signal-to-noise ratio in squeezed-light laser radar

Signal-to-noise ratio in squeezed-light laser radar

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

Massachusetts 02420-9185, USA

*Corresponding author: [email protected]

Received 7 April 2009; revised 1 July 2009; accepted 10 July 2009;posted 13 July 2009 (Doc. ID 109773); published 4 August 2009

The formalism for computing the signal-to-noise ratio (SNR) for laser radar is reviewed and applied to thetasks of target detection, direction finding, and phase-change estimation with squeezed light. The SNRfor heterodyne detection of coherent light using a squeezed local oscillator is lower than that obtainedusing a coherent local oscillator. This is true for target detection, for phase estimation, and for directionfinding with a split detector. Squeezing the local oscillator also lowers SNR in balanced homodyne andheterodyne detection of coherent light. Loss places an upper bound on the improvement that squeezingcan bring to direct-detection SNR. © 2009 Optical Society of America

OCIS codes: 270.0270, 270.6570, 280.5600, 040.2840.

1. Introduction

Squeezed light was proposed in [1] as a means ofreducing noise in interferometers used as gravi-tational-wave detectors. In that article, the impactof loss on performance improvement due to squeezingwas noted. In laser radar applications, loss fromspreading and absorption in the target-return beamcan be severe [2]. It is, therefore, natural to investi-gate the potential of using squeezed light in thelocal oscillator of a heterodyne laser radar system[3–5], since the loss suffered by the local-oscillatorbeamwill beminimal. Herewe review relevant signaldetection theory and quantum optics formalism,and compute the signal-to-noise ratio (SNR) for sev-eral application scenarios. In all the caseswe examinewe find that squeezed light offers no advantages forlaser radar.This paper is organized as follows. Sections 2 and 3

review relevant aspects of signal detection theory.Section 4 obtains explicit forms for quantum opera-tors that will be used in the subsequent analysis, andSection 5 shows how they fit into the heterodyne de-tection approach. The SNR for heterodyne target de-tection using a squeezed local oscillator is derived in

Section 6 and is shown to be lower than that obtain-able with heterodyne target detection with the usualcoherent local oscillator. The expression for SNR isgiven in terms of conventional parameters inSection 7, and is extended to include nonunity quan-tum efficiency in Section 8. Section 9 presents andanalyzes models of heterodyne and direct-detectiondirection finding using a split detector, and showsthat here, too, squeezing the local oscillator of theheterodyne system yields no advantage. Section 10looks at heterodyne detection of phase. Section 11 ex-tends the result of Section 6 to the case of balanceddetection. Appendix A examines direct target detec-tion with squeezed light and derives limits on SNRimprovement.

2. Signal-to-Noise Ratio

From [6], Chap. IV, “Quantum Hypothesis Testing:”“Suppose that when the signal is present, it is re-

peated during some number M of observation inter-vals of duration T. Let ~xk be a set of data samplestaken in the kth interval, k ¼ 1; 2;…;M. We assumethese are statistically independent from one another.Let gð~xkÞ be a statistic formed from the data~x, and letthe choice between hypotheses H0 and H1 be basedon the sum

0003-6935/09/234597-13$15.00/0© 2009 Optical Society of America

10 August 2009 / Vol. 48, No. 23 / APPLIED OPTICS 4597

G ¼XMk¼1

gð~xkÞ ð3:2Þ

of the statistics for each interval…Hypothesis H1 isselected if G exceeds a certain decision level G0.“When M is very large, the p.d.f. [probability

density function] of the statistic G is very nearlyGaussian, by virtue of the central limit theorem,and the false alarm and detection probabilities areapproximately

Q0 ≈ erfcðξÞ; Qd ≈ erfcðξ −DgM1=2Þ; ð3:3Þ

where

D2g ¼ ½EðgjH1Þ − EðgjH0Þ�2

Var0gð3:4Þ

with

Var0g ¼ Eðg2jH0Þ − ½EðgjH0Þ�2 ð3:5Þ

the variance of the statistic g under hypothesis H0.[EðgjHiÞ, i ¼ 1; 2, is the “expected value” or “meanvalue” [7] of g when hypothesis Hi holds.] We callD2

g the equivalent signal-to-noise ratio (e.s.n.r.).”For direct detection Var0g ¼ 0, so Var1g, the

variance of the signal when it is present (i.e., whenH1 holds) will be used. See Appendix A.

3. Quantum Hypotheses and Statistical Measures

When hypothesisHi holds, the quantum state is jψ ii,i ¼ 0; 1. The mean value of S when Hi holds is

EðSjHiÞ ¼ hψ ijSjψ ii; ð1Þ

where S is the operator corresponding to thequantity S. The variance of S when Hi holds is

VariS ¼ hψ ijS2jψ ii − hψ ijSjψ ii2: ð2Þ

Using Eqs. (1) and (2) in Eqs. (3.4) and (3.5) ofSection 2, and taking g ¼ S,

D2S ¼ ðhψ1jSjψ1i − hψ0jSjψ0iÞ2

hψ0jS2jψ0i − hψ0jSjψ0i2: ð3Þ

4. Operators

The positive-frequency part of the electric-fieldoperator is

EðþÞμ ðtÞ ¼

X~k;ζ

i

�ℏω~k

2ε0V

�1=2

a~k;ζe−iω~ktε~k;ζ;μ; ð4Þ

where ζ ¼ 1; 2 is the polarization index, μ ¼ 1; 2; 3 isthe Cartesian index, and ε~k;ζ;μ is the (real) polariza-tion vector [8]. For suitable broadband detectors, theoperator corresponding to the photoelectric currentat time t (in units of the electron charge) is

IðtÞ ¼Xν;μ

sνμEð−Þν ðtÞEðþÞ

μ ðtÞ; ð5Þ

where

Eð−Þμ ðtÞ ¼ ðEðþÞ

μ ðtÞÞ†; ð6Þ

and where the sensitivity function sνμ is constant andsymmetric. Using Eqs. (4) and (6) in Eq. (5),

IðtÞ ¼X

~k;ζ;~l;ρ;μ;ν

ℏ2ε0V

ðω~kω~kÞ1=2a†~l;ρa~k;ζe

iðω~l−ω~kÞtε~l;ρ;νε~k;ζ;μsνμ:

ð7Þ

The following functional of IðtÞ will play a key role(see Section 5):

