Patent Publication Number: US-2009224962-A1

Title: Apparatus And Method For Sensors Having Improved Angular Resolution

Description:
RELATED APPLICATIONS 
     The present application claims the benefit of priority to Provisional Application Ser. No. 60/976,318 filed Sep. 28, 2007, which is incorporated herein by reference. 
    
    
     FIELD 
     The present application relates to the field of echo-location and echo-imaging systems, including radar, sonar, and lidar systems, medical ultrasound, and other imaging systems that use coherent electromagnetic or acoustic waves. 
     BACKGROUND 
     Existing echo-location and echo-imaging systems, including radar, sonar, and lidar systems, medical ultrasound, forward imaging systems, such as transmission imaging, scattering imaging, and diffraction, and other imaging systems using coherent electromagnetic or acoustic waves, such as those that may typically have a transmitter for emitting coherent waves for “illuminating” one or more targets. This transmitter may incorporate one or more of radio frequency or microwave transmitters, infrared or optical lasers, or may include ultrasonic transducers. 
     The coherent waves are reflected by the one or more targets towards receiving and/or imaging apparatus, hereinafter receiver, that may, but need not, be collocated with the transmitter. These reflected waves are an echo or echoes. 
     It is desirable to determine the number and locations, and other qualities such as speed, of targets, or to produce quality images from, information embedded in reflected waves and echoes. For example, a warship&#39;s crew may respond quite differently if it can be determined that echoes are being received from a single, large, transport aircraft instead of several small aircraft flying in a tight formation. 
     Radar, lidar, active sonar, and medical ultrasound systems may use round-trip “time-of-flight” information to determine distance from the receiving and/or imaging apparatus, they may also use Doppler-shift of echoes to determine target speed and the velocity of blood flow. It is also desirable to discriminate between, or image, targets based upon the direction, or angle, from which echoes are received—for which good angular resolution is required. The minimum angle that must separate two targets for the system to reliably determine that echoes are from two, and not one larger target, is the angular resolution of the system. Good angular resolution is of importance in medical imaging, and sonar, as well as radar, since imaging of a large target is equivalent to studying many smaller, closely spaced, targets. 
     Classically, a limit for angular resolution of a receiving and/or imaging system is related to the wavelength of the waves and the aperture size, or the greatest distance between elements, of the receiver. 
     Resolution 
     Resolution refers to the ability to distinguish closely spaced signal sources. The angular resolution of the classical sensor is given by the diffraction angle λ/D of the array aperture; the field of view is Nλ/D for N elements. To see this, consider a plane wave incident on a one-dimensional antenna array with N elements and aperture D, which we assume is the limiting aperture in the system. The signal received at the array aperture in angular space ψ from a point source far away has the form: 
     
       
         
           
             
               
                 
                   
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     Where θ is the angle of incident on the detector, k=2π/λ, λ is the wavelength, ω is the operating angular frequency, d is the separation between the elements. A typical angular signal strength distribution is plotted in  FIG. 3 . A target is imaged as a finite-sized spot by the conventional imaging system. The minimum spot dimension obtained for point-like objects is determined by two zero signal strength angular positions adjacent the maximum signal strength. This means that the argument of the sine term in the numerator of equation (1) should span an integral multiple of π. They are at Nkd(sin ψ−sin θ)=π, and Nkd(sin ψ-sin θ)=−π. Consequently the spot size is: 
     
       
         
           
             
               
                 
