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September 15, 1996 / Vol. 21, No. 18 / OPTICS LETTERS 1427
Optical heterodyne imaging and Wignerphase space
distributions
A. Wax and J. E. Thomas
Department of Physics, Duke University, Durham, North Carolina
27708-0305
Received May 6, 1996
We demonstrate that optical heterodyne imaging directly measures
smoothed Wigner phase space distributions.This method may be
broadly applicable to fundamental studies of light propagation and
tomographic imaging.Basic physical properties of Wigner
distributions are illustrated by experimental measurements.
1996Optical Society of America
In 1932 Wigner1 introduced a wave-mechanical phasespace
distribution function that plays a role closelyanalogous to that of
a classical phase space distributionin position and momentum. For a
wave field varyingin one spatial dimension, E sxd, the Wigner phase
spacedistribution is given by2
W sx, pd Z de
2pexpsiepdkE psx 1 ey2dE sx 2 ey2dl ,
(1)
where x is the position, p is a wave vector (mo-mentum), and
angle brackets denote a statisticalaverage. Despite their frequent
use in theory andpotential practical importance to imaging,3 – 5
Wignerphase space distributions have received relativelylittle
attention in optical measurements. Becauserigorous transport
equations can be derived for Wignerdistributions, these
distributions are important forfundamental studies of light
propagation and tomo-graphic imaging.
In this Letter we demonstrate that the mean-squareheterodyne
beat signal, which we measure in real time,is proportional to the
overlap of the Wigner phase spacedistributions for the local
oscillator and signal f ields.This remarkable result, which seems
not to have beenexploited previously in heterodyne detection,6,7
permitsus to measure Wigner phase space distributions forthe signal
field directly as contour plots with high dy-namic range. The
measured phase space contours aresmoothed Wigner distributions for
the signal f ield; i.e.,the phase space resolution is determined by
the diffrac-tion angle and the spatial width of the local
oscilla-tor.8 We measure Wigner distributions for cases
thatillustrate their basic physical properties.
The scheme of the heterodyne method, Fig. 1, em-ploys a
helium–neon laser beam that is split at BS1into a 1-mW local
oscillator (LO) and a 1-mW sig-nal beam. One can introduce a sample
into the sig-nal path to study the transmitted field. The
signalbeam is mixed with the LO at a 50–50 beam split-ter (BS2).
Technical noise is suppressed by use of astandard balanced
detection system.9 The beat signalat 10 MHz is measured with an
analog spectrum ana-lyzer. An important feature of the experiments
is thatthe analog output of the spectrum analyzer is squaredby a
low-noise multiplier.10 The multiplier output is
0146-9592/96/181427-03$10.00/0
fed to a lock-in amplif ier, which subtracts the mean-square
signal and noise voltages with the input beamon and off.11 In this
way the mean-square electronicnoise and the LO shot noise are
subtracted in real time,and the lock-in output is directly
proportional to themean-square beat amplitude kjVB j2l.
The beat amplitude VB is determined in the paraxialray
approximation by the spatial overlap of the LOand signal f ields in
the detector planes, z zD .6 Thefields in the detector planes can
be related to the f ieldsin the source planes at input lenses L1
and L2 sz 0d,which have equal focal lengths f . L2 is translated
offaxis by a distance dp, and mirror M1 is translated offaxis a
distance dx. The mean-square beat amplitudeis obtained in the
Fresnel approximation as
kjVB j2l ~*É Z
dx0E pLOsx0, zD dES sx0, zD dÉ2+
*ÉZdxE pLOsx 2 dx, z 0dES sx, z 0d
3 expµik
dpf
x∂É2+
. (2)
Fig. 1. Scheme for heterodyne measurement of Wignerphase space
distributions. The displacement dx of mirrorM1 determines the
position x, and the displacement dp oflens L2 determines the
momentum p. AyO’s, acousto-opticmodulators.
1996 Optical Society of America
8981 (DLC)
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1428 OPTICS LETTERS / Vol. 21, No. 18 / September 15, 1996
Here E is a slowly varying f ield amplitude (band
centerfrequency phase factor removed) and k 2pyl. Forsimplicity the
corresponding y integral in the detec-tor plane is suppressed. It
is assumed here that theRayleigh and coherence lengths of the LO
field arelarge compared with dx, so the translation of M1 sim-ply
shifts the center of the input LO field without sig-nificantly
altering the LO optical path length beforeL1. When this is not the
case, a variable LO pathlength can be introduced to compensate for
the path-length change that is due to moving M1. The detec-tors, D1
and D2, are located in the Fourier planeszD f of both lenses L1 and
L2, so the LO positionin the detector planes remains fixed as dx is
scanned.
