Abstract

The use of quantum resources—squeezed-vacuum injection (SVI) and noise-free phase-sensitive amplification (PSA)—at the receiver of a soft-aperture homodyne-detection LAser Detection And Ranging (LADAR) system is shown to afford significant improvement in the receiver’s spatial resolution. This improvement originates from the potential for SVI to ameliorate the loss of high-spatial-frequency information about a target or target complex that is due to soft-aperture attenuation in the LADAR’s entrance pupil, and the value of PSA in realizing that potential despite inefficiency in the LADAR’s homodyne detection system. We show this improvement quantitatively by calculating lower error rates—in comparison with those of a standard homodyne detection system—for a one-target versus two-target hypothesis test. We also exhibit the effective signal-to-noise ratio (SNR) improvement provided by SVI and PSA in simulated imagery.

© 2010 Optical Society of America

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  1. B.Bhanu, D.E.Dudgeon, E.G.Zelnio, A.Rosenfeld, D.Casasent, and I.S.Reed, eds., Special Issue on Automatic Target Recognition, IEEE Trans. Image Process.6 (1997).
  2. G.W.Kamerman, ed., Selected Papers on LADAR, SPIE Milestone Series, Vol. MS133 (SPIE, 1997).
  3. G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).
  4. C. M. Caves, “Quantum mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
    [CrossRef]
  5. D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
    [CrossRef]
  6. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [CrossRef] [PubMed]
  7. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
    [CrossRef] [PubMed]
  8. G. Gilbert, M. Hamrick, and Y. S. Weinstein, “On the use of photonic N00N states for practical quantum interferometry,” J. Opt. Soc. Am. B 25, 1336-1340 (2008).
    [CrossRef]
  9. P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” presented at the 14th Coherent Laser Radar Conference, Snowmass, Colorado, 9-13 July 2007.
  10. M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52, 4930-4940 (1995).
    [CrossRef] [PubMed]
  11. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938-1941 (1999).
    [CrossRef]
  12. J. H. Shapiro, B. A. Capron, and R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292-3313 (1981).
    [CrossRef] [PubMed]
  13. J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Networking 4, 501-516 (2005).
    [CrossRef]
  14. Strictly speaking, the soft aperture must be embedded inside a hard-aperture pupil, e.g., A(ρ′)=e−2|ρ′|2/R2 for |ρ′|≤D/2 and A(ρ′)=0 otherwise. Without appreciable loss of generality we shall ignore that constraint here and in Section because: (1) we are interested in soft apertures whose transmission at the hard-aperture limit is <1%; and (2) we will not be assuming so much SVI and PSA that the hard-aperture limit will constrain the soft-aperture resolution improvement afforded by these quantum enhancements. We shall, however, impose the hard-aperture limit in Section , when we employ MTF analysis to generate simulated baseline and quantum-enhanced imagery.
  15. One-to-one imaging corresponds to setting F=L in Fig. , a choice that simplifies the notation. In reality, of course, the target range will satisfy L≫F, however, this merely introduces a minification factor into the analysis.
  16. J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547-1569 (2009).
  17. This equal-strength assumption makes the LADAR's task of distinguishing between the one-target and two-target hypotheses entirely a matter of the spatial pattern in the image plane rather than detected target-return strength.
  18. Strictly speaking, |T(x″)|≤1 is required. However, because of the quasi-Lambertian nature of the target reflection, the simple expressions we have provided lead to appropriate statistics for the classical target-return field arriving at the LADAR receiver's entrance pupil.
  19. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, 1968).
  20. H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493-507 (1952).
    [CrossRef]
  21. Strictly speaking π/2 is the maximum value of θ0. It is only because we are employing paraxial optics that θ0—>∞ appears to be possible. In practice, however, we will never be concerned with angular separations that take us outside the realm of paraxial optics.
  22. Our Gaussian soft-aperture definition for Rayleigh resolution is chosen so that the depth of the trough between the average photon-flux density of the two point targets on the detector array is equal to the trough depth present for the same two targets when they are imaged through an unobscured hard pupil of length D and they are separated by the hard-aperture Rayleigh angle 2θ0=λ/D.
  23. Alternatively, for stationary targets, we can use a two-pulse illumination sequence, with the first pulse employed to image the real quadrature and the second pulse employed to image the imaginary quadrature, to achieve similar results.
  24. Once again we are violating |T|≤1, and once again our field-reflection model does not pose a problem in that it gives physically reasonable statistics for the Fraunhofer-diffracted target return that is collected by the LADAR's entrance pupil.
  25. M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56, 2029-2033 (2009).
    [CrossRef]

2009 (2)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547-1569 (2009).