S ¼ τ−1Z τ

0dt cosðωHtþ θHÞIðtÞ: ð8Þ

Using Eq. (7) in Eq. (8),

S ¼ ℏ4ε0V

X~k;ζ;~l;ρ;μ;νs:t:jω~l−ω~kj¼ωH

ðω~lω~kÞ1=2a†~l;ρ

× a~k;ζe−iεðω~l−ω~kÞθHε~l;ρνε~k;ζμsνμ; ð9Þ

where

εðxÞ ¼ sign of x: ð10Þ

From Eq. (9),

S2 ¼�

ℏ4ε0V

�2 X~k;ζ;~l;ρ;μ;ν s:t: jω~l−ω~kj¼ωH

X~k0;ζ0;~l0;ρ0;μ0;ν0 s:t: jω~l0−ω~k0 j¼ωH

× ðω~lω~kω~l0ω~k0 Þ1=2a†~l;ρa~k;ζa

†~l0;ρ0

a~k0;ζ0e−i½εðω~l−ω~kÞþεðω~l0−ω~k0 Þ�θHε~l;ρ;νε~k;ζμsνμε~l0;ρ0;ν0ε~k0;ζ0;μ0sv0μ0 :

ð11Þ

4598 APPLIED OPTICS / Vol. 48, No. 23 / 10 August 2009

5. Heterodyne Signal

As pointed out in Section 1, our motivation for inves-tigating the use of squeezed light in a heterodyne la-ser radar system is the presence of the essentiallyloss-free local-oscillator beam. In the heterodyne ap-proach to target detection [2], the mixing of the localoscillator with the light reflected from the target gen-erates a current in the photodetector that oscillateswith angular frequency ωH and phase θH, where ωHand θH are, respectively, the frequency difference andthe phase difference (at the detector location) be-tween the light from the local oscillator and the lightreflected from the target.Let IðtÞ denote the total photoelectric current (in

units of the electron charge) produced by the photo-detector at time t of a given experimental run. Thenthe statistic g we will use for heterodyne detection isthe Fourier component of IðtÞ at an angular fre-quency ωH and phase θH, which we will denote by S:

S ¼ τ−1Z τ

0dt cosðωHtþ θHÞIðtÞ: ð12Þ

The average of this signal is

hSi ¼ τ−1Z τ

0dt cosðωHtþ θHÞhIðtÞi: ð13Þ

Here and subsequently, mean value is denoted by an-gle brackets h i. We assume that ωH and θH do notvary from one run of the experiment to another, i.e.:

hωHi ¼ ωH ; ð14Þ

hθHi ¼ θH : ð15Þ

6. Heterodyne Target Detection with Squeezed LocalOscillator

Let j0i~k;ζ denote the vacuum state for mode~k; ζ of theelectromagnetic field. We consider the class ofsqueezed states [8] parameterized by two complexnumbers, α, ξ, and defined as

jα; ξi~k;ζ ¼ DðαÞ~k;ζQðξÞ~k;ζj0i~k;ζ; ð16Þ

where the displacement operator DðαÞ~k;ζ and squeez-ing operator QðξÞ~k;ζ for mode ~k; ζ are, respectively,

DðαÞ~k;ζ ¼ expðαa†~k;ζ

− α�a~k;ζÞ; ð17Þ

QðξÞ~k;ζ ¼ exp�12ðξ�a2

~k;ζ − ξða†~k;ζ

Þ2Þ�: ð18Þ

For ξ ¼ 0 the squeezed state jα; ξi~k;ζ reduces to thecoherent state jαi~k;ζ:

jαi~k;ζ ¼ jα; 0i~k;ζ ¼ DðαÞ~k;ζj0i~k;ζ: ð19Þ

So, in the absence of the coherent signal reflectedfrom the target (null hypothesis, H0), the full quan-tum state is

jψ0i ¼ jα; ξiLOY

~k;ζ≠LO

j0i~k;ζ: ð20Þ

When the signal is present (alternative hypothesis,H1), the state is

jψ1i ¼ jβiT jα; ξiLOY

~k;ζ≠T;LO

j0i~k;ζ: ð21Þ

Here T and LO are shorthand for ~k; ζ for the target-return signal and the local oscillator, respectively.

Taking for concreteness ωT − ωLO ¼ ωH > 0, andusing Eqs. (9), (20), and (21)

hψ0jSjψ0i ¼ 0; ð22Þ

since the only possible nonzero term, ayLOaLO, is

forbidden by the restriction on the summation inEq. (9), and

hψ1jSjψ1i ¼κℏ

2ε0VðωTωLOÞ1=2jαjjβj cosðθT − θLO þ θHÞ:

ð23Þ

Assuming

ωH ≪ ωT ; ωH ≪ ωLO; ð24Þ

soωT ≈ ωLO ≡ ω; ð25Þ

Eq. (23) becomes

hψ1jSjψ1i ¼κℏω2ε0V

jαjjβj cosðθT − θLO þ θHÞ: ð26Þ

Here

θT ¼ arg β; θLO ¼ arg α; ð27Þ

and

κ ¼Xν;μ

εLO;νεT;μsνμ: ð28Þ

The analysis presented here has left out phase fac-tors in directions normal to the detector surface. Hadthose been included, the argument of the cosine inEq. (23) would have terms dependent on the locationon the surface of the detector, unless both wave vec-tors, T and LO, were normal to the detector surface.So, unless both wave vectors are sufficiently close tonormal so that the respective wave fronts are paral-lel to the detector surface to within less than a quar-ter wavelength over the surface, the net signal from


the entire detector surface will add to zero (i.e., zeromixing efficiency). Assuming that the polarization ofthe target-return signal is also parallel to that of thelocal oscillator, we can rewrite Eq. (28) as

κ ¼Xν;μ

εLO;νεLO;μsνμ: ð29Þ

Using Eqs. (11) and (20),

hψ0jS2jψ0i ¼�

ℏ4ε0V

�2 X~k;ζ;μ;ν s:t: jωLO−ω~kj¼ωH

X~l0;ρ0;μ0;ν0 s:t: jω~l0−ωLOj¼ωH

ωLOðω~kω~l0 Þ1=2

× hψ0ja†LOa~k;ζa

†~l0;ρ0

aLOjψ0ie−i½εðωLO−ω~kÞþεðω~l0−ωLOÞ�θHεLO;νε~k;ζμsνμε~l0;ρ0;ν0εLO;μ0sv0μ0 : ð30Þ

Neither~k; ζ nor~l0; ρ can be LO, due to the restrictionsin the summations arising from the heterodyning. If~k; ζ ≠~l0; ρ, then a~k;ζ and a†