Using Eq. (1), we can rewrite relation (2) (suppress-ing the y
integration) as
kjVBsdx, dpdj2l ~Z
dxdp WLOsx 2 dx, p
1 kdpyf dWSsx, pd . (3)
WSsx, pd fWLOsx, pdg is the Wigner distribution of thesignal
(LO) field in the plane of L2 (L1) given byEq. (1). Relation (3)
shows that the mean-square beatsignal yields a phase space contour
plot of WS sx, pdwith phase space resolution determined by WLO.8
Thecurrent system measures position over 61 cm andmomentum over
60.1 k (i.e., 6100 mrad).
First we review the basic properties of Wignerdistributions for
Gaussian signal beams and demon-strate their measurement as phase
space contours. AGaussian beam has a slowly varying field of the
formE sxd ~ expf2x2ys2w2d 1 ikx2ys2Rdg. Equation (1)yields the
corresponding Wigner distribution (normal-ized to unity):
W sx, pd s1ypdexps2x2yw2d
3 exph2w2sp 2 kxyRd2
i. (4)
Here the intensity 1ye width is w and the wave-frontradius of
curvature is R.
Wigner distributions obey a simple propagation lawin free space:
The convective derivative is zero, whichfollows from the wave
equation in the slowly vary-ing amplitude approximation. For a
time-independentWigner distribution propagating paraxially in the z
di-rection with wave vector pz . k the distribution in theplane z L
then is given in terms of that for z 0
according to W sx, p, z Ld W sx 2 pLyk, p, z 0d. Hence the x
argument propagates in straightlines. For propagation through a
lens of focal lengthf it is easy to show that the quadratically
varyingphase of the lens, fsxd 2kx2y2f , leads to a changein the
momentum argument: p ! p 1 kxyf . Theseresults easily yield the
ABCD law of Gaussian beamoptics.12 Hence, for example, suppose that
WG is theWigner distribution for a Gaussian beam at a waist,i.e.,
Eq. (4), with w a and R `. Then it is easy toshow that W sx, p, z
Ld WG sx 2 pLyk, pd takes theform of Eq. (4), with w and R given by
the usual Gauss-ian beam results that properly include
diffraction.12
In the experiments we begin with Gaussian signalfields, and WS
sx, pd takes the form of Eq. (4). TheLO beam is chosen to be
Gaussian with its waist inthe plane of L1. Then WLOsx, pd WG sx, pd
is givenby Eq. (4) with w a 380 mm and R `. Withthe sample removed
(Fig. 1), the signal beam waistand radius of curvature are
determined by a lens(not shown) that focuses the input beam to a
waist,as 35 mm, at a plane located a distance L behind thesignal
input plane at L2.
Figure 2 shows measured phase space contours,kjVBsdx, dpdj2l,
obtained by scanning dx and dp withstepper motors. The position
axis denotes the LOcenter position dx. The momentum axis denotes
theLO center momentum pc in units of the optical wavevector: pcyk
2dpyf . The contours rotate as thedistance L is changed. For L 0
the waist is at L2and the curvature R `. The phase space ellipse
hasits principal axes oriented vertically and horizontally.The
position width of the distribution is dominatedby the LO width in
this case, and the momentumwidth is dominated by the signal beam.
The phasespace ellipse rotates clockwise (counterclockwise) forL 5
cm (L 25 cm), indicating positive (negative)curvature, i.e., R . 0
sR , 0d at L2. The rotation ofthe phase space ellipse is a simple
consequence of thecorrelation between the momentum and the position
fora beam with curvature, Eq. (4). As one would expectfor a
diverging beam, the mean momentum shifts tothe right for x . 0.
These results clearly demonstratehow the measured phase space
contours are sensitiveto the spatially varying phase of the
field.
It is instructive to measure phase space contours fora source
consisting of two mutually coherent, spatiallyseparated Gaussian
beams. The input beams are
Fig. 2. Measured Wignerphase space contours forGaussian signal
beams:(a) beam waist (f lat wavefront), (b) diverging (posi-tive
wave-front curvature),(c) converging (negativewave-front
curvature).