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56, 2029-2033 (2009).
[CrossRef]

2008 (1)

2006 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

2005 (1)

J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Networking 4, 501-516 (2005).
[CrossRef]

2000 (1)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

1999 (1)

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938-1941 (1999).
[CrossRef]

1995 (1)

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52, 4930-4940 (1995).
[CrossRef] [PubMed]

1994 (1)

D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
[CrossRef]

1981 (2)

1952 (1)

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493-507 (1952).
[CrossRef]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Bollinger, J. J.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
[CrossRef]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Capron, B. A.

Caves, C. M.

C. M. Caves, “Quantum mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Chernoff, H.

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493-507 (1952).
[CrossRef]

Choi, S.-K.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938-1941 (1999).
[CrossRef]

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Erkmen, B. I.

J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Networking 4, 501-516 (2005).
[CrossRef]

Gilbert, G.

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

Grigoryan, V.

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” presented at the 14th Coherent Laser Radar Conference, Snowmass, Colorado, 9-13 July 2007.

Guha, S.

J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Networking 4, 501-516 (2005).
[CrossRef]

Hamrick, M.

Harney, R. C.

Itano, W. M.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
[CrossRef]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Kolobov, M. I.

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52, 4930-4940 (1995).
[CrossRef] [PubMed]

Kumar, P.

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56, 2029-2033 (2009).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938-1941 (1999).
[CrossRef]

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” presented at the 14th Coherent Laser Radar Conference, Snowmass, Colorado, 9-13 July 2007.

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

Lugiato, L. A.

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52, 4930-4940 (1995).
[CrossRef] [PubMed]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

Moore, F. L.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
[CrossRef]

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).

Shapiro, J. H.

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547-1569 (2009).

J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Networking 4, 501-516 (2005).
[CrossRef]

J. H. Shapiro, B. A. Capron, and R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292-3313 (1981).
[CrossRef] [PubMed]

Stelmakh, N.

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56, 2029-2033 (2009).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, 1968).

Vasilyev, M.

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56, 2029-2033 (2009).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938-1941 (1999).
[CrossRef]

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” presented at the 14th Coherent Laser Radar Conference, Snowmass, Colorado, 9-13 July 2007.

Weinstein, Y. S.

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Wineland, D. J.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
[CrossRef]

Ann. Math. Stat. (1)

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493-507 (1952).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547-1569 (2009).

J. Mod. Opt. (1)

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56, 2029-2033 (2009).
[CrossRef]

J. Opt. Networking (1)

J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Networking 4, 501-516 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Rev. A (2)

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52, 4930-4940 (1995).
[CrossRef] [PubMed]

D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1994).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, “Quantum mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Phys. Rev. Lett. (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938-1941 (1999).
[CrossRef]

Other (13)

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” presented at the 14th Coherent Laser Radar Conference, Snowmass, Colorado, 9-13 July 2007.

B.Bhanu, D.E.Dudgeon, E.G.Zelnio, A.Rosenfeld, D.Casasent, and I.S.Reed, eds., Special Issue on Automatic Target Recognition, IEEE Trans. Image Process.6 (1997).

G.W.Kamerman, ed., Selected Papers on LADAR, SPIE Milestone Series, Vol. MS133 (SPIE, 1997).

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).

Strictly speaking, the soft aperture must be embedded inside a hard-aperture pupil, e.g., A(ρ′)=e−2|ρ′|2/R2 for |ρ′|≤D/2 and A(ρ′)=0 otherwise. Without appreciable loss of generality we shall ignore that constraint here and in Section because: (1) we are interested in soft apertures whose transmission at the hard-aperture limit is <1%; and (2) we will not be assuming so much SVI and PSA that the hard-aperture limit will constrain the soft-aperture resolution improvement afforded by these quantum enhancements. We shall, however, impose the hard-aperture limit in Section , when we employ MTF analysis to generate simulated baseline and quantum-enhanced imagery.

One-to-one imaging corresponds to setting F=L in Fig. , a choice that simplifies the notation. In reality, of course, the target range will satisfy L≫F, however, this merely introduces a minification factor into the analysis.

This equal-strength assumption makes the LADAR's task of distinguishing between the one-target and two-target hypotheses entirely a matter of the spatial pattern in the image plane rather than detected target-return strength.