~l0; ρ0 can commute, yielding

zero since the non-LO modes are in thevacuum state. So the only surviving terms are thosefor which ~k; ζ ¼~l0; ρ:

hψ0jS2jψ0i ¼�

ℏ4ε0V

�2 X~k;ζ; s:t: jωLO−ω~kj¼ωH

× ωLOω~k�nLO

�Xν;μ

εLO;νε~k;ζ;μsμν�

2; ð31Þ

or, using Eqs. (24) and (25),

hψ0jS2jψ0i ¼ 2

�ℏωLO

4ε0V

�2�nLO

X~k;ζ; s:t:ω~k¼ωLO

×�X

ν;μεLO;νε~k;ζ;μsμν

�2; ð32Þ

where

�nLO ¼LOhα; ξjnLOjα; ξiLO; ð33Þ

nLO ¼ ayLOaLO: ð34Þ

In going from Eq. (31) to Eq. (32), the expressionfor hψ0jS2jψ0i picked up a factor of 2. This is as a re-sult of contributions in the sum in Eq. (31) comingnot only from modes with ~k such that ω~k ¼ωLO þ ωH ¼ ωT , but also frommodes with~k such thatω~k ¼ ωLO − ωH . The latter modes are termed “imageband” modes [9].

The argument about mixing efficiency does not ap-ply here to the sum in parentheses in Eq. (31) sincethe nonzero terms come from raising and loweringoperators corresponding to the same mode. Define

ðκ0Þ2 ¼X

~k;ζ; s:t:ω~k¼ωLO

�Xν;μ

εLO;νε~k;ζ;μsμν�

2: ð35Þ

Since the sum over ~κ; ζ contains the term ~k; ζ ¼ LO

ðκ0Þ2 ≥ ðκÞ2; ð36Þ

where κ is as given in Eq. (29). If the field of view ofthe detector is such that the only modes satisfyingthe constraint ω~k ¼ ωLO have ~k colinear with thewave vector of the local oscillator, then

ðκ0Þ2 ¼ κ2: ð37Þ

Let Ltr be the transverse size of the quantization vo-lume (here taken to be equal to the size of the detec-tor) and let Ω be the solid angle of the detector’s fieldof view. Then, since the transverse components of thewave vector are quantized in units of 2π=Ltr and areassumed to be much smaller than the othercomponent (in the direction of the local-oscillatorwave vector),

ðκ0Þ2 ≈ κ2N ; ð38Þ

where

N ¼X

~ks:t:ω~k¼ωLO

≈ maxðj~kj2Ω=ðð2πÞ=LtrÞ2; 1Þ

¼ maxððLtr=λÞ2Ω; 1Þ; ð39Þ

with

λ ¼ 2πc=ω ð40Þ

the wavelength of the local oscillator and

½x� ¼ integral part of x: ð41Þ


E.g., for λ ¼ 10−6 m, Ltr ¼ 10−5 m, and Ω ¼ ð1mradÞ2,N equals unity. Changing Ltr to 10−3 m and Ω toð10mradÞ2 changes N to 100. For the remainder ofthis paper we will set

N ¼ 1; ð42Þimplying that Eq. (37) holds.Using Eqs. (35) and (37) in Eq. (32),

hψ0jS2jψ0i ¼ 2

�κℏωLO

4ε0V

�2�nLO: ð43Þ

Using Eqs. (22), (26), and (43) in Eq. (3),

D2S ¼ 2jαj2jβj2cos2ðθT − θLO þ θHÞ

�nLO

¼ 2

�1 −

sinh2ðrÞ�nLO

��nTcos2ðθT − θLO þ θHÞ; ð44Þ

where�nT ¼ThβjnT jβiT ; ð45Þ

nT ¼ a†TaT ; ð46Þ

and r is the squeezing parameter, nonnegative bydefinition:

r ¼ jξj: ð47ÞThese quantities and Eqs. (33) and (34) satisfy therelations Eq. [8]:

�nLO ¼ jαj2 þ sinh2ðrÞ; ð48Þ

�nT ¼ jβj2; ð49Þwhich have been used in obtaining Eq. (44). FromEq. (44) it is clear that squeezing the local oscillatordecreases the SNR.As can be seen from Eq. (48), the value of r is

constrained by

sinh2ðrÞ ≤ �nLO: ð50ÞThe case of equality in Eq. (50) is termed the“squeezed vacuum.”

7. Signal-to-Noise Ratio in Terms of ConventionalParameters

For a mode with expectation value of the number �n,the total energy is E ¼ ℏω�n. Hence the averageenergy density is ρE ¼ E=V ¼ ℏω�n=V, the averageenergy flux is Φ ¼ ρEc ¼ ℏω�nc=V, and the averagepower is P ¼ ΦEA ¼ ℏω�ncA=V, where A ¼ L2

tr isthe area of the quantization region (transverse tothe wave vector of the mode in question). Using these

with Eq. (23),

hψ1jSjψ1i2 ¼ 1

c2A2

� κ2ε0

�2PLOPTcos2ðθT − θLO

þ θHÞ�1 −

sinh2ðrÞ�nLO

�: ð51Þ

Let L be the dimension of the quantization regionparallel to the wave vector,

L ¼ V=A; ð52Þ

and, following [9], define the quantization time T tobe the time light would traverse this length,

T ¼ L=c ¼ V=cA: ð53Þ

Taking the effective bandwidth B to be [2]

B ¼ 1=2T; ð54Þ

Eq. (43) becomes

hψ0jS2jψ0i ¼� κ4ε0

�2 4Bℏωc2A2 PLO; ð55Þ

so

D2S ¼ PT

ℏωB cos2ðθT − θLO þ θHÞ�1 −

sinh2ðrÞ�nLO

�: ð56Þ

8. Quantum Efficiency

Quantum efficiency is a measure of the extent towhich photons are lost to the detection process. Thisloss is incorporated in the model by allowing both thetarget-return beam T and the local-oscillator beamLO to pass through a beam splitter before reachingthe detector [10]. (Naively, one might think that thequantum efficiency would involve the sensitivityfunction. As seen in Section 6, this is not the case,since both the signal and the noise scale with the sen-sitivity function.)

Let a~k;ζ be the lowering operator at the detector.This is related to the operator at the input port tothe beam splitter by

a~k;ζ ¼ t~k;ζaðinÞ~k;ζ þ r~k;ζaðvacÞ~k;ζ; ð57Þ

where aðinÞ~k;ζ and aðvacÞ~k;ζ are, respectively, the opera-tors at the input and vacuum ports of the beam split-ter, and t~k;ζ and r~k;ζ are the (possibly wavelength orpolarization dependent) c-number transmission andreflection coefficients.