                   
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     Nd in equations (2) and (3) is called numerical aperture (NA) and is the size of the array aperture D. The spot size (3) is called point-spread function (PSF), it can be used as a convention criterion to define a limit to the minimum angular separation below which two nearby objects can not be distinguished as clearly providing two peaks, see  FIG. 4 . It has been known for some time that this criterion, the Rayleigh Limit, is the resolution limit of a classical system. 
     In past two decades, parameter estimation has been an area of focus by applied statisticians and engineers. As applications expanded the interest in accurately estimating relevant temporal as well as spatial parameters grew. Sensor array signal processing emerged as an active area of research and was centered on the ability to fuse, that is, to process, analyze, and/or synthesize, data collected at several sensors in order to carry out a given estimation task (space-time processing). This framework has the advantage of prior information on the data acquisition system (i.e. array geometry, sensor characteristics). The methods have proven useful for solving several real world problems. One of most notable is for source location. It demonstrated the possibility that the processing developed such as MUltiple SIgnal Classification (MUSIC) algorithm, which uses the eigenvector decomposition method or signal subspace approach, might be a superresolution algorithm useful to locate closely spaced multiple emitters (targets) with high resolution (smaller than the Rayleigh Limit). 
     However, the Cramer-Rao Bound principle, 
     
       
         
           
             
               
                 
                   
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     named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, expresses a lower bound on the variance of estimators of a deterministic parameter. It is the “best” in a minimum error variance sense (lower bound) that an estimator can achieve. In a statistical setting, assumptions can be made regarding statistical properties of the signal and/or noise 
     In conclusion, the resolution obtained in classical sense might, with MUSIC, be better than Rayleigh Limit, but never better than Cramer-Rao Bound. 
     Since the Raleigh Limit has been known for many years, prior systems for improving angular resolution of a system have often involved increasing operating frequency, thereby decreasing wavelength λ, or alternatively increasing aperture size D. There are often practical limitations to either. For example, waves, whether sonic or electromagnetic, of differing wavelengths may propagate differently—for example short radar wavelengths may be limited to line of sight while atmospheric ionization may allow longer radar wavelengths to follow the earth&#39;s curvature thereby allowing detection of targets at greater distances from the imaging system. Similarly, receivers having a large physical aperture size D may be unwieldy. 
     SUMMARY 
     An imaging or echolocation system has a source of coherent waves, such as acoustic and electromagnetic waves, that are transmitted towards any target or targets of interest. Any waves reflected or echoed by the target or targets are received by a receiver further having many sensor elements spaced across a surface. A reference signal of the same frequency of the waves as received from received waves. A least one phase amplifier receives signals from at least one sensor element, and amplifies phase differences between the reference signal and the received waves. In imaging systems, signals from the phase amplifier(s) enter image construction apparatus and are used for constructing an image; in echolocation systems, signals from the phase amplifiers are used to distinguish between and identify targets. In various embodiments, phase amplifiers may be implemented in analog or digital form. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is a block diagram illustrating transmitter, receiver, and angular separation between two targets as viewed by a device of present invention. 
