June 1, 2007 / Vol. 32, No. 11 / OPTICS LETTERS 1369

Squeezing the local oscillator does not improvesignal-to-noise ratio in heterodyne laser radar

Mark A. Rubin* and Sumanth KaushikLincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington,

Massachusetts 02420-9185, USA*Corresponding author: rubin@LL.mit.edu

Received December 1, 2006; revised February 27, 2007; accepted March 12, 2007;posted March 15, 2007 (Doc. ID 77721); published April 24, 2007

The signal-to-noise ratio for heterodyne laser radar with a coherent target-return beam and a squeezedlocal-oscillator beam is lower than that obtained using a coherent local oscillator, regardless of the methodemployed to combine the beams at the detector.

OCIS codes: 270.0270, 270.6570, 280.5600, 040.2840.

Squeezed light holds promise for reducing noise inoptical-detection applications below the level obtain-able using coherent light such as that emitted by la-sers [1]. However, squeezing is degraded by loss [2].In laser radar applications, the loss in the target-return beam—the beam received by the radar systemafter reflection from a target—is severe [3]. Thereforesqueezing is not useful in laser radar, at least notwhen applied to the target-return beam.

This still leaves open the possibility that squeezedlight could be profitably employed as the local oscil-lator (LO) in a heterodyne laser radar. In such a sys-tem the target-return beam is combined with an LObeam of a different frequency on a photosensitive de-tector, and the presence of a target is inferred fromobservation of oscillation of the detector response atthe difference frequency of the two beams [3]. A het-erodyne laser radar system combining the target-return and LO beams on a single detecting elementhas been proposed by Li et al. [4]. Their work hasbeen criticized by Ralph [5] on the grounds that themethod they employ to combine the target-returnbeam and LO beam on the detector introduces suffi-cient noise to cancel out any improvement in signal-to-noise ratio (SNR) due to squeezing. However, onemight envision employing other methods of combin-ing the beams which do not add noise; e.g., using aFabry–Perot etalon [6] that reflects the LO frequencyand transmits the target-return frequency.

This approach is to be distinguished from a “bal-anced” heterodyne system in which the two beamsare directed to two detectors using a beam splitter.Quantum noise in balanced heterodyne detectionwith squeezed light has been examined by Yuen andChan [7] and by Annovazzi-Lodi et al. [8]. We wouldnot expect such a system to benefit from squeezing,since both beams must pass through the beam split-ter and thus suffer squeezing-destroying loss. In fact,a calculation of the SNR in the balanced case givesthe same result as that obtained below in the presentcase, Eq. (25), except for the change of cos2 to sin2 dueto the � /2 phase change of the reflected light at thebeam splitter. The detection of a Doppler beat signalusing a squeezed LO, as proposed by Li et al. [9], isalso an example of a heterodyne measurement to

which the considerations of the present paper apply.

Here, we show that a heterodyne detection schemecombining a coherent target-return beam and asqueezed LO beam on a single detector will fail to im-prove SNR regardless of the method used to combinethe beams.

For target detection using the statistic S [10],

SNR = ���1�S��1� − ��0�S��0��2/Var0S, �1�

where ��1�S��1� is the mean value of S in that quan-tum state, ��1�, in which the target is present,��0�S��0� is the mean value of S when the target is ab-sent, and Var0S is the variance of S in the target-absent condition,

Var0S = ��0�S2��0� − ��0�S��0�2. �2�

In choosing pure quantum states to correspond to thetarget-present and target-absent conditions we areassuming the absence of additional nonquantumsources of noise, e.g., thermal noise, which wouldhave to be treated using the density operator formal-ism [11].

For heterodyne detection,

S = �−1�0

�

dt cos��Ht + �H�I�t�, �3�

where I�t� is the quantum operator corresponding tothe photoelectric current produced by the detector attime t and � is the fixed time interval during whichthe target-present/absent decision is to be made.That is, S is the Fourier component of the photoelec-tric current at frequency �H and phase �H. We take�H to be equal to the difference between the respec-tive frequencies of the target-return and LO beams,

�H = �T − �LO. �4�

(For simplicity we will always take �T−�LO�0.) So Scorresponds to detection of the beat frequency be-tween the LO and the target-return beam.

