectroscopy has been poorly developed, only at low spectral 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
resolution, mostly due to its experimental complexity. FM-FTS should represent an easy 
manner of getting this information over extended spectral domains, which may induce new 
interest to the experimental investigation of dispersion profiles. 
References 
[1] G.C. Bjorklund, Frequency-modulation spectroscopy: a new method for measuring weak 
absorptions and dispersions, Optics Letters 5, 15-17 (1980). 
[2] G.C. Bjorklund, M.D. Levenson, W. Lenth, C. Ortiz, Frequency-modulation (FM) 
spectroscopy. Theory of  lineshapes and signal-to-noise analysis, Applied Physics B 32, 145-
152 (1983). 
[3] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley and H. 
Ward, Laser phase and frequency stabilization using an optical resonator, Applied Physics B 
31, 97-105 (1983).  
[4] W. Zapka, M. D. Levenson, F. M. Schellenberg, A. C. Tam, G. C. Bjorklund, Continuous-
wave Doppler-free two-photon frequency-modulation spectroscopy in Rb vapor Optics 
Letters 8, 27-29 (1983) 
[5] J.L. Hall, L. Hollberg, T. Baer, H.G. Robinson, Optical heterodyne saturation 
spectroscopy, Applied Physics Letters 39, 680-682 (1981). 
[6] P. Maddaloni, P. Malara, G. Gagliardi, P. De Natale, Two-tone frequency modulation 
spectroscopy for ambient-air trace gas detection using a portable difference-frequency source 
around 3 µm, Applied-Physics-B-Lasers-and-Optics B85, 219-22 (2006). 
[7] N. Picqué, G. Guelachvili, High-information time-resolved Fourier transform 
spectroscopy at work, Applied Optics 39, 3984-3990 (2000). 
[8] J. Mandon, G. Guelachvili, N. Picqué, Frequency Comb Spectrometry with Frequency 
Modulation, in preparation, 2007. 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Figure captions 
Fig. 1. 
Schematic of the experimental setup. 
Fig. 2. 
Absorption interferogram using in-phase RF detection with FM-FTS. Maximum path 
difference is 40 cm corresponding to 12.5 10-3 cm-1 unapodized resolution. 
Fig. 3. 
FM-FTS dispersion and absorption spectra of the acetylene molecule at 1528.6 nm. The 
middle plot represents the line relative intensities taken from the HITRAN database. 
Frequency modulation Fourier transform spectroscopy  Mandon, Guelachvili, Picqué, 2007 
Broadband 
source
FTS EOM Sample Cell
Driver
Lock-In
Detector
Fig. 1. Schematic of the experimental setup. 
0 10 20 30 40
Path difference (cm)
FM-FTS Interferogram (absorption channel)
Fig. 2. Absorption interferogram using in-phase RF detection with FM-FTS. Maximum path 
difference is 40 cm corresponding to 12.5 10-3 cm-1 unapodized resolution. 
6541.50 6541.75 6542.00
Dispersion
Absorption
(2) R
ν1+ν3+ν51-ν51
ν1+ν3
(cm-1)
Fig. 3. FM-FTS dispersion and absorption spectra of the acetylene molecule at 1528.6 nm. 
The middle plot represents the line relative intensities taken from the HITRAN database. 
	Frequency modulation Fourier transform spectroscopy