         FIG. 2  is an illustration showing details of targets and a detector, according to an embodiment. 
         FIG. 3  is an illustration of an angular signal strength from a single target in prior art or the present systems. 
         FIG. 4  is an illustration of an angular signal strength from a pair of closely spaced targets in prior art systems. 
         FIG. 5  illustrates the effect of phase-difference amplification on angular resolution in a complex phase/amplitude diagram. 
         FIG. 6  illustrates phase amplification in field-quadrature phase space. 
         FIG. 7  is a block diagram of an individual phase amplifier. 
         FIG. 8  is a block diagram of an echolocation or imaging system embodying the phase amplifier of  FIG. 7 . 
         FIG. 9  is a block diagram of an echolocation or imaging system having a separate transmitter. 
         FIG. 10  is a block diagram of an alternative embodiment of an individual phase amplifier. 
         FIG. 11  is an illustration of the effect of a phase amplifier in field-quadrature phase space. 
         FIG. 12  illustrates simulated performance of normal radar using a 10 GHz, 64-element phased array with 0.003 radian separation between targets. 
         FIG. 13  illustrates simulated performance of a radar system using the same array and angular separation between targets, but with an 8× phase amplifier. 
         FIG. 14  illustrates an embodiment using digital signal processing, such as may be used for Sonar or Ultrasonic Imaging, or adapted to RF-frequency radar And applications to LIDAR. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
       FIG. 1  illustrates an imaging or target identification system  100  in use. Targets, such as targets  102 ,  104 , are illuminated with coherent waves of a known wavelength λ by a transmitter  106 , waves reflected by targets  102 ,  104  are received by a receiver  108 . In the event that there is more than one target  102 ,  104 , waves from the targets  110 ,  112 , arrive at the receiver  108  from directions separated by an angle θ  114 . 
     Phase Difference Corresponds to Angle from Perpendicular 
     Arriving waves from the two targets  102 ,  104  in  FIGS. 1 and 2  strike receiver  108  at two or more places  116 ,  118 , separated by a distance D that corresponds to the aperture. In ultrasonic, radio frequency, and microwave applications a sensor element  119 , such as a piezoelectric transducer or a phased-array antenna element, may be located at each of places  116 ,  118  of the receiver. The path  120  length from a target  102  to one of these places  116  on receiver  108  is equal to the path length from target  102  to a different place  118  on receiver  108  only if the target  102  is located at a point perpendicular to the midpoint of a line between the places  116 ,  118  on the receiver. As an angle  122  from the perpendicular to the path  120  increases, a phase difference between the signals received by the receiver  108  at the first place  116  and second place  118  from that target increases. Each target  102 ,  104  will produce signals at the sensors  119  at first  116  and second  118  places at the receiver  108  that differ in phase by different amounts for each target. The angular resolution of a system is equivalent to the ability to distinguish between arriving signals having differing phase-differences at separated points  116 ,  118  on receiver  108 . 
     The devices we propose exploits the fact that the coherent detection on the focal plane converts a problem in spatial or angular resolution of a target to one of resolution in phase, and the fact that faster phase variation implies higher resolution. Our approach does this by adding to the classical sensor described above a quantum phase amplifier (QPA). 
     Suppose we could increase the phase differences of the incident plane wave by a scale factor g prior to detection: this would have the effect of increasing the fringe spatial frequency across the array. Then equation (1) would become: 
     