For suitable broadband detectors, the operator cor-responding to the photoelectric current at time t is

[12]

1370 OPTICS LETTERS / Vol. 32, No. 11 / June 1, 2007

I�t� = �E�−��t�E�+��t�, �5�

where � is a constant, E�−��t�= �E�+��t��†, and E�+��t� isthe positive-frequency part of the time-dependentelectric field operator at the detector,

E�+��t� = �k

i� ��k

20V1/2

ak exp�− i�kt�. �6�

The mode frequencies are �k=ck where the wave-number k runs over the values k=2�n /V1/3, n=1,2, . . .. In writing E�+��t� as in Eq. (6), we are as-suming that the detector is sensitive to only a singledirection of polarization (which is the direction inwhich both the LO and target-return beam will be po-larized) and that the optical system is such that, foreach frequency, only a single spatial mode need beconsidered (that mode with wave vector normal tothe detector surface). The annihilation operators aksatisfy the usual commutation relations,

ak,al� = ak†,al

†� = 0, ak,al†� = kl. �7�

Using Eqs. (3)–(6) we obtain, in the limit �→�,

S =��

40V �l,k st ��l−�k�=�H

��l�k�1/2al†ak

�exp�− i��l − �k��H�, �8�

where

�x� = sign of x. �9�

In the target-absent state, all modes but the LOare in the vacuum state:

��0� = � ,��kLO �k�kLO

�0�k. �10�

Here � ,��kLOis the squeezed LO-frequency ��LO�

mode parameterized by complex numbers and �[13]. In the target-present case an additional mode isin a nonvacuum state, specifically the coherent state���kT

at the target-return frequency �T:

��1� = ���kT� ,��kLO �

k�kT,kLO

�0�k. �11�

Using Eqs. (4) and (8)–(11) and the relations [13]

ak�0�k = k�0�ak† = 0, �12�

kT���akT

���kT= �, kT

���akT

† ���kT= �*, �13�

kLO� ,��akLO

� ,��kLO= ,

kLO� ,��akLO

† � ,��kLO= *, �14�

we find that

��0�S��0� = 0, �15�

since the only possible nonzero term, akLO

† akLO, is for-

bidden by the restriction on the summation in Eq. (8),and

��1�S��1� =��

20V��T�LO�1/2�a����cos��T − �LO + �H�,

�16�

where

�T = arg �, �LO = arg . �17�

Using Eqs. (7)–(10) and (12),

��0�S2��0� = � ��

40V2

�k s.t. ��LO−�k�=�H

�l s.t. ��l−�LO�=�H

��LO��k�l�1/2��0�akLO

† akal†akLO

��0�

�exp�− i��LO − �k� + ��l − �LO���H�.

�18�

Neither k nor l can be equal to kLO, due to the restric-tions in the summations in Eq. (18). If k� l then akand al

† commute, yielding zero since the non-LOmodes are in the vacuum state. So the only survivingterms are those for which k= l. Using Eq. (7),

��0�S2��0� = � ��

40V2

�k s.t. ��LO−�k�=�H

�LO�knLO,

�19�

where

nLO = kLO� ,��akLO

† akLO� ,��kLO

. �20�

Using Eqs. (2), (15), and (19),

Var0S = � ��

40V2

�k s.t. ��LO−�k�=�H

�LO�knLO. �21�

The contribution to Eq. (21) from the term �k=�LO−�H is termed the “image band” contribution [14].

In practice �H��T ,�LO, so we can take

�T �LO � �. �22�

Using Eqs. (22), Eqs. (16) and (21) become

��1�S��1� =���

20V� ����cos��T − �LO + �H�, �23�

Var0S = 2� ���

40V2

nLO. �24�

Using Eq. (15), (23), and (24), the SNR (1) is

June 1, 2007 / Vol. 32, No. 11 / OPTICS LETTERS 1371

SNR =2� �2���2 cos2��T − �LO + �H�

nLO

= 2�1 −sinh2�r�

nLOnT

�cos2��T − �LO + �H�, �25�

using the relations [13]

���2 = nkT= kT

���akT

† akT���kT

, �26�

nLO = � �2 + sinh2�r�. �27�

The parameter r= ��� is termed the “squeezing param-eter.” The value r=0 corresponds to no squeezing (co-herent state). From Eq. (25) it is clear that squeezingthe LO mode, i.e., letting the LO be in a state withr�0, can only reduce the SNR.