Using Eq. (57) in Eq. (9), we find the heterodynesignal operator to be


SB ¼ ℏ4ε0V

X~k;ζ;~l;ρ;μ;ν s:t: jω~l−ω~kj¼ωH

ðω~lω~kÞ1=2ðt�~l;ρa†

ðinÞ~l;ρ

þ r�~l;ρa†

ðvacÞ~l;ρÞðt~k;ζaðinÞ~k;ζ

þ r~k;ζaðvacÞ~k;ζÞe−iεðω~l−ω~lÞθHε~l;ρ;νε~k;ζ;μsνμ: ð58Þ

The null-hypothesis and alternative-hypothesisstates are, respectively,

jψB;0i ¼ jψ ðinÞ;0ijψ ðvacÞ;0i; ð59Þ

jψB;1i ¼ jψ ðinÞ;1ijψ ðvacÞ;0i; ð60Þ

where

jψ ðvacÞ;0i ¼Y~k;ζ

j0iðvacÞ; ð61Þ

jψ ðinÞ;0i ¼Y

~k;ζ≠LO

j0iðinÞ;~k;ζjα; ξiðinÞ;LO; ð62Þ

jψ ðinÞ;1i ¼Y

~k;ζ≠LO;T

j0iðinÞ;~k;ζjα; ξiðinÞ;LOjβiðinÞ;T : ð63Þ

Using Eqs. (58)–(63), defining

θB ¼ argðt�LOtTÞ; ð64Þ

and taking the polarizations of the local oscillatorand target beam to be the same, we obtain

hψB;0jSBjψB;0i ¼ 0; ð65Þ

hψB;1jSBjψB;1i ¼ jtLOtT j� κℏω2ε0V

�ð�n − sinh2ðrÞÞ1=2�n1=2

T

× cosðθT − θLO þ θH þ θBÞ; ð66Þ

hψB;0jS2BjψB;0i ¼ 2jtLOj2

� κℏω4ε0V

��nLO: ð67Þ

From Eqs. (65) and (67)

Var0SB ¼ hψB;0jS2BjψB;0i − hψB;0jSBjψB;0i2

¼ 2jtLOj2� κℏω4ε0V

��nLO: ð68Þ

Using Eqs. (65), (66), and (68), the SNR is

D2SB

¼ ðhψB;1jSBjψB;1i − hψB;0jSBjψB;0i2=Var0SB

¼ jtT j22�1 −

sinh2ðrÞ�nLO

��nT

× cos2ðθT − θLO þ θH þ θBÞ: ð69Þ

Comparing Eq. (69) with Eq. (44), we see that thenorm-squared of the transmission coefficient entersas the quantum efficiency η:

η ¼ jtT j2: ð70Þ

9. Direction Finding With a Split Detector

Since the SNR for direct detection of a light beam isimproved by squeezing (see Appendix A), even whenonly a single mode is excited, one might expect that itis possible to improve the SNR of directional mea-surements using a single-mode squeezed beam anda split detector. As pointed out in [11], this is notthe case, and one must use a beam with at leasttwo transverse modes.

A. Transverse Modes

1. Expansions of Operators

However many modes are in nonvacuum states, if weare to examine a detector with a response that variesin the transverse direction, we should expand theelectric field in transverse modes. (If nothing else,we may find in some situations that the expansionin tranverse modes is unnecessary.)

We take the positive-frequency part of the electric-field operator to be

EðþÞμ ðt; xÞ ¼ i

X~k;ζ;m

�ℏω~k

2ε0V

�1=2

a†~k;ζ;m

ðxÞe−iω~ktu~k;ζ;mðxÞε~k;ζ;μ;

ð71Þ

where x is the transverse coordinate—we will onlyconsider modes depending on a single transversecoordinate—and the transverse modes are indexedby m. The current operator is

Iðt; xÞ ¼Xνμ

~sν;μEð−Þν ðt; xÞEðþÞ

μ ðt; xÞ; ð72Þ

where

~sν;μ ¼ sν;μ=W; ð73Þ

with W the beam width at the detector and

Eð−Þμ ðt; xÞ ¼ ðEðþÞ

μ ðt; xÞÞ†: ð74Þ


Using Eqs. (71), (72), and (74),

Iðt; xÞ ¼X

l;ρ;n;k;ζ;m;ν;μ

ℏ2ε0V

ðω~lω~kÞ1=2a†~l;ρ;n

a~k;ζ;m

× eiðω~l−ω~kÞtu�~l;ρ;n

ðxÞu~k;ζ;mðxÞε~l;ρ;νε~k;ζ;μs∼

νμ: ð75Þ

2. Mode Expansion for Split Detector Scenario

The following approach is taken in [12]: “Let us con-sider a beam of light with an electric field distribu-tion given by EðxÞ. We can build an orthonormalbasis of the transverse plane fuig such that u0 ¼EðxÞ=∥EðxÞ∥ is the first vector; u1 is a “flipped”mode,given by −u0ðxÞ for x < 0 and u0ðxÞ for x > 0… andthe other modes are chosen in order to form a basis.”They go on to conclude that, in computing the noisein a split detector, it is sufficient to consider only u0and u1, ignoring the higher transverse modes.Since the beam width at the detector is W,

u0ðxÞ ¼ 1; −W=2 ≤ x ≤ W=2

¼ 0 otherwise: ð76Þ

u1ðxÞ ¼ −1; −W=2 ≤ x ≤ 0

¼ 1; 0 < x ≤ W=2

¼ 0 otherwise

: ð77Þ

B. Direct-Detection Direction Finding

In direct detection, the local-oscillator beam, whichinduces oscillations in the photoelectric currentwhen mixed with the reflected light from that target,is absent. Therefore, we do not include the factorcosðωHtþ θHÞ in the definition of the signal operator.Using Eq. (10), the operator corresponding to di-

rect detection of the direction of beam arrival withthe split detector can be written as

S0sp ¼

ZdxεðxÞ1τ

Z τ

0dtIðt; xÞ: ð78Þ

Using Eq. (75) in Eq. (78),

S0sp ¼

X~l;ρ;n;~k;ζ;m;ν;μ s:t:ω~l¼ω~k

ℏω~l2ε0V

a†~l;ρ;n

a~k;ζ;m

ZdxεðxÞ

× u�~l;ρ;n

ðxÞu~k;ζ;mðxÞε~l;ρ;nε~k;ζ;μ~sνμ ð79Þ

in the limit τ → ∞.