Strictly speaking, |T(x″)|≤1 is required. However, because of the quasi-Lambertian nature of the target reflection, the simple expressions we have provided lead to appropriate statistics for the classical target-return field arriving at the LADAR receiver's entrance pupil.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, 1968).

Strictly speaking π/2 is the maximum value of θ0. It is only because we are employing paraxial optics that θ0—>∞ appears to be possible. In practice, however, we will never be concerned with angular separations that take us outside the realm of paraxial optics.

Our Gaussian soft-aperture definition for Rayleigh resolution is chosen so that the depth of the trough between the average photon-flux density of the two point targets on the detector array is equal to the trough depth present for the same two targets when they are imaged through an unobscured hard pupil of length D and they are separated by the hard-aperture Rayleigh angle 2θ0=λ/D.

Alternatively, for stationary targets, we can use a two-pulse illumination sequence, with the first pulse employed to image the real quadrature and the second pulse employed to image the imaginary quadrature, to achieve similar results.

Once again we are violating |T|≤1, and once again our field-reflection model does not pose a problem in that it gives physically reasonable statistics for the Fraunhofer-diffracted target return that is collected by the LADAR's entrance pupil.

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Figures (5)

Fig. 1
Fig. 1

Diagram of our quantum-enhanced LADAR receiver. At the far left we show the targets (at range L from the receiver) considered in the resolution analysis—one point target on axis and two point targets symmetrically disposed at ± θ 0 in angle about the optical axis—that will be presented in Section 3. The baseband photon-units field operator E ̂ R of the target return is transmitted through a spatially dependent (soft) aperture A ( ρ ) and a squeezed-vacuum field operator E ̂ S is injected according to Eq. (5). The resulting field operator E ̂ R then undergoes PSA before being mixed with a LO and homodyne detected. We assume a continuum homodyne-detection array. This quantum-enhanced receiver will be compared with a classical baseline LADAR system in which E ̂ S is in its vacuum state and no PSA is employed.

Fig. 2
Fig. 2

(a) Error probability, P E , for minimum error-probability decisions between equally likely one-target and two-target hypotheses at three different n ¯ values plotted versus the angular separation θ 0 : n ¯ = 21 dB (circles), n ¯ = 16 dB (squares), n ¯ = 11 dB (diamonds). The angular separations are normalized to the Rayleigh resolution θ 0 ( Ray ) 0.32 λ R . The error probabilities were found via 10,000 Monte-Carlo trials of the sufficient statistic vector r under each hypothesis. The LADAR’s resolution, defined to be the angle θ 0 ( res ) at which P E = 0.03 occurs, are the points where the curves cross the dashed line. All the P E curves asymptote to nonzero constants with increasing θ 0 , and at n ¯ = 17 dB this value is above the P E = 0.03 resolution threshold. (b) Resolution (relative to the Rayleigh resolution) plotted versus n ¯ for the classical baseline (open circles) and SVI-enhanced systems (filled circles) both with unity homodyne efficiency η = 1 and with 15 dB of SVI, i.e., e 2 r = 10 1.5 . There is an 8 dB SNR shift between the baseline and SVI curves at high n ¯ values. (c) Classical baseline resolution (open circles) for an inefficient detection system ( η = 0.25 ) , for that system enhanced with 15 dB SVI but no PSA (filled circles), and for it enhanced with 15 dB of SVI plus G eff = 15 dB of PSA (filled squares). This latter case shows an 11 dB SNR shift at high n ¯ values. Also shown is the case η = 1 with SVI (open squares), which corresponds to the performance obtained in the limit G eff .

Fig. 3
Fig. 3

(a) SNR shift versus homodyne efficiency η for several different PSA gain values (from bottom to top) G eff = 4.8 dB (open squares), 7.0 dB (open circles), 10 dB (filled diamonds), and 15 dB ( filled squares). The top curve (filled circles) is 1 η , the limit for G eff . (b) The same curves with 15 dB of SVI in addition to PSA. Here we also plot the case with no PSA G eff = 0 dB ( open diamonds). Note that for η = 1 all curves converge to the 8 dB SVI SNR shift. For low values of η and G eff , the vacuum noise introduced by inefficient detection destroys the SVI advantage. For higher G eff values, PSA preserves the SVI advantage. At low η values, the SVI plus PSA advantage is primarily due to the PSA stage, as the curves converge to the same values seen in (a) for the non-SVI case.

