Document text
Improving the sensitivity of FM spectroscopy using
nano-mechanical cantilevers
B.M. Chernobrod, G.P. Berman, and P.W. Milonni
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
It is suggested that nano-mechanical cantilevers can be employed as high-
er filters to circumvent laser noise limitations on the sensitivity of frequency
modulation spectroscopy. In this approach a cantilever is actuated by the
radiation pressure of the amplitude modulated light that emerges from an
absorber. Numerical estimates indicate that laser intensity noise will not
prevent a cantilever from operating in the thermal noise limit, where the high
Q's of cantilevers are most advantageous.
Frequency modulation spectroscopy (FMS) [1,2] has proven to be one of the most sensi-
tive absorption-based spectroscopic techniques. The essential idea is that when a frequency
modulated laser beam enters an absorption cell the emerging, partially absorbed beam is am-
plitude modulated (AM). If the modulation index of the frequency modulation is sufficiently
small, the spectrum of the incident FM beam consists primarily of the peak at the carrier
frequency uo c and sidebands at uo c ± Q, where Q is the modulation frequency. In addition to
the component at the carrier frequency, the intensity of the output beam has a component
that oscillates sinusoidally at Q with an amplitude proportional to the modulation index
and to the difference in the attenuation coefficients of the absorber at the frequencies uj c + Q
and uj c — Q. The output of a photodetector is electronically filtered and amplified, and the
signal of interest that oscillates at Q is extracted by a mixer. An absorption spectrum is
obtained by scanning the laser frequency over the spectral feature of interest.
It is important for FMS that there be little spectral overlap between the carrier and the
sidebands. If the modulation frequency is not high enough, the spectral wings of the side-
bands and the carrier will overlap, making the exact amplitude and phase balance required
for full FM beat cancellation impossible. Thus, one of the major limiting factors in FMS is
laser noise, which requires that the modulation frequency be large compared with the laser
bandwidth. For laser bandwidths of 10-100 MHz, for instance, modulation frequencies in
the 100-1000 MHz range are desirable. However, electronic detection systems, consisting
of a photodetector and several amplification cascades, produce an additional noise which
increases with increasing frequency, so that the shift to higher modulation frequency could
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be inefficient. The highest sensitivity of FMS is usually achieved with well stabilized, low-
power semiconductor lasers. While it is possible to substantially reduce the laser technical
noise and bandwidth in these lasers, this is generally incompatible with the relatively high
laser powers required, for instance, for remote sensing or long-length multipass cells.
In this letter we suggest the use of nano-mechanical cantilevers (nano-resonators) as
filters with much higher Q factors than are currently possible by conventional methods. In
this approach the AM signal that emerges from the absorption cell (or is backscattered in
the case of remote sensing) actuates a cantilever by resonant light pressure or by optical
gradient forces. The cantilever functions both as a high-frequency detector and as a high-Q
filter. As discussed below, the use of cantilevers in FMS in this way could offer the possibility
of detecting molecules with unprecedented sensitivity.
The lower limit on aL, where a is the absorption coefficient and L is the total propagation
length, can be estimated from the condition that the signal-to-noise ratio (SNR C ) be unity.
SNR C can be written as
„2
cy\rp _ El (])
x s
,J SN) K x ¥ J ~>~ K^N
where x S i g is the vibrational amplitude of the cantilever, is the root mean square (rms)
vibration noise induced by the laser shot noise, x™ s is the rms vibrational thermal noise,
and £™ s is the vibrational noise induced by the laser intensity noise. Substitution in (1) of
the expressions for the vibrational amplitudes gives
9/V7? = Q\l + R)\aLfPj
c 4k B Tkc 2 + Q(l + R) 2 cu [P hu; + P N (u ) 2 ] ' 1 '
where R is the reflection coefficient, k is the spring constant, k B is Boltzmann's constant,
T is the temperature, Pq is the incident laser power, u; is the fundamental frequency of
the cantilever, and Pn(u) = £(u)P is the laser intensity noise, where £(a>) is the relative
intensity noise (RIN). Consider as an example the following parameter values: T = 4 K,
k = 0.3 N/m, Q = 2 x 10 5 , R = 0.5, P = 100 fiW, and u = 20 MHz. In this case
the laser noise dominates if the RIN satisfies the inequality £(u) > 1.8 x 10~ 5 Hz -1 / 2 .
This value of the RIN is typical for solid state lasers [3]. Neglecting the shot noise and
thermal noise, we obtain SNR C ~ (aL) 2 /(lu £ 2 (lu)). The condition SNR C = 1 then gives
(aL) cantilever = Z(u} )yju /Q = 1.8 X 10~ 4 .
Let us compare this estimate of the minimal aL with the sensitivity of conventional
electronic detection. The photodetection usually involves at least three electronic stages [4] .
The first stage is the photodetector and preamplifier or the photomultiplier, or avalanche
photodiode; the second stage is the lock-in-amplifier; and the third stage is the output
amplifier. Each stage produces noise. The noise of the two first stages increases significantly
at higher modulation frequencies. We can characterize the noise by the noise figure NF =
2
10\og[SNR out /SNR in } (where SNR in and SNR out are the SNR in the input and output
signals, correspondingly). For a modulation frequency greater than 10 MHz the noise figure
for the photodetector plus preamplifier is about NFx = 2 dB; for the lock-in-amplifier
NF 2 = 3 dB - 5 dB; and for the output amplifier NF 3 = 4 dB. Thus the total noise figure is
NF = NFi x NF 2 x NF 3 ~ 10 dB. The SNR for electronic photodetection can be written
as
o NE = m
Afe(gP + 2k B T/R) + Afg*& ff (uo)P§ ' W
where g = er]/hu . The symbols in this equation are electric charge e, detector quantum
efficiency rj, photon energy tuu, bandwidth Af, resistance R, and £ e // = x NF. The
first term in the denominator corresponds to the laser shot noise, the second term to thermal
noise, and the third term to the laser intensity noise. For the parameter values i? = 50 SI,
i] = 0.8, = 1.8 x 10~ 5 Hz~ 1/2 , and NF = lOdB, the laser intensity noise dominates,
and the condition SNR C = 1 for the minimum detectable absorption gives («L) c i cc t r0 mc =
e(a;)7VFA/ 1 /2.