1. Single Nonvacuum Transverse Mode

The “alternative-hypothesis” state, corresponding tothe single-mode beam being displaced by an amountδ in the x direction (we will always take δ > 0) is

jψ 0sp;1i ¼

Y~l00;ρ00;n00

≠T

j0i~l00;ρ00;n00 jα; ξiT : ð80Þ

The transverse-mode function for the target-returnmode is the even function u0ðxÞ displaced in thepositive-x direction a distance δ:

uT−1ðxÞ ¼ u0ðx − δÞ: ð81Þ

The null-hypothesis state is the same, but with δ ¼ 0:

jψ 0sp;0i ¼ jψ 0

sp;1iδ¼0: ð82Þ

From Eqs. (76) and (81),

ZdxεðxÞjuT−1ðxÞj2 ¼ 2δ: ð83Þ

Using Eqs. (79)–(83),

hψ 0sp;0jS0

spjψ 0sp;0i ¼ 0; ð84Þ

and

hψ 0sp;1jS0

spjψ 0sp;1i ¼ 2δ ~κℏωT

2ε0V�nT ; ð85Þ

where~κ ¼ κ=W: ð86Þ

Using Eqs. (79)–(83), and keeping in mind that onlythe T is in a nonvacuum state,

hψ 0sp;0jS02

spjψ 0sp;0i ¼

X~k;ζ;m;ν;μ s:t:ω~k¼ωT

X~l0;ρ0;n0;ν0;μ0 s:t:ω~l0 ¼ωT

Y~l00;ρ00;n00

≠T

~l00;ρ00;n00 h0jThα; ξja†Ta~kζ;ma

†~l0ρ0;n0 aT

×Y

~k00;ζ00;m00≠T

j0i~k00;ζ00;m00 jα; ξiT�Z

dxεðxÞu�TðxÞu~k;ζ;mðxÞ

��Zdx0εðxÞu�

~l0;ρ0;n0 ðxÞuTðxÞ�

× εT;νε~k;ζ;μ~sνμε~l0;ρ0;ν0εT;μ0~sν0μ0 : ð87Þ


By virtue of the constraints in the summations aswell as the x; x0 integrals [recall Eqs. (82) and (83)]the only mode that can make a nonzero contributionis the odd transverse mode u1ðxÞ with frequency ωT.Since this mode is in the vacuum state, we obtain

hψ 0sp;0jS02

spjψ 0sp;0i ¼ W2

�~κℏωT

2ε0V

��nT : ð88Þ


Var0S0sp0 ¼ hψ 0

sp;0jS02spjψ 0

sp;0i − hψ 0sp;0jS0

spjψ 0sp;0i2

¼ W2

�~κℏωT

2ε0V

�2�nT : ð89Þ


D2S0sp0

¼�2δW

�2�nT : ð90Þ

In terms of the angular displacement Δθ, thewavelength λ of the beam, and the focal length fand aperture d of the detector optics,

δ ¼ fΔθ; ð91Þ

W ¼ f λ=d; ð92Þ(see, e.g., [13]), so Eq. (90) can be written as

D2S0sp0

¼�2dΔθ

λ

�2�nT : ð93Þ

The minimum discernable angular beam displace-ment is defined as that angle for which theSNR ¼ 1. From Eq. (93),

Δθmin0 ¼ 1

2�n1=2T

�λd

�: ð94Þ

Both Eqs. (90) and (94) are seen to be unaffected bysqueezing. (Note that this analysis, as well as theones that follow, use the “top-hat” form u0ðxÞ forthe average beam profile. A more realistic form foru0ðxÞ and, consequently, for u1ðxÞwould likely changeconstant factors, such as the “1/2” in Eq. (94), butprobably not the dependence on occupation numberor squeezing parameters.)

2. Two Nonvacuum Transverse Modes

Wenow examine amodel along the lines of the experi-ment of [12]. That experiment uses a beam with a“flipped” (i.e., odd function of x) coherent mode andan even-in-x squeezed vacuum mode. The beams arecombined using a beam splitter, which reflects mostof the squeezed vacuum and transmits a small partof the flippedcoherentbeam.Thecoherentbeamshifts

in the transverse direction, while the squeezed beamremains fixed, sowewill refer to the squeezedbeamasthe “local oscillator” and the flipped coherent beam asthe “target-return beam.” Both modes have the samewavelength. So, the alternative-hypothesis state is

jψsp;1i ¼Y

~l00;ρ00;n00≠LO;T

j0i~l00;ρ00;n00 jα; ξiLOjβiT : ð95Þ

The transverse-mode functions are

u0LOðxÞ ¼ u0ðxÞ; ð96Þ

u0TðxÞ ¼ u1ðx − δÞ: ð97Þ

As before, the null-hypothesis state is the same withδ ¼ 0:

jψsp;0i ¼ jψ sp;1iδ¼0: ð98Þ


ZdxεðxÞu0�

LOðxÞu0LOðxÞ ¼ 0; ð99Þ

ZdxεðxÞu0�

LOðxÞu0TðxÞ ¼

ZdxεðxÞu0�

T ðxÞu0LOðxÞ

¼ W − 3δ; ð100Þ

ZdxεðxÞu0�

T ðxÞu0TðxÞ ¼ 2δ: ð101Þ

Using Eqs. (78), (95), and (98)–(101),

hψ sp;0jS0spjψsp;0i ¼ 2W

�~κℏω2ε0V

�ð�nLO − sinh2ðrÞÞ1=2�n1=2

× cosðθT − θLOÞ; ð102Þ

hψsp;1jS0spjψsp;1i ¼

�~κℏω2ε0V

�ð2δ�nT þ 2ðW − 3δÞ

× ð�nLO − sinh2ðrÞÞ1=2�n1=2

× cosðθT − θLOÞÞ; ð103Þ

Var0S0sp ¼ hψsp;1jS02

spjψsp;1i − hψsp;1jS0spjψsp;1i2

¼�~κℏω2ε0V

�W2ð�nLO þ �nTf1þ 2 sinhðrÞ½sinhðrÞ

− coshðrÞ cosð2θT − θsqÞ�gÞ: ð104Þ


Choosing θsq so as to minimize Eq. (104), i.e., so that

cosð2θT − θsqÞ ¼ 1; ð105Þ

Eq. (104) becomes

Var0S0sp ¼

�~κℏω2ε0V

�W2ð�nLO þ �nTe−2rÞ: ð106Þ


D2S0sp ¼

�2δW

�2

×

��nT − 3

��nLO − sinh2ðrÞ

�1=2

�n1=2T cosðθT − θLOÞ

�2

ð�nLO þ �nTe−2rÞ:

ð107Þ

In this experiment, the LO mode is a squeezedvacuum,

�nLO ¼ sinh2ðrÞ: ð108Þ

For

�nLO ≫ 1; ð109Þ

Eq. (108) implies

e−2r ≈1

4�nLO: ð110Þ

Using Eqs. (108)–(110) in Eq. (107),

D2S0sp ≈

�2δW

�2�nT

��nT

�nLO

�: ð111Þ

This particular experiment is not relevant to laserradar since it involves transmitting squeezed light,but the variation in Subsection 9.C is.