       
         
           
             
               
                 
                   
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     immediately leading to the angular resolution (analogously to equation (3)), 
     
       
         
           
             
               
                 
                   
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     The QPA does not increase the operating frequency, but introduce a phase shift in the incident field proportional to its local phase as compared to a reference phase φref, i.e., Δφ=(g−1)(φ−φ ref ). 
     We can picture the effect of the QPA by referring to  FIG. 5 . A plane wave is incident on the planar surface of the phase amplifier, its phase varying linearly across the surface according to the angle between the perpendicular and path from the target from which the waves are arriving. Referring to point A in the figure, suppose the local phase is equal to the reference phase, φ=φ ref =0 (dashed line phase front). At point B, a distance d to the right, the local relative phase is larger, φ=kdθ (dotted line phase front). Below the phase amplifier, the phase at A is unchanged (φ=gφ=0), but the phase at B experiences a shift, advancing to φ=gkdθ. To find the direction of the transmitted wave, we form the line of constant phase φ=gkdθ, indicated by the dot-dashed line. We see that the wavefront of the transmitted wave is tilted away from the normal direction. The net result is as if the wave has entered a medium of smaller refractive index, of magnitude 1/g, but without inducing a shift in wavelength. 
     Effect of Phase Amplification 
     The effect of the phase amplifier in the coherent imaging system is depicted in  FIG. 1 , where we illustrate phase amplification&#39;s increasing the rate of change of phase at the detector. The magnification of the incident angle increases the apparent position of the off-axis target  102 , producing an apparent target image  124  separated by a greater angle from the nearly on-axis target  104 . Thus, we can summarize by stating that we achieve resolution enhancement by magnification of the angular separation of targets. 
     To visualize the phase amplification process, we can look at its action on a coherent state in the phase plane whose coordinates are the real and imaginary parts of the electric field ( FIG. 6 ). The initial coherent state can be depicted as a circle, which represents the uncertainty area (quantum noise) of the complex field amplitude. The squared magnitude of the field amplitude, A 2 , is equal to the mean photon number in the state. Under phase amplification, the mean photon number is diminished, while the phase, which is canonically conjugate to the photon number, is increased by the same factor. The final state is nonclassical, having been squeezed in amplitude and antisqueezed in phase; the uncertainty area is now elongated in the phase direction. 
     Note that both the phase and the phase noise have been amplified. However, phase amplification may preserve or improve the overall SNR, as mentioned above. 
     In order to build the phase amplifier for the frequency of interest, we must figure out how to generate the squeeze state realized by this frequency. 
     Active Approach 
     An active phase amplifier or QPA  600  is illustrated in  FIG. 7 . 
     A signal at QPA  600  input  602  is representable as cos(ωt+kdθ). 
     The input signal is applied to a first frequency doubler  604  that operates by mixing the input with itself, with output taken through a filter as the upper harmonic, giving cos(2ωt+2 kdθ). A source  606  of a reference signal having frequency co, the fundamental frequency of the signal arriving from the targets, is provided. The signal from the first frequency doubler  604  is mixed with the reference  606  signal at the second mixer  608 , and take the lower harmonic is selected by a filter. The filtered signal at the second mixer  608  output is cos(ωt+2kdθ). Phase differences from the reference to the input signal are now doubled. 
     The filtered signal at the second mixer  608  output is applied to a second frequency doubler  610  that operates by mixing the input with itself, with output taken through a filter as the upper harmonic, giving cos(2χt+4kdθ). We then mix this signal again with reference  606  signal at the fourth mixer  612 , and take the lower harmonic, we have the signal cos(ωt+4kdθ). We now have the 4× gain phase gain desired in this particular embodiment. Every two mixers complete one phase doubling operation, we call this one multiply. If there are M multiples we have the gain of 2 M . Necessary amplifiers and filters have been omitted from  FIG. 7  for simplicity. Although this embodiment features phase multiples in the form 2 M , one of ordinary skill in the art, after reading and comprehending the present disclosure, will understand that the present invention is not limited to only this form. Other indirect methods are available to estimate the phase multiples, as well as other terms. 
       FIG. 8  illustrates a phased-array echolocation or imaging radar system  700  embodying the phase amplifier of  FIG. 7 . The system has an array of N, N at least two and chosen for cost and good resolution, sensor elements, or diplexer-antenna elements,  702  in an array. For simplicity only three of the N sensor elements  702  are shown in  FIG. 8 . Transmitter circuitry  704  is provided as known in the art. In the embodiment of  FIG. 8 , the same antennas are used for transmitting illumination as for receiving echoes, so duplexing circuitry is incorporated in diplexer-antenna elements  702  to prevent receiver burnout. 
     In the embodiment of  FIG. 8 , coherent radiation is transmitted in pulses, once a transmit pulse is ended the diplexer circuitry permits sensor elements  702  to receive any echoes from the targets. In an alternative embodiment, chirp-modulated pulses alternate with constant-frequency pulses, echoes from the chirp-modulated pulses being processed as known in the art for high resolution in range, echoes from the constant-frequency pulses being processed as described herein for high angular resolution. 
