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)

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).

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).