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Imaging resolution signal-to-noise ratio in transverse phase amplification from classical information theory
Physics Letters A 373 (2009) 999–1000
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Imaging resolution signal-to-noise ratio in transverse phase amplification from classical information theory Doug French ∗,1 , Zun Huang, Hsueh-Yuan Pao 2 , Igor Jovanovic School of Nuclear Engineering, Purdue University, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
Article history: Received 16 December 2008 Accepted 11 January 2009 Available online 21 January 2009 Communicated by V.M. Agranovich
a b s t r a c t A quantum phase amplifier operated in the spatial domain can improve the signal-to-noise ratio in imaging beyond the classical limit. The scaling of the signal-to-noise ratio with the gain of the quantum phase amplifier is derived from classical information theory. © 2009 Published by Elsevier B.V.
PACS: 42.50.-p 42.65.Lm 42.68.Wt 42.79.Pw Keywords: Resolution Phase amplification Classical information theory
Regardless of the design of the imaging system, detector performance, and the technique used to analyze the signal, there is a limit to the detectability of a luminous object. This limit is set by the random nature of both the signal and the interfering background. All efforts to reduce and eliminate the errors in detection ultimately reach a limit determined by the nature’s underlying randomness, which infects all observations of physical phenomena with an innate uncertainty. The probabilities of error in detection incurred by this uncertainty can be used to reveal and quantify the ultimate limits on the distinguishability of signals. We shall seek out that ultimate limit in a domain where observations and their inevitable errors can be analyzed with some clarity and ease. The most important characteristic of an imaging system is resolution. The Rayleigh criterion limits the angular resolution of a sensor aperture to the value r0 ≈ λ/a, where λ is the operating wavelength, and a is the aperture size. Helstrom derived the limit on the resolution by analyzing the detectability of two weak, closely spaced point sources of radiation observed against a uniform background [1]. He applied the classical detection and estimation formalism to test two alternative hypotheses and quantified the reliability of the system in terms of
* 1 2
Corresponding author. Tel.: +1 765 496-1147; fax: +1 765 494 9570. E-mail address: [email protected] (D. French). 400 Central Dr., West Lafayette, IN 47907, USA. On sabbatical leave from Lawrence Livermore National Laboratory.
0375-9601/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.physleta.2009.01.020
the signal-to-noise ratio (SNR). In Helstrom’s analysis, the SNR is a function of the angular separation of the two sources and the signal power. He treated the resolution of two sources located at (u 1 , R ) and (−u 1 , R ), as shown in Fig. 1, as a choice between an illuminance J 0 (x, ω) due to a single source at the zenith and its background, and a second illuminance J 1 (x, ω) arising from two close sources and a background, where ω is the signal frequency. The threshold SNR associated with the problem of resolving the two point sources is 2 D r0 =
3(45π 2 − 272)
ka|u 1 |
2(3π 2 − 16)
2R
4
D 2θ0 = C
θ r0
4 D 2θ0 ,
(1)
where k = 2π /λ is the wavenumber, θ is half of the angular separation of two sources, C is a constant, and D 2θ0 is the maximum threshold SNR which is D 2θ0 =
3π 2 − 16 ητ P s2 k2 a4 96π 3
P b h¯ ω R 4
(2)
for circular aperture, or D 2θ0 =
ητ P s2k2 a4 72π 3 P b h¯ ω R 4
(3)
for square aperture, where η is the quantum efficiency, P s is the signal power, and P b is the background noise power density. The maximum threshold SNR is associated with the problem of detection of the signal above the background when the source and the
1000
D. French et al. / Physics Letters A 373 (2009) 999–1000
amplifier, for which many experimental demonstrations have been made. Before phase amplification, the non-oscillating component of the signal is E 0 = Ae i θ .
(4)
After phase amplification, the signal can be approximated as A E 1 = √ ei H θ , H
(5)
where H is the gain of the phase amplification. The signal amplitude and phase experience the following changes after the phase amplifier: A
|E1| = √
H
|E0| =√ ,
(6)
H
and Fig. 1. Two point sources positions relative to the imaging system aperture.