C. Heterodyne Direction Finding

To convert the model of Subsection 9.B.2 into some-thing that might be relevant for laser radar, we needto insure that (a) squeezed light is only used in thelocal oscillator and (b) the target-return mode is not“flipped,” i.e., does not contain a sharp edge in thetransverse direction where the phase of the fieldchanges abruptly. Such an edge would become diffuseover distance, due to diffraction; and it would be im-probable to have it well aligned with the boundarybetween the two halves of the split detector. In addi-tion, (c) the frequencies of LO and T must differ, sothat they can be combined using an etalon ratherthan a beam splitter. (Cases with smooth transitionof the flipped mode are analyzed in [11]. These onlyyield factors of 0.60 and 0.94 in the minimum mea-surable distance compared to coherent light. “Thismodest improvement with respect to the standard

quantum limit is due to the fact that the variationof the odd squeezed mode amplitude is too slow whenone crosses the edge x ¼ 0” [11].)

According to [12], switching which mode, squeezedor coherent, is the flipped one makes no difference (inthe homodyne situation considered in that reference;see Subsection 9.B.2). So here we consider a squeezedlocal oscillator with an odd transverse mode functionfixed so that the abrupt phase transition always co-incides with the split in the detector (x ¼ 0). The tar-get-return beam is taken to be coherent with a flattransverse profile, which is displaced from symmetryabout x ¼ 0 by an amount δ > 0. That is,

uLOðxÞ ¼ u1ðxÞ; ð112Þ

uTðxÞ ¼ u0ðx − δÞ: ð113ÞThe signal operator is Ssp ¼ R

dxεðxÞ 1τR τ0

dt cosðωHtþ θHÞIðtÞ: Using Eq. (75) in Eq. (114),

Ssp ¼X

~l;ρ;n;~k;ζ;m;ν;μ s:t: jω~l−ω~kj¼ωH

×�

ℏ4ε0V

�ðω~lω~kÞ1=2a†

~l;ρ;na~k;ζ;me

−iεðω~l−ω~kÞθH

×�Z

dxεðxÞu�~l;ρ;n

ðxÞu~k;ζ;mðxÞ�ε~l;ρ;νε~k;ζ;μsνμ: ð115Þ

The expressions for the states are the same as in thehomodyne case [Eqs. (95) and (98)], but with differenttransverse mode functions [Eqs. (112) and (113)] and,of course, different frequencies for the LO andT modes.

An analysis along the lines of that in Subsec-tion 9.B.2 shows that

hψ sp;0jSspjψsp;0i ¼ W~κℏω2ε0V

ð�nLO − sinh2ðrÞÞ1=2�n1=2t

× cosðθt − θLO þ θHÞ; ð116Þ

hψsp;1jSspjψsp;1i ¼ ðW − δÞ ~κℏω2ε0V

ð�nLO − sinh2ðrÞÞ1=2�n1=2t

× cosðθt − θLO þ θHÞ; ð117Þ

Var0Ssp ¼ W2

�~κℏω4ε0V

�2f2�nLO þ 2�nT ½1 − sinhðrÞ

× ð− sinhðrÞ þ coshðrÞ× cosð2θ − θsq þ 2θHÞÞ�g: ð118Þ

Minimizing this by setting θsq so that

2θ − θsq þ 2θH ¼ 2πn; n ¼ 0;�1;�2;…; ð119Þ

we obtain


Var0Ssp ¼ 2W2

�~κℏω4ε0V

�2��nLO þ �nT

2ð1þ e−2rÞ

�:

ð120Þ

The SNR is then

D2Ssp

¼ ðhψ sp;1jSspjψsp;1i − hψsp;0jSspjψsp;0iÞ2=Var0Ssp

¼ 2

�dΔθλ

�2� 1 −

sinh2ðrÞ�nLO

1þ �nT2�nLO

ð1þ e−2rÞ

��nT

× cos2ðθT − θLO þ θHÞ; ð121Þ

using Eqs. (91), (92), (116), (117), and (120).The largest value of the second factor in parenth-

eses in Eq. (121) is unity, which can be reached onlyby increasing �nLO so that

�nT=�nLO → 0; ð122Þ

sinh2ðrÞ=�nLO → 0: ð123Þ

The latter limit will be approached faster for smallerr. There is presumably no practical limit to how largethe local-oscillator signal can be made relative to thetarget-return signal in a laser radar system. So,there is no reason to do squeezing.Essentially the same result is obtained when

taking the transverse functions for both LO and Tto be even (u0ðxÞ):

D2Ssp−2e

¼ 2

�dΔθλ

�2� 1 −

sinh2ðrÞ�nLO

1þ �nT�nLO

ð1þ e−2rÞ

��nT

× cos2ðθT − θLO þ θHÞ: ð124Þ

10. Heterodyne Phase-Change Estimation

Consider again the heterodyne target scenario ofSection 6, and take the null-hypothesis state to bethe alternative-hypothesis state of that section:

jψ0;phi ¼ jβiT jα; ξiLOY

~k;ζ≠T;LO

j0i~k;ζ: ð125Þ

Take the alternative-hypothesis state to be

jψ1;phi ¼ jβeiδθT iT jα; ξiLOY

~k;ζ≠T;LO

j0i~k;ζ; ð126Þ

i. e., the null hypothesis is that there is a targetpresent at a distance such that the phase of thetarget-return signal at the detector is arg β; the al-ternative hypothesis is identical to the null hypoth-esis, but with a value of the phase differing by anamount δθT .