Fig. 4
Fig. 4

Simulated intensity images for our quantum-enhanced LADAR receiver when the planar target is the USAF resolution chart, shown in (a), that gives rise to fully developed laser speckle. We have assumed a target range L = 1 km , a 15 m × 15 m square target region, λ = 1.55 μ m , an R = 4 mm soft-aperture receive pupil inside a D = 8 mm diameter hard aperture imaged onto a continuum-detector homodyne array. (b) Image of the resolution chart after blurring by transmission through the soft aperture. This corresponds to the image in the limit of high SNR and averaging a large number of intensity images with statistically independent speckle. Images (c)–(f) show detected images averaging over M = 100 intensity images assuming independent speckle fluctuations and homodyne efficiency η = 0.25 . (c) Image obtained by the baseline homodyne LADAR. (d) Image obtained by the SVI-enhanced LADAR with 15 dB squeezing. Little image improvement is seen in comparison with the baseline case owing to homodyne inefficiency. (e) Image obtained by the PSA-enhanced LADAR with G eff = 15 dB . Some image improvement is seen in comparison with the baseline case. (f) Image obtained using SVI plus PSA enhanced LADAR. Substantial image improvement over the baseline case in (c) is seen.

Fig. 5
Fig. 5

1D slices showing the spatial-frequency content (magnitude-squared Fourier transforms) of the signal and noise from an M = 100 simulated intensity-image average using the USAF resolution chart from Fig. 4a as the target. In each plot the signal contribution is the solid curve and the noise component is the dashed curve. (a) Spatial-frequency content for the baseline sensor (no quantum enhancement) with perfect homodyne efficiency, η = 1 . (b) Spatial-frequency content for the baseline sensor with η = 0.25 ; the signal is reduced but the noise level is unaffected. (c) Spatial-frequency content with η = 0.25 and PSA with G eff = 10 dB employed; the SNR level is restored to that obtained with perfect homodyne efficiency. (d) Spatial-frequency content when η = 0.25 and 15 dB of SVI employed in to addition G eff = 10 dB of PSA; the noise is suppressed at spatial frequencies that are attenuated by the soft aperture, giving an improved SNR relative to that in (c) and (a).

Equations (39)

