Volume 154, number 5,6
PHYSICS LETTERS A
8 April 1991
Optical sub-wavelength displacement sensors Stephan Schiller and Roberto Onofrio’ Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA Received 17 September 1990; accepted for publication 7 February 1991 Communicated by J.P. Vigier
We propose the use of techniques developed for sub-wavelength optical microscopy to monitor displacements of macroscopic objects at levels better than 10—’ m. Models ofoptomechanical transducers that are based on photon tunneling and near-zone field effects and the sensitivity ofsome realistic designs are discussed.
Monitoring displacements of macroscopic objects with high sensitivity is of crucial importance in some high-precision experiments, e.g. in gravitational-wave detection [1] and for studying the dynamics of macroscopic bodies in the quantum regime [2]. Displacement sensors such as piezoelectric [3], induc-
A simple model ofa photon-tunneling sensor based on frustrated total internal reflection is shown in fig. 1. A monochromatic plane wave of intensity I,, polarized parallel to the interface between homogeneous, dielectric media 1 and 2 (indices of refraction n1 > n2), is totally internally reflected for angles of
tive [4], capacitive [5] transducers and optical interferometers [6] are based on classical phenomena. The sensitivity of the first three types is independent ofany characteristic length while for optical interferometers it is wavelength-dependent. Recently a novel class of transducers, electron tunneling probes (ETP5), has been developed which makes use of a purely quantum phenomenon in which the sensitivity is determined by the de Broglie wavelength of the tunneling electrons, of the order of 1— 10 A [7]. Spatial resolution on the nanometer scale has also been achieved in sub-wavelength scanning optical microscopy [8,9]. Here we discuss applications of this lund of microscopy for the detection of one-dimensional displacements First we analyze the analog of ETPs tunneling sensors based on the de tection of evanescent waves transmitted through a gap. Secondly we discuss near-zone field phenomena which exhibit spatial sensitivity determined by the size of the probe (apertures or antennae) rather than the optical wavelength.
incidence O~>arcsin( n2In1). An evanescent wave is present in medium 2 with an exponentially decaying electric field having a decay length [10] 2O ,~ [sin 1 (n2/n1 ) 2j I/2 (1)
On leave from Scuola del Dottorato di Ricerca, Dipartimento di Fisica dell’Università di Roma “La Sapienza” and INFN, Sezione di Roma, Rome 00185, Italy. 0375-960l/91/$ 03.50 © 1991
—
—
—
/ /
~ St
‘-~~
mass
laser
1~ ~ ::::: x ut__~~83n~ f ~
n~
/
/ /
/
~7
Fig. 1. Detection of displacements of a macroscopic body (test mass) through evanescent waves. Changes in the gap width are monitored through variations in the intensity of the transmitted light.
Elsevier Science Publishers B.V. (North-Holland)
221
Volume 154, number 5,6
PHYSICS LETTERS A
Here A, is the wavelength in medium 1. In the presence of a third medium (index of refraction n3) energy flux through the gap occurs by conversion of the evanescent field into a propagating wave. The transmission coefficient is given by T=~= —2sinØ,2sinØ23 I, cosh(x/c~)+cos(Ø12+Ø23) where —2n, cos 0, ~,/‘n~sin20, —n~ Ø, 2O 2O 2=arctan n (cos 1 sin 1 ) + n ~ ~
023
—
(2)
=
(SNR
~
SNR =
1/h v
~
L~X opt
=
A 222
(4)
— —
55/2
16a
~
(~/2—0,), 0,~m/2, 0, ~arcsin(n
A
2/n,).
~
______
=
mm
~,
(3)
The most fundamental contribution to SNR is the quantum shot noise inherent in the laser field and the photodetection process. It is given by P
~°~‘
0pl
(5)
—,
=
~
—n~’ (9)
Due to the divergence oflaser beams, in practice one is limited to angles O~distinctly smaller than 90°.As an example we take 0, =85°,A=600 nm, P,= 1 mW, 5m. = 0.8, i41=10 A~OPt 2xl0~Hz, obtaining x°”~5.5 nm and To achieve this sensitivity, frequency fluctuations mm of the laser must be less than mSi’/v=&°°’ /x°~’ 2 x 10—v, which is well within current laser technology [11]. Obviously laser amplitude fluctuations directly map into apparent displacements, according to ~ /x°” 3~I~/I~. Shot noise limited displacement sensitivity therefore also requires amplitude stability at the shot-noise limit; this can be accomplished by external electro-optic amplitude control [121 of diode pumped solid state lasers, which ex-
6)
where 41 is the detection bandwidth, P, and P, are
0.3~ 0 08
the transmitted and ij is the quantum and incident efficiency powers, of the respectively, detector. Combining (5) and (6) we obtain (7) The factor a = ,.Ji1~, /h v4f depends on photon source and detector properties only. For given 0,, &mifl exhibits a minimum for a gap width
+~,,/~os2(oI2
222
+O23)+15]/3}
~
~
‘_,
0.06 .