The SNR for discrimination between these twohypotheses—i.e., for detecting a small change inthe phase of the target-return signal—is

D2ph ¼ ðhψ1;phjSjψ1;phi − hψ0;phjSjψ0;phiÞ2=Var0;phS;

ð127Þwhere

Var0;phS ¼ hψ1;phjS2jψ1;phi − hψ1;phjSjψ1;phi2: ð128Þ

Using Eqs. (9), (21), (26), (125), and (126),

hψ1;phjSjψ1;phi − hψ0;phjSjψ0;phi

¼ κℏω2ε0V

jαjjβjðcosðθT þ δθT − θLO þ θHÞ

− cosðθT − θLO þ θHÞÞ

≈ −δθTκℏω2ε0V

jαjjβj sinðθT − θLO þ θHÞ: ð129Þ

for small δθT . Using Eqs. (11), (125), and (128),

Var0;phS ¼ 2

� κℏω4ε0V

�2ð�nLO

þ �nTf1þ sinhðrÞ½sinhðrÞ− coshðrÞ cosð2θT − θsq þ 2θHÞ�gÞ: ð130Þ

Minimizing this by taking θsq to satisfy

2θT − θsq þ θH ¼ 2πn; n ¼ 0;�1;�2;…; ð131Þ

we obtain

Var0;phS ¼ 2

� κℏω4ε0V

�2��nLO þ �nT

2ð1þ e−2rÞ

�: ð132Þ

Using Eqs. (127), (129), and (132),

D2ph ¼ 2ðδθTÞ2

� 1 −sinh2ðrÞ

�nLO

1þ �nT2�nLO

ð1þ e−2rÞ

��nT

× sin2ðθT − θLO þ θHÞ: ð133Þ

As per the discussion at the end of Subsection 9.C,there is nothing to be gained by squeezing the LO.

11. Balanced Detection

In balanced detection [14,15], the LO and T beamseach enter through one of the input ports of a50=50 beam splitter, and the signal is the differenceof the photoelectron currents at detectors at the twooutput ports.

Operators corresponding to modes of the beamsentering the input ports of the beam splitter will


be denoted by subscripts ðLOÞ and ðTÞ. Operatorscorresponding to modes of beams leaving the exitports will be denoted by subscripts tr and ref (respec-tively, “transmitted” and “reflected” relative to thetarget-return beam). Then

aðref Þ;~k;ζ ¼ raðTÞ;~k;ζ þ t0aðLOÞ;~k;ζ; ð134Þ

aðtrÞ;~k;ζ ¼ taðTÞ;~k;ζ þ r0aðLOÞ;~k;ζ: ð135Þ

Using Eqs. (134) and (135) in Eq. (7), the current op-erators at the two photodetectors are, respectively,

IðtrÞðtÞ ¼X

~lρ;~kζ;νμ

ℏ2ε0V

ðω~lω~lÞ1=2eiðω~l−ω~kÞtε~l;ρ;νε~k;ζ;μ

× sνμðjtj2a†

ðTÞ~l;ρaðTÞ~k;ζ þ r0�ta†

ðLOÞ~lρaðTÞ~kζ

þ t�r0a†

ðTÞ~lρaðLOÞ~kζ þ jr0j2a†

ðLOÞ~l;ρaðLOÞ~k;ζÞ; ð136Þ

Iðref ÞðtÞ ¼X

~lρ;~kζ;νμ

ℏ2ε0V

ðω~lω~lÞ1=2eiðω~l−ω~kÞtε~l;ρ;νε~k;ζ;μ

× sνμðjrj2a†

ðTÞ~l;ρaðTÞ~k;ζ þ t0�ra†

ðLOÞ~lρaðTÞ~kζ

þ r�t0a†

ðTÞ~lρaðLOÞ~kζ þ jt0j2a†

ðLOÞ~l;ρaðLOÞ~k;ζÞ: ð137Þ

The operator corresponding to the differencebetween the currents at the two detectors is

Iðdif f ÞðtÞ ¼ IðtrÞðtÞ − Iðref ÞðtÞ: ð138Þ

From Eqs. (136)–(138),

Iðdif f ÞðtÞ ¼X

~lρ;~kζ;νμ

ℏ2ε0V

ðω~lω~lÞ1=2eiðω~l−ω~kÞtε~l;ρ;νε~k;ζ;μsνμððjtj2

− jrj2Þa†

ðTÞ~l;ρaðTÞ~k;ζ þ ðr0�t − t0�rÞ× a†

ðLOÞ~lρaðTÞ~kζðt�r0 − r�t0Þa†

ðTÞ~lρaðLOÞ~kζ

þ ðjr0j2 − jt0j2Þa†

ðLOÞ~l;ρaðLOÞ~k;ζÞ; ð139Þ

which simplifies to

IðbalÞðtÞ ¼ −iX

~lρ;~kζ;νμ

ℏ2ε0V

ðω~lω~kÞ1=2eiðω~l−ω~kÞtε~l;ρ;νε~k;ζ;μ

× sνμða†

ðLOÞ~lρaðTÞ~kζ − a†

ðTÞ~lρaðLOÞ~kζÞ ð140Þ

when the transmission and reflection coefficients arechosen appropriate to the balanced case, specifically,

t ¼ t0 ¼ 1=ffiffiffi2

p; ð141Þ

r ¼ r0 ¼ i=ffiffiffi2

p: ð142Þ

The heterodyne signal operator is

Sbal−het ¼1τ

Z τ

0dt cosðωH þ θHÞIðbalÞðtÞ

¼ −iX

~lρ;~kζ;νμ s:t: jω~l−ω~kj¼ωH

ℏ4ε0V

ðω~lω~kÞ1=2e−iεðω~l−ω~kÞθH

× ε~l;ρ;νε~k;ζ;μsνμða†

ðLOÞ~l;ρaðTÞ~k;ζ

− a†

ðTÞ~l;ρaðLOÞ~k;ζÞ: ð143Þ

The null-hypothesis and alternative-hypothesisstates are, respectively,

jψbal;0i

¼�Y~l00;ρ00

j0iðTÞ~l00;ρ00�� Y

~k00;ζ00≠LO

j0iðLOÞ~k00;ζ00

�jα; βiðLOÞLO;

ð144Þ

jψbal;1i ¼� Y~l00;ρ00≠T

j0iðTÞ~l00;ρ00�� Y

~k00;ζ00≠LO

j0iðLOÞ~k00;ζ00

�

× jα; βiðLOÞLOjβiðTÞT : ð145Þ

Using Eqs. (143)–(145) the SNR is found to be

D2bal−het ¼ 2

�1 −

sinh2ðrÞ�nLO

��nTsin2ðθT − θLO þ θHÞ:

ð146Þ

For homodyne detection, the signal operator is

Sbal−hom ¼ 1τ

Z τ

0dtIðbalÞðtÞ ¼ −i

X~lρ;~kζ;νμ s:t:ω~l¼ω~k

×ℏω~l2ε0V

ε~l;ρ;νε~k;ζ;μ

× sνμða†

ðLOÞ~l;ρaðTÞ~k;ζ − a†

ðTÞ~l;ρaðLOÞ~k;ζÞ: ð147Þ

The states jψbal;0i and jψbal;1i are the same as in theheterodyne case, with of the course ωH ¼ ωT. Theresulting SNR is

D2bal−hom ¼ 4

�1 −

sinh2ðrÞ�nLO

��nTsin2ðθT − θLOÞ: ð148Þ

Neither Eq. (146) nor Eq. (148) is improved bysqueezing.