This result is consistent with the observation byYuen and Chan [7], in the context of balanced detec-tion, that while “quantum noise is frequently sup-posed to arise from local-oscillator (LO) shot noise…itactually arises from the signal quantum fluctuation.”The reasonable but incorrect expectation thatsqueezing the LO will improve SNR arises from thefact that the variance of the zero-frequency signal,i.e., the time-averaged photoelectric current corre-sponding to the operator

S� = �−1�0

�

dtI�t� =���k

20Vak

†ak �28�

(the second equality holding in the limit �→�), doeschange with squeezing. In the target-absent state,

Var0S� = ��0�S�2��0� − ��0�S���0�2, �29�

which for �→� has the value

Var0S� = ����LO

2V 2

var nLO, �30�

where

var nLO = kLO� ,���akLO

† akLO�2� ,��kLO

− �kLO� ,��akLO

† akLO� ,��kLO

�2�. �31�

For suitable choice of the phase of �, Eq. (31) can in-deed be lower than nLO, the value it takes in a coher-ent state [13].

However, statistical decision theory [10] indicatesthat if a quantity S computed from measurementsmade by a detector is used as the decision criterion ina target-detection task, then it is the variance of thatsame quantity S that is relevant in evaluating thesuitability of S for the task. In heterodyne radar the

computed quantity S is the Fourier component of the

instantaneous photocurrent induced at the detectorby the combined target-return and LO beams.[3] Theoperator corresponding to the instantaneous detectorresponse is I�t�, so the operator corresponding to therequired Fourier component is S as defined in Eq. (3).It is thus the variance of S, not that of S�, that mustbe used for computing SNR.

The general expression for the signal operator (3)for � not necessarily infinite, �H not necessarily equalto ��k−�l� for any k , l, is

S =��

20V�l,k

��l�k�1/2al†ak exp�− i��l − �k��H�

�1

2i�� exp�i�H�

�l − �k + �Hexp�i��l − �k + �H��� − 1�

+exp�− i�H�

�l − �k − �Hexp�i��l − �k − �H��� − 1�� . �32�

This reduces to Eqs. (8) and (28) for the appropriatelimiting values of �, �H and �H.

Mark A. Rubin thanks Jonathan Ashcom and JaeKyung for a helpful discussion on mixing efficiency.This work was sponsored by the Air Force under AirForce Contract FA8721-05-C-0002. Opinions, inter-pretations, conclusions, and recommendations arethose of the authors and are not necessarily endorsedby the U.S. Government.

References

1. D. F. Walls and G. J. Milburn, Quantum Optics(Springer, 1994).

2. C. M. Caves, Phys. Rev. D 23, 1693 (1981).3. R. H. Kingston, Optical Sources, Detectors and

Systems: Fundamentals and Applications (Academic,1995).

4. Y.-Q. Li, D. Guzun, and M. Xiao, Phys. Rev. Lett. 82,5225 (1999).

5. T. C. Ralph, Phys. Rev. Lett. 85, 677 (2000).6. G. Hernandez, Fabry-Perot Interferometers

(Cambridge, 1986).7. H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983).8. V. Annovazzi-Lodi, S. Donati, and S. Merlo, Opt.

Quantum Electron. 24, 258 (1992).9. Y.-Q. Li, P. Lynam, M. Xiao, and P. J. Edwards, Phys.

Rev. Lett. 78, 3105 (1997).10. C. W. Helstrom, Quantum Detection and Estimation

Theory (Academic, 1976).11. K. Gottfreid and T.-M. Yan, Quantum Mechanics:

Fundamentals, 2nd ed. (Springer, 2003).12. R. J. Glauber, in Quantum Optics and Electronics, C.

DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds.(Gordon and Breach, 1965).

13. C. G. Gerry and P. L. Knight, Introductory QuantumOptics (Cambridge, 2005).

14. H. A. Haus, Electromagnetic Noise and Quantum

Optical Measurements (Springer, 2000).