0.2
-~
0.1
p.,
:~
0.02 0.0c
__________
x°°’ (0, ) = çe arccosh { [cos (Ot 2
cal-
culated from (7) and the corresponding gap are shown in fig. 2 as a function of incidence angle, with the choice n, =n3=2, n2= 1. Both decrease to zero for grazing incidence of the beam. Expanding (7) in the limits 01—~arcsin(n2/n,)and 0,—+it/2 one obtains (for n,=n3) T=
and x is the distance between interfaces 1—2 and 2— 3. If either medium 1 or 3 is attached to a macroscopic body, a displacement of the latter results in a change of transmitted intensity I,. The sensitivity of I~to changes in the gap width is obtained by performing the derivative of (2) with respect to x. To determine of thetheminimum detectable displacement knowledge x-independent fluctuations of I~is required. For a signal-to-noise ratio SNR of I,, the displacement sensitivity L~Xmjn may then be estimated from ~mmn
The minimum detectable displacement ~
‘
_2fl3COS03~fl~Sifl20i_fl~ =arctan n~cos203—n~sin2O, +n~
8 April 1991
30
40
50
60
70
angle of incidence
80
~
90
8,
Fig. 2. Optimal gap width (solid curve, left ordinate scale) and minimum detectable displacement (dashed curve, right scale) versus the incident angle 0,. The indices of refraction are
+ 023) .
0.0
(8)
n,=n3=2, n2=l.
Volume 154, number 5,6
PHYSICS LEfl’ERS A
hibit high intrinsic amplitude and frequency stability [13]. An alternative scheme consists in detecting both the transmitted and reflected intensities with two photodiodes and subtracting the signals to obtam a discriminator output. The electronic gain for the reflected beam channel is chosen such that the effect of changes in I, is suppressed to first order in the difference photocurrent. Johnson noise in the detector is smaller than the shot-noise for transmitted powers larger than 10—i W (room-temperature photodiode with low input impedance) [14]. Finally, the Brownian noise of the macroscopic mass to which the transducer is attached can be reduced below the shot noise level if low-dissipation materials such as sapphire single crystals at cryogenic temperatures are used, as in resonant gravitational wave antennae [15]. A careful design of the attachment of the first optical medium to the test mass is required to preserve the high mechanical quality factor. Another class of sub-wavelength optical phenomena has been studied in connection to near-field optical scanning microscopy. It has been shown that the radiation emitted from antennae (aperture or probes) with diameters d smaller than A remains spatially collimated up to distances ofthe order of d away from the aperture (the proximity zone). To achieve very high lateral and vertical spatial resolution imaging of a surface of the object, it is placed at a distance x d from the antenna so that it interacts with the radiation field only over an area of the
I
r~
laser ~
test mass x
L~J
j,
~::~~:_I
8 April 1991
order of d2. The effects of this near-field antenna— object interaction on the far field radiation pattern are measured by means of a photodetector. Because the strength of the localized interaction has a strong dependence of the distance between antenna and object, surface mapping is possible. While a quantitative description of the antenna radiation in the presence of an object is not yet available, an estimate for the sensitivity to vertical displacements can be deduced from the spatial dependence of the field strength around a circular aperture [161, dT 0 1 (10) —~-.
In this rough estimate we neglect the modifications in the field induced by the presenceof the object. On the other hand the detected signal intensity for a given intensity illuminating the aperture decreases with decreasing aperture size. In a typical experiment a laser beam ofa few mW power incides on the end of a metal-coated pipette with aperture diameter d~10— 100 nm and transmitted flux of about l0~photons! s is obtained [8]. Using (5) a sensitivity Ax~1 nm is obtained assuming an integration time of 0.1 s and an aperture width d= 10 nm. Values of this order of magnitude have recently been demonstrated [17]. Improvements in sensitivity seem possible by increasing transmitted power through the use of pipettes whose tips contain an excitonic material that channels the laser light more effectively to the aperture [18]. Resonance phenomena occurring between antenna and receiver at very small separation provide another path to sub-nanometer sensitivity. It has been shown that surface plasmon resonances of thin metallic films coated on asperities can be excited by the presence of a dielectnc surface at a particular dis tance The sensitivity of the excitation to vertical
J j.
P PC sample~”~” ~
tte
detector Fig. 3. Example of a configuration for measuring displacements through near-zone field effects. The onfice of the pipette has a diameter ofthe order ofnanometers and the sample is located in its near-zone. The transmitted light intensity 1, is sensitive t~ changes in the relative position ofpipette and sample.
displacements may be estimated from the spatial width ofthe resonance peak. Widths as low as 15 nm have been measured with high SNR (—~100) [19], indicating that displacements of the order of 1 A or less are measurable. While the sensitivity dT/dx of these near-field techniques is comparable to photon tunneling probes, their throughputs P~are very low, .
.
.
.
.