Appendix A: Direct Target Detection

1. Signal-to-Noise Ratio

The states for H0 and H1 are, respectively,

jψ 00i ¼

Y~k;ζ

j0i~k;ζ; ðA1Þ

jψ 01i ¼ jα; ξiT

Y~k;ζ≠T

j0i~k;ζ: ðA2Þ

The signal operator is

S0 ¼ τ−1Z τ

0dtIðtÞ

¼X

~l;ρ;~k;ζ;μ;ν s:t:ω~l¼ω~k

ℏω~k

2ε0Va†~l;ρa~k;ζε~l;ρνε~k;ζμsνμ: ðA3Þ

(See comments at the beginning of Subsection 9.B.)Since the variance of the signal Eq. (A3) vanishesin the state Eq. (A1), the definition of the equivalentSNR must be changed to

D02g ¼ ½EðgjH1Þ − EðgjH0Þ�2

Var1g; ðA4Þ

with

Var1g ¼ Eðg2jH1Þ − ½EðgjH1Þ�2: ðA5Þ

This a reasonable definition, since variance in thesignal under either hypothesis will contribute to de-tection errors. Taking g ¼ S0 and using Eq. (37) andEqs. (A1)–(A4),

D02S0 ¼ �n2

T

varsqðnTÞ; ðA6Þ

where varsqðnTÞ is the variance of nT in the squeezedstate:

varsqðnTÞ ¼T hα; ξjn2T jα; ξiT − ðThα; ξjnT jα; ξiTÞ2:

ðA7Þ

For suitable choice of parameters of the squeezedstate, varsqðnTÞ < �nT . So squeezing can improvedirect-detection SNR. For jα; ξiT,

varsqðnTÞ ¼ ð�nT − sinh2ðrÞÞj coshðrÞ− eiðθsq−2θTÞ sinhðrÞj2 þ 2cosh2ðrÞsinh2ðrÞ

ðA8Þ

(see, e.g., [8]), where θT is as given in Eq. (27) and θsqis defined by

ξ ¼ reiθsq : ðA9Þ

To minimize Eq. (A8), set θsq so that

θsq − 2θT ¼ 2π; n ¼ 0;�1;�2;…; ðA10Þ

so

varsqðnTÞ ¼ ð�nT − sinh2ðrÞÞe−2r þ 2cosh2ðrÞsinh2ðrÞ;ðA11Þ

D02S0 ¼ �n2

T

ð�nT − sinh2ðrÞÞe−2r þ 2cosh2ðrÞsinh2ðrÞ :

ðA12ÞSuppose the amount of squeezing, as quantified

by the values of r, is much less than the maximumallowed:

sinh2ðrÞ ≪ �nT : ðA13Þ

Then Eq. (A12) becomes

D02S0 ¼ �nTe2r: ðA14Þ

For large �nT and r,

sinh2ðrÞ ≈ cosh2ðrÞ ≈ 14e2r; ðA15Þ

so, with Eq. (A13),

e2r ≪ 4�nT : ðA16Þ

An upper bound on the SNR is, therefore, usingEqs. (A14) and (A16),

D02S0 ≪ 4�n2

T : ðA17Þ

2. Quantum Efficiency in Direct Detection

Using Eq. (57) in Eq. (A3), the direct-detection signaloperator is

S0B ¼

X~l;ρ;~k;ζ;μ;ν s:t:ω~l¼ω~k

�ℏω~k

2ε0V

�ðt�~l;ρa

†

ðinÞ~l;ρ

þ r�~l;ρa†

ðvacÞ~l;ρÞðt~k;ζaðinÞ~k;ζ þ r~k;ζaðvacÞ~k;ζÞε~l;ρ;νε~k;ζ;μsνμ:ðA18Þ

The states are

jψ 0B;0i ¼ jψ 0

ðinÞ;0ijψ ðvacÞ;0i; ðA19Þ

jψ 0B;1i ¼ jψ 0

ðinÞ;1ijψ ðvacÞ;0i; ðA20Þwhere


jψ 0ðinÞ;0i ¼

Y~k;ζ

j0iðinÞ;~k;ζ; ðA21Þ

jψ 0ðinÞ;1i ¼

Y~k;ζ≠T

j0iðinÞ;~k;ζjα; ξiðinÞ;T ; ðA22Þ

and with jψ ðvacÞ;0i as given in Eq. (61).Using Eqs. (A4) and (A5), and Eqs. (A18)–(A22),

D02S0

B¼

jtT j2�n2ðinÞ;T

jtT j2varsqnðinÞ;T þ ð1 − jtT j2Þ�nðinÞ;T; ðA23Þ

where varsqnðinÞ;T is given by Eq. (A7) withnT → nðinÞ;T . Comparing with Eq. (A6), we see thatnonunity quantum efficiency, η ¼ jtT j2 < 1; shiftsthe statistics of the noise toward those of coherent-state light (variance=nðinÞ;T).

3. Loss Limits Squeezed-Light Improvementin Direct-Detection Signal-to-Noise Ratio

The analysis of Subsection 1 of Appendix A showsthat the use of squeezed light can improve direct-detection SNR compared to that obtained with coher-ent light. From Subsection 2 of Appendix A we cansee that the presence of loss places a limit on theamount of improvement that is possible.For coherent light,

varsqnðinÞ;T → varcohnðinÞ;T ¼ �nðinÞ;T ; ðA24Þ

so, from Eq. (A23),

D02S0

B→ D02

S0B−coh

¼ jtT j2�nðinÞ;T : ðA25Þ

So the improvement in SNR obtained by usingsqueezed-light is

D02S0

B

D02S0

B−coh

¼ �nðinÞ;TjtT j2varsqnðinÞ;T þ ð1 − jtT j2Þ�nðinÞ;T

≤1

1 − jtT j2; ðA26Þ

since varsqnðinÞ;T ≥ 0, or

D02S0

B

D02S0

B−coh

≤1L; ðA27Þ

where the loss L is

L ¼ 1 − jtT j2 ¼ 1 − η ðA28Þby Eq. (70).

M. A. R. thanks Jonathan Ashcom and Jae Kyungfor a helpful discussion on mixing efficiency. Thiswork was sponsored by the Air Force under Air ForceContract FA8721-05-C-0002. Opinions, interpreta-tions, conclusions, and recommendations are thoseof the author and are not necessarily endorsed bythe U.S. Government.

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signal-to-noise ratio in squeezed-light laser radar

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