To avoid additional loss of signal, the effective bandwidth Af should exceed the band-
width of modulation. For the modulation frequency u m /27i = 20 GHz and the (highest)
quality factor Q rn = 2 x 10 8 , the bandwidth Af = 100 Hz. The assumption that the
cantilever bandwidth Uo/Q = Af implies (aL) cant iiever = 0.1(o;L) c i cc t r onic, i-e., the smallest
measurable absorption using the cantilever is less than the corresponding value for electronic
detection by at least an order of magnitude.
By choosing the cantilever resonance frequency appropriately, the proposed sensor can be
made to operate in the thermal noise limit. To see this, let us assume a laser noise spectrum
P/v(c<-0 — Po£ [r 2 /(r 2 + (ujl — uj) 2 )] 1 ^ 2 , where £ is the spectral density of relative intensity
noise (RIN) at the center of the spectral distribution around the peak of laser intensity noise
at the frequency ujl- If the cantilever is to operate in the thermal noise limit the following
condition must be satisfied: x™ s > x™ s , or
1/2
c A7ck B Tk . .
_l + /i 2 J ?< (l + i?)P V QT U
where /i = (lu — cj l )/T. For uj l = 0.3 MHz, T = 1 MHz, and for the other parameters
assumed above, the inequality (4) gives /i > 5, or lu > 5 MHz.
We have assumed that radiation pressure rather than the photothermal effect produces
the dominant force in exciting the cantilever vibrations. Experimental evidence suggests
that this is indeed the case for Si cantilevers [5,6]; namely, the fact that the cantilever could
be actuated at very high temperatures, where thermal gradients are much smaller than the
surface temperature, suggests that photothermal effects are relatively small compared with
radiation pressure.
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Large resonance frequencies uq, and therefore small cantilevers, are generally desirable
for increasing the sensitivity. If the cantilever is smaller than the spot size of the beam
incident upon it, actuation of the cantilever vibrations may be inefficient. In this case one
could use a scheme of apertureless near-field microscopy. The tip is put in close proximity to
the cantilever surface, and focused light illiminates the tip-surface region. The light intensity
near the tip apex exceeds the external intensity by an enhancement factor that could be
~ 10 6 in the case of a plasmon resonance with a metallic tip. Near the tip the field is very
inhomogeneous, implying that the gradient force could exceed the force of light pressure,
depending on the geometry.
mid-IR laser
/—{ —
cantilever! ;
abs. cell
R
two-tone frequency modulator
modulator
low power frequency-
stabilized laser
FIG. 1. A frequency modulated laser beam is passed through an absorption cell and causes a
cantilever to vibrate near its resonant frequency. The cantilever vibrations are detected interfero-
metrically, as indicated in the right side of the figure.
Optical actuation of cantilevers by light pressure has been demonstrated [5,6]. In the
experiments of Yang et ai, deflections of a 60 x 6 /im 2 cantilever with 680- nm, 40-//W
laser radiation of beam size ~ 300 x 100 /im 2 were observed. Deflections were observed for
temperatures as high as 780 C. Their cantilever had a Q factor m 10 5 and a spring constant
k = 4.4 x 10~ 3 N/m. For these parameters we estimate a force sensitivity Ft ~ 10~ 16 N, in
rough agreement with the value quoted by Yang et al. The condition (4) for the cantilever
to operate in the thermal noise limit is found to be easily realized for these parameters. Such
numerical estimates support the viability of the proposed cantilever sensor. Note also that
in Reference [7] a cantilever quality factor up to Q ~ 10 5 was achieved for driven oscillations
of the cantilever for a laser power of a few hundred microwatts.
In conclusion, we have presented estimates indicating that nano-mechanical cantilevers
can be employed as high-Q filters to circumvent laser noise limitations on the sensitivity of
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frequency modulation spectroscopy.
ACKNOWLEDGMENTS
We thank Professor Umar Mohideen and Dr. C. E. Strauss for helpful discussions. This
work was supported by the Department of Energy under the contract W-7405-ENG-36 and
DOE Office of Basic Energy Sciences, and by the Defense Advanced Research Projects
Agency.
[1] G.C. Bjorklund, Opt. Lett. 5, 15 (1980).
[2] Reviews of the subject are given, for instance, by P. Werle, Infra. Phys. Tech. 37, 59 (1996);
P. Kluczynski et al, Spectrochim. Acta A 51, 2211 (2001); and K. Song and E.C. Jung, Appl.
Spectrosc. Rev. 38, 395 (2003).
[3] S. Taccheo et al, IEEE Phot. Tech. Lett. 13, 19 (2001).
[4] J. Silver, Appl. Opt. 31, 707 (1992).
[5] O. Marti et al, Ultramicrosc. 42-44, 345 (1992).
[6] J. Yang, T. Ono, and M. Esashi, Appl. Phys. Lett. 77, 3860 (2000).
[7] M. Zalalutdinov et al, Appl. Phys. Lett., 79, 695 (2001).
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