.
limiting the actual sensitivity &mjfl. As a possible solution, it has been suggested [18] that the energy transfer from an antenna made of an excitonic ma223
Volume 154, number 5,6
PHYSICS LETTERS A
terial (anthracene) to a surface containing appropriate acceptors (rhodamine) can be orders of magnitude higher than in the case of dielectric receivers. Furthermore, displacements measurements require spatial resolution only in one dimension. Therefore it should be possible to use slits instead of the tiplike antennae employed for microscopy. An antenna with macroscopic extension along one dimension but nanometer size in an orthogonal direction still exhibits a laterally focused field necessary for spatial sensitivity [201 but has an enormously higher light throughput. It is useful to compare the optical transducers discussed so far with electron tunneling displacement sensors. The advantage of optical transducers with appropriately stabilized laser sources lies in the possibility ofoperating in the shot-noise limited regime. This regime has not yet been reached in ETPs because of the presence of I/f and “popcorn” noise [21] due to tunneling electron—surface interactions. Such low-frequency noise, absent in optical sensors, may present a remarkable difficulty in using roomtemperature bar antennae instrumented with ETPs for long term observation of gravitational waves emitted from stellar collapses in our galaxy, as recently proposed [22]. A further difference between ETP and photon tunneling transducers is the much smaller momentum transferred to the receiver for each tunneling quantum. At equal tunneling currents the associated momentum noise can therefore be neglected in the case of optical transducers while it must be taken into account in ETP. With respect to the fundamental limits of sensitivity of optical transducers discussed here, a complete quantum-mechanical analysis analogous to that performed for ETPs [23], taking into account the quantized nature of the electromagnetic field in the near-zone, remains an open and challenging problem. We would like to thank M.F. Bocko, F. Bordoni, M. Fejer and M. Karim for useful discussions.
8 April 1991
[2] M.F. Bocko and W.W. Johnson, in: New techniques and ideas in quantum measurement theory, ed. D.M. Greenberger (New York Academy of Sciences, Vol. 480, New York, 1986) p. 250; V.B. Braginskii, Soy. Phys. Usp. 31(1988) 836; R. Onofrio, Europhys. Lett. 11(1990) 695. [3] J. Weber, Phys. Rev. Lett. 17 (1966) 1228. [4] H.J. Paik, J. App!. Phys. 47 (1976)1168. [5] P. Rapagnani, Nuovo Cimento C 5 (1982) 385. [6] M. Gertsenstein and V. Pustovoit, Soy. Phys. JETP 16 (1962) 433; P. Meystre and MO. Scully, eds., Quantum optics, expenmental gravitation, and measurement theory (Plenum, New York, 1983). [7] M. Niksch and G. Binnig, J. Vac. Technol. A 6 (1988) 470; M.F. Bocko, R.H. Koch and K.A. Stephenson, Phys. Rev. Lett. 61(1988) 726; F. Bordoni, F. Fuligni, M.F. Bocko and R.H. Koch, in: Proc. Int. Workshop on Gravitational wave signal analysis and processing, ed. I. Pinto (World Scientific, Singapore, 1990). [8] E.C. Teague, ed., SPIE, Vol. 897. Scanning microscopy technologies and applications (1988). [9] R.C. Reddick, R.J. Warmack andT.L. Ferrell, Phys. Rev. B 39 (1989) 767; D. Coui:jon, K. Sarayeddine and M. Spajer, Opt. Commun. 71(1989) 23. [10] M. Born and E. Wolf, Principles of optics (Pergamon, Oxford, 1980). [1!] M.W. Hamilton, Contemp. Phys. 30 (1989) 21. [12] J.L. Hall, in: Quantum optics IV, eds. J.D. Harvey and D.F. Walls (Springer, Berlin, 1986) p.273. [13] R.L. Byer, Science 239 (1988) 742. [14] W. Demtröder, Laserspectroscopy (Springer, Berlin, 1982). [151YB. Braginskii, V.P. Mitrofanov and VI. Panov, Sistemi s maloi dissipatsei (Nauka, Moscow, 198!) [English translation: Systems with small dissipations (University of Chicago Press, Chicago, 1985)]. [16] Y. Leviatan, J. App!. Phys. 60 (1986) 1577; A. Roberts, J. App!. Phys. 65(1989) 2896. [171U.Ch. Fischer, U.T. Dung and D.W. Pohl, Scanning Microsc. 3 (1989) 1; D.W. PohI, U.Ch. Fischer and U.T. Dung, in: SPIE, Vol. 897. Scanning microscopy technologies and applications, ed. E.C. Teague (1988) p. 84. [18] K. Liebennan, S. Harush, A. Lewis and R. Kopelman, Science 24 (1990) 59. [19] U.Ch. Fischer and D.W. PohI, Phys. Rev. Lett. 62 (1989) 458. [20] E. Betzig, A. Harootunian, A. Lewis and M. Isaacson, App!.
References [1] K.S. Thorne, in: Three hundred years ofgravitation, eds. S. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1987) p. 330.
224
Opt. 25 (1986) 1890. [21]F. Bordoni, M. Karim, D.P.E. Smith and G. Binnig, submitted to Phys. Rev. A. [22] F. Bordoni, M. Karim, M.F. Bocko and T. Mengxi, to be published in Phys. Rev. D. [23] B.YurkeandG.P.Kochanski, Phys.Rev. B41 (1990) 8184.