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E R ( ρ , t ) = d ρ I T τ p ω λ s ( t 2 L c ) T ( ρ ) e i k ρ ρ L + i k | ρ | 2 2 L i λ L ,
E R ( ρ , t ) = A ( ρ ) E R ( ρ , t ) ,
y d ( ρ ) = η I T τ p ω λ Re [ d ρ T ( ρ ) m ( ρ ρ ) ] + n d ( ρ ) ,
m ( ρ ) = d f A ( λ L f ) e i 2 π f ρ = π 2 ( R λ L ) 2 e k 2 | ρ | 2 R 2 8 L 2
E ̂ R ( ρ , t ) = A ( ρ ) E ̂ R ( ρ , t ) + 1 A 2 ( ρ ) E ̂ S ( ρ , t ) .
E ̂ S ( ρ , t ) = cosh ( r ) E ̂ in ( ρ , t ) sinh ( r ) E ̂ in ( ρ , t ) ,
E ̂ ( ρ , t ) = G E ̂ R ( ρ , t ) + G 1 E ̂ R ( ρ , t ) ,
y d ( ρ ) = G eff η I T τ p ω λ Re [ d ρ T ( ρ ) m ( ρ ρ ) ] + n d ( ρ ) ,
S n d , n d ( f ) = η G eff 4 ( A 2 ( λ L f ) + [ 1 A 2 ( λ L f ) ] e 2 r ) + 1 η 4 .
Y d ( f ) d ρ y d ( ρ ) e i 2 π f ρ , for < f x , f y < ,
Re [ Y d ( f ) ] = G eff η I T τ p ω λ Ev [ T ̃ r ( f ) ] A ( λ L f ) + n r ( f ) ,
Im [ Y d ( f ) ] = G eff η I T τ p ω λ Od [ T ̃ r ( f ) ] A ( λ L f ) + n i ( f ) ,
T ̃ r ( f ) d f Re [ T ( ρ ) ] e i 2 π f ρ
n j ( f 1 ) n j ( f 2 ) = S n d n d ( f 1 ) 2 δ ( f 1 f 2 ) , for j = r , i .
T ( x ) = { λ d T v 0 δ ( x ) , under H 1 λ d T 2 [ v + δ ( x θ 0 L ) + v δ ( x + θ 0 L ) ] , under H 2 }
Y d ( f x ) 2 S n d n d ( f x ) Y d ( f x ) ,
Re [ Y d ( f x ) ] = 2 G eff η I T τ p ω λ S n d n d ( f x ) Ev [ T ̃ r ( f x ) ] A ( λ L f x ) + n r ( f x ) ,
Im [ Y d ( f x ) ] = 2 G eff η I T τ p ω λ S n d n d ( f x ) Od [ T ̃ r ( f x ) ] A ( λ L f x ) + n i ( f x ) ,
Ev [ T ̃ r ( f x ) ] = { λ d T Re ( v 0 ) , under H 1 λ d T 2 Re ( v + + v ) cos ( 2 π θ 0 L f x ) , under H 2 }
Od [ T ̃ r ( f x ) ] = { 0 , under H 1 λ d T 2 Im ( v + v ) sin ( 2 π θ 0 L f x ) , under H 2 . }
r c d f x Re [ Y d ( f x ) ] A ( λ L f x ) 2 S n d n d ( f x ) ( 4 ( λ L ) 2 π R 2 ) 1 4 cos ( 2 π θ 0 L f x ) ,
r d f x Re [ Y d ( f x ) ] A ( λ L f x ) 4 S n d n d ( f x ) ( 4 ( λ L ) 2 π R 2 ) 1 4 ,
r s d f x Im [ Y d ( f x ) ] A ( λ L f x ) 2 S n d n d ( f x ) ( 4 ( λ L ) 2 π R 2 ) 1 4 sin ( 2 π θ 0 L f x ) ,
Λ 1 = [ 8 G eff η n ¯ C 1 2 + C 0 + C 2 2 ( 4 G eff η n ¯ C 0 + 1 ) C 1 0 2 ( 4 G eff η n ¯ C 0 + 1 ) C 1 ( 4 G eff η n ¯ C 0 + 1 ) C 0 0 0 0 C 0 C 2 ] .
Λ 2 = [ Λ 0 0 T [ 2 G eff η n ¯ ( C 0 C 2 ) + 1 ] ( C 0 C 2 ) ] ,
Λ = [ [ 2 G eff η n ¯ ( C 0 + C 2 ) + 1 ] ( C 0 + C 2 ) 2 [ 2 G eff η n ¯ ( C 0 + C 2 ) + 1 ] C 1 2 [ 2 G eff η n ¯ ( C 0 + C 2 ) + 1 ] C 1 4 G eff η n ¯ C 1 2 + C 0 ] .
C 0 = d f x A 2 ( λ L f x ) 4 S n d n d ( f x ) 4 ( λ L ) 2 π R 2 ,
C 1 = d f x A 2 ( λ L f x ) 4 S n d n d ( f x ) 4 ( λ L ) 2 π R 2 cos ( 2 π θ 0 L f x ) ,
C 2 = d f x A 2 ( λ L f x ) 4 S n d n d ( f x ) 4 ( λ L ) 2 π R 2 cos ( 4 π θ 0 L f x ) ,
n ¯ = I T d T τ p ω λ L π R 2 4 .
r T [ ( Λ 1 ) 1 ( Λ 2 ) 1 ] r decide H 2 < decide H 1 2 ln ( | Λ 2 | 1 2 | Λ 1 | 1 2 ) ,
y 1 ( ρ ) = G eff η I T τ p 2 ω λ Re [ d ρ T ( ρ ) m ( ρ ρ ) ] + n 1 ( ρ ) ,
y 2 ( ρ ) = G eff η I T τ p 2 ω λ Im [ d ρ T ( ρ ) m ( ρ ρ ) ] + n 2 ( ρ ) .
A ( ρ ) = { e 2 | ρ | 2 R 2 , for | ρ | D 2 0 , otherwise . }
y ( ρ ) = y 1 ( ρ ) + i y 2 ( ρ ) = G eff η I T τ p 2 ω λ d ρ T ( ρ ) m ( ρ ρ ) + n ( ρ ) = s ( ρ ) + n ( ρ ) ,
y ( ρ ) = d ρ y ( ρ ) h D ( ρ ρ ) = s ( ρ ) + n ( ρ )
h D ( ρ ) | f | D 2 λ L d f e i 2 π f ρ = π D 2 4 ( λ L ) 2 J 1 ( π D | ρ | λ L ) π D | ρ | 2 λ L .
T * ( ρ 1 ) T ( ρ 2 ) = λ 2 T ( ρ 1 ) δ ( ρ 1 ρ 2 ) ,
SNR I ( ρ ) = SNR ( ρ ) 2 1 + SNR ( ρ ) 2 + 1 2 SNR ( ρ ) 1 ,

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