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September 15, 1996 / Vol. 21, No. 18 / OPTICS LETTERS 1429
centered at positions x 6dy2 with d 1 mm andintensity 1ye radii
as 110 mm at the waist in theplane of L2. The Wigner distribution
for this signalfield is given by Eq. (1) as
WS sx, pd WG sx 2 dy2, pd 1 WG sx 1 dy2, pd
1 2WGsx, pdcossdp 1 wd , (5)
where WG denotes the Wigner distribution for eitherGaussian beam
at its waist. An interesting feature ofthis distribution is that
for d .. as, as used here, thecosine term is dominant at x 0 and
negative valuesare obtained as p is varied.
Figure 3 shows the measured contour plots. Inthe central region
the intensity oscillates with nearly100% modulation but is positive
definite, as it mustbe.8 Note that the orientation of the phase
spaceellipses indicates beam waists. The right-hand ellipseis
centered at a higher momentum than the left,indicating a small
angle between the two input beams.The two-peaked position profile
for p 0 is shownalong with the oscillatory momentum profile for x
0midway between the two intensity peaks. The solidcurve shows the
theoretical f it to the momentumdistribution, with a signal beam
1ye width of 103 mm,which is consistent with diode array
measurementswithin 10%.
In conclusion, we have demonstrated direct hetero-dyne
measurement of smoothed Wigner phase spacedistributions. This
method achieves high dynamicrange13 and is applicable to light from
arbitrarysamples. Study of Wigner distributions may be usefulfor
placing biological imaging methods, such as opticalcoherence
tomography14 and potential high-resolutionoptical biopsy
techniques, on a rigorous theoreticalfooting.
This research was supported by the U.S. Air ForceOff ice of
Scientific Research and the National ScienceFoundation. We are
indebted to M. G. Raymer formany stimulating conversations
regarding it and to S.John for a preprint of his paper.
Fig. 3. Measured Wignerphase space contours fortwo spatially
separated,mutually coherent beams:(a) Phase space contour,(b)
position profile formomentum p 0, (c) mo-mentum profile at
positionx 0. Dotted curves,data; solid curve, theory.
References
1. E. P. Wigner, Phys. Rev. Lett. 40, 749 (1932).2. M. Hillery,
R. F. O’Connel, M. O. Scully, and E. P.
Wigner, Phys. Rep. 106, 121 (1984).3. D. F. McAlister, M. Beck,
L. Clarke, A. Mayer, and M.
G. Raymer, Opt. Lett. 20, 1181 (1995).4. M. G. Raymer, C. Cheng,
D. M. Toloudis, M. Anderson,
and M. Beck, in Advances in Optical Imaging and Pho-ton
Migration (Optical Society of America, Washington,D.C., 1996), pp.
236–238.
5. S. John, G. Pang, and Y. Yang, Proc. SPIE 2389, 64(1995).
6. See, for example, V. J. Corcoran, J. Appl. Phys. 36,
1819(1965); A. E. Siegman, Appl. Opt. 5, 1588 (1966); S.Cohen,
Appl. Opt. 14, 1953 (1975); A. Migdall, B. Roop,Y. C. Zheng, J. E.
Hardis, and G. J. Xia, Appl. Opt. 29,5136 (1990).
7. Recent heterodyne studies in turbid media includeK. P. Chan,
M. Yamada, B. Devaraj, and H. Inaba, Opt.Lett. 20, 492 (1995); M.
Toida, M. Kondo, T. Ichimura,and H. Inaba, Appl. Phys. B 52, 391
(1991).
8. The mean-square beat is positive definite and takesthe form
of a smoothed Wigner distribution. See N. D.Cartwright, Physica
83A, 210 (1976).
9. H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983).10.
This method has been used in light beating spec-
troscopy; see H. Z. Cummins and H. L. Swinney, inProgress in
Optics, E. Wolf, ed. (North-Holland, NewYork, 1970), Vol. VIII,
Chap. 3, pp. 133–200.
11. This method has been used by G. L. Abbas, V. W. S.Chan, and
T. K. Yee, IEEE J. Lightwave Technol. 3,1110 (1985).
12. A. Yariv, Introduction to Optical Electronics
(Holt,Rinehart, & Winston, New York, 1976), Chap. 3, p. 35.
13. A. Wax and J. E. Thomas, in Advances in OpticalImaging and
Photon Migration (Optical Society ofAmerica, Washington, D.C.,
1996), pp. 238–242.
14. The magnitude of the mean beat amplitude (ratherthan the
mean square) is usually measured in this case.See, for example, J.
A. Izatt, H. R. Hee, G. M. Owen,E. A. Swanson, and J. G. Fujimoto,
Opt. Lett. 19, 590(1994).