detector are located exactly on the axis of the optical system. It is obvious that D 2θ0 is proportional to P s2 , and by simple inspection
2 is proportional to θ 4 and P s2 . This means of (1) we find that D r0 that an increase in the angular separation produces a significantly greater improvement in the SNR than the same relative increase in the signal power. In [1], the ultimate limitations on the detection of optical signals are considered, which stem from the random nature of the signals themselves. The noise of the detection and processing circuitry adds to the basic data, but is not included in this consideration. In order to achieve system performance beyond the classical limit, the quantum noise, which is the result of the quantum nature of the light, has to be manipulated. In general, the properties of the quantum noise depend on both the quantum state of light and the measurement process. The quantum state control is one of the most efficient methods to reduce the quantum noise. It has been suggested that a quantum phase amplifier can be used to control the quantum state [2] in a fashion which is advantageous to the problem of transverse imaging resolution. The phase amplifier deamplifies the photon number (power) with gain 1/ H while amplifying the phase by a gain H . Formally, phase amplification has the same effect as scaling the frequency of the received signal by a factor of H , but with no concomitant wavelength change. Nearly ideal phase amplification can be accomplished through common nonlinear processes; it can be well approximated by a phase sensitive amplifier such as a degenerate optical parametric
arg E 1 = H θ = H arg E 0 .
(7)
The squared magnitude of the signal amplitude, A 2 , is proportional to the signal power. Phase amplification diminishes the signal power, while increasing the phase, which is canonically conjugate to the power, by the same factor. The final state is nonclassical, having been squeezed in amplitude and antisqueezed in phase. Substituting (6) and (7) into (1), we find the SNR after phase amplification to be 2 D r1 =
3(45π 2 − 272)
=C
2(3π 2 − 16) 4 2 D θ0 Hθ r0
H2
Hka|u 1 | 2R
2 = H 2 D r0 .
4
D 2θ0 H2 (8)
We conclude from (8) that the SNR for resolving two closely spaced, weak point sources improves by a factor of H 2 after the 2 quantum phase amplifier. If H = 10, for example, D r1 increases 2 100 times compared to D r0 , thus making it possible to resolve two point sources separated only one tenth of the Rayleigh limit r0 = λ/(10a) with the same SNR as the two point sources separated by one Rayleigh limit when no quantum phase amplifier is used. This approach has a potential to offer significant improvements in resolution-limited imaging problems. References [1] C. Helstrom, IEEE Trans. Inform. Theory IT-10 (1964) 275. [2] G.M. D’Ariano, C. Macchiavello, N. Sterpi, H.P. Yuen, Phys. Rev. A 54 (1996) 4712.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Imaging resolution signal-to-noise ratio in transverse phase amplification from classical information theory Doug French ∗,1 , Zun Huang, Hsueh-Yuan Pao 2 , Igor Jovanovic School of Nuclear Engineering, Purdue University, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
Article history: Received 16 December 2008 Accepted 11 January 2009 Available online 21 January 2009 Communicated by V.M. Agranovich
a b s t r a c t A quantum phase amplifier operated in the spatial domain can improve the signal-to-noise ratio in imaging beyond the classical limit. The scaling of the signal-to-noise ratio with the gain of the quantum phase amplifier is derived from classical information theory. © 2009 Published by Elsevier B.V.
PACS: 42.50.-p 42.65.Lm 42.68.Wt 42.79.Pw Keywords: Resolution Phase amplification Classical information theory
Regardless of the design of the imaging system, detector performance, and the technique used to analyze the signal, there is a limit to the detectability of a luminous object. This limit is set by the random nature of both the signal and the interfering background. All efforts to reduce and eliminate the errors in detection ultimately reach a limit determined by the nature’s underlying randomness, which infects all observations of physical phenomena with an innate uncertainty. The probabilities of error in detection incurred by this uncertainty can be used to reveal and quantify the ultimate limits on the distinguishability of signals. We shall seek out that ultimate limit in a domain where observations and their inevitable errors can be analyzed with some clarity and ease. The most important characteristic of an imaging system is resolution. The Rayleigh criterion limits the angular resolution of a sensor aperture to the value r0 ≈ λ/a, where λ is the operating wavelength, and a is the aperture size. Helstrom derived the limit on the resolution by analyzing the detectability of two weak, closely spaced point sources of radiation observed against a uniform background [1]. He applied the classical detection and estimation formalism to test two alternative hypotheses and quantified the reliability of the system in terms of
* 1 2
Corresponding author. Tel.: +1 765 496-1147; fax: +1 765 494 9570. E-mail address: [email protected] (D. French). 400 Central Dr., West Lafayette, IN 47907, USA. On sabbatical leave from Lawrence Livermore National Laboratory.
0375-9601/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.physleta.2009.01.020
the signal-to-noise ratio (SNR). In Helstrom’s analysis, the SNR is a function of the angular separation of the two sources and the signal power. He treated the resolution of two sources located at (u 1 , R ) and (−u 1 , R ), as shown in Fig. 1, as a choice between an illuminance J 0 (x, ω) due to a single source at the zenith and its background, and a second illuminance J 1 (x, ω) arising from two close sources and a background, where ω is the signal frequency. The threshold SNR associated with the problem of resolving the two point sources is 2 D r0 =
3(45π 2 − 272)
ka|u 1 |
2(3π 2 − 16)
2R
4
D 2θ0 = C
θ r0
4 D 2θ0 ,
(1)
where k = 2π /λ is the wavenumber, θ is half of the angular separation of two sources, C is a constant, and D 2θ0 is the maximum threshold SNR which is D 2θ0 =
3π 2 − 16 ητ P s2 k2 a4 96π 3
P b h¯ ω R 4
(2)
for circular aperture, or D 2θ0 =
ητ P s2k2 a4 72π 3 P b h¯ ω R 4
(3)
for square aperture, where η is the quantum efficiency, P s is the signal power, and P b is the background noise power density. The maximum threshold SNR is associated with the problem of detection of the signal above the background when the source and the
1000
D. French et al. / Physics Letters A 373 (2009) 999–1000
amplifier, for which many experimental demonstrations have been made. Before phase amplification, the non-oscillating component of the signal is E 0 = Ae i θ .
(4)
After phase amplification, the signal can be approximated as A E 1 = √ ei H θ , H
(5)
where H is the gain of the phase amplification. The signal amplitude and phase experience the following changes after the phase amplifier: A
|E1| = √
H
|E0| =√ ,
(6)
H
and Fig. 1. Two point sources positions relative to the imaging system aperture.
detector are located exactly on the axis of the optical system. It is obvious that D 2θ0 is proportional to P s2 , and by simple inspection
2 is proportional to θ 4 and P s2 . This means of (1) we find that D r0 that an increase in the angular separation produces a significantly greater improvement in the SNR than the same relative increase in the signal power. In [1], the ultimate limitations on the detection of optical signals are considered, which stem from the random nature of the signals themselves. The noise of the detection and processing circuitry adds to the basic data, but is not included in this consideration. In order to achieve system performance beyond the classical limit, the quantum noise, which is the result of the quantum nature of the light, has to be manipulated. In general, the properties of the quantum noise depend on both the quantum state of light and the measurement process. The quantum state control is one of the most efficient methods to reduce the quantum noise. It has been suggested that a quantum phase amplifier can be used to control the quantum state [2] in a fashion which is advantageous to the problem of transverse imaging resolution. The phase amplifier deamplifies the photon number (power) with gain 1/ H while amplifying the phase by a gain H . Formally, phase amplification has the same effect as scaling the frequency of the received signal by a factor of H , but with no concomitant wavelength change. Nearly ideal phase amplification can be accomplished through common nonlinear processes; it can be well approximated by a phase sensitive amplifier such as a degenerate optical parametric
arg E 1 = H θ = H arg E 0 .
(7)
The squared magnitude of the signal amplitude, A 2 , is proportional to the signal power. Phase amplification diminishes the signal power, while increasing the phase, which is canonically conjugate to the power, by the same factor. The final state is nonclassical, having been squeezed in amplitude and antisqueezed in phase. Substituting (6) and (7) into (1), we find the SNR after phase amplification to be 2 D r1 =
3(45π 2 − 272)
=C
2(3π 2 − 16) 4 2 D θ0 Hθ r0
H2
Hka|u 1 | 2R
2 = H 2 D r0 .
4
D 2θ0 H2 (8)
We conclude from (8) that the SNR for resolving two closely spaced, weak point sources improves by a factor of H 2 after the 2 quantum phase amplifier. If H = 10, for example, D r1 increases 2 100 times compared to D r0 , thus making it possible to resolve two point sources separated only one tenth of the Rayleigh limit r0 = λ/(10a) with the same SNR as the two point sources separated by one Rayleigh limit when no quantum phase amplifier is used. This approach has a potential to offer significant improvements in resolution-limited imaging problems. References [1] C. Helstrom, IEEE Trans. Inform. Theory IT-10 (1964) 275. [2] G.M. D’Ariano, C. Macchiavello, N. Sterpi, H.P. Yuen, Phys. Rev. A 54 (1996) 4712.