     A local reference source  706  is coupled to at least one sensor element  702 . In order to prevent Doppler effects from affecting the QPA  708 , this reference source  706  may be a local oscillator phase-locked to the echo as received by one predetermined sensor element  702  of sensor elements  702 , or alternatively to a signal derived from an average of several sensor elements. In another embodiment, the reference signal source  706  buffers echo received by one predetermined sensor element  702 . In other embodiments, such as those where targets are stationary, the reference source may be tapped from the transmitter  704 . The output of reference source  706  is applied as a common reference to the reference  606  ( FIG. 7 ) of each QPA  708 . 
     Each sensor element  702  feeds one of identical QPAs  708  with phase gain g. The input signals at each of the QPAs  708  are effectively 1, e −j(ωt+kdθ) , e −j(ωt+2kdθ) , . . . , e −j(ωt+Nkdθ) . The N outputs of the QPAs  708  are 1, e −j(ωt+kdgθ) , e −j(ωt+kdgθ) , . . . , e −j(ωt+Nkdgθ) . 
     The QPAs therefore operate as phase-difference amplifiers, amplifying a phase shift between reference  706  and the signals received through sensor elements  702 . 
     Outputs from QPAs  708  feed a resolver and/or imager  710 . Resolver and/or imager  710  uses conventional beam forming techniques or parameter estimating algorithms such as MUSIC to resolve any targets  712 , or form images of any targets  712 , that may be present. Resolver and/or imager  710  provide information to a display system  716  as known in the art. Resolver and/or imager  710  may act to resolve separate targets directly, or may act to form a narrow beam that may then be scanned by other apparatus to identify the targets. 
       FIG. 8  can be viewed as illustrating a pulsed active sonar system by replacing diplexer-antenna elements as sensor elements  702  with piezoelectric transducers and transmit-receive switching circuitry as sensor elements  702 , and adjusting operating frequencies appropriately. 
     In an alternative embodiment, as illustrated in  FIG. 9 , the source of coherent acoustic or electromagnetic illumination may be separated from the receiving array. A system of this type may use either continuous standing-wave illumination or pulsed illumination. In this embodiment, a transmitter  804  feeds a transmit antenna or transducer  802  to emit coherent waves towards any target or targets that may be present. Signals reflected from the target or targets are received by receive sensor elements  805 . The remainder of blocks in  FIG. 9  greatly resemble equivalent blocks in  FIG. 8  and will not be separately described herein. 
     In an alternative embodiment of the phase amplifier as a degenerate squeeze state generator is illustrated in  FIG. 10 . This embodiment uses an approximation of the action of a phase amplifier in field quadrature phase space as illustrated in  FIG. 11 . 
     It is desirable that only one field quadrature will be amplified, while the other will be deamplified. We see that for small angles θ˜X 2 /X 1 , the degenerate squeeze state generator provides gain to the phase and deamplifies the amplitude, i.e., it behaves like a quantum phase amplifier. 
     In the embodiment of phase amplifier  900  ( FIG. 10 ), an IF signal, such as may be derived from an antenna-diplexer-downconverter element  702  ( FIG. 8 ), e j(wt−kdθ)  inputs to the squeeze state generator through a frequency divider  902  first. The signal is e j(ωt/kd−θ)  at the frequency divider  902  output. In this balanced configuration, this divider output passes through splitter  904  into two equal amplitude and two equal phase signals. The reference signal  906  is combined with the signal at two mixers  908 ,  910  with a 90° phase difference between them, here induced by phase shifter  912 . Two outputs from the mixers  908 ,  912  at baseband represented the real part and the imaginary part of the incoming signal which associate with the X 1  and X 2  quadrature in  FIG. 11 , respectively. The imaginary part signal passes through a amplifier  918  by providing gain to the X 2  quadrature, while the real part signal path cascades a deamplifier  920  (attenuator) which squeezes the X 1  quadrature. These signals may then be used by resolver and imager  710  to form an effective beam and/or further processing to derive an image. 
     The alternate embodiment of  FIG. 10  could be realized in both analog domain and digital domain, which opens a wide door for this invention&#39;s validity in metrology (instrument, CCD), remote sensing (RADAR, microwave and RF), and imaging (Lithography, Ultrasound, CT, MRI, PET and nuclear scanning). Taking the advantage of the digital implementation will allow existing systems to be usable with only a small portion of software code added. 
     An alternate embodiment of the system  1300  is illustrated in  FIG. 14 . In this embodiment, transmitter circuitry  1302  generates a pulse of coherent acoustic or electromagnetic waves, these are transmitted to any targets that may be present  1304 ,  1306  through two or more duplexer-transducer elements  1308 . Received signals, such as reflections and echoes from targets  1304 ,  1306  enter through duplexer-transducer elements  1308 . These signals are then amplified, down converted by mixers if necessary, sampled, and digitized by multichannel amplifier, sampler, digitizer  1310 . Digital signals representative of signals received by each duplexer-transducer element  1308  are passed from digitizer  1310  to a digital signal processor  1312 . 
     Digital signal processor  1312  implements reference signal recovery  1314 , similar to the function of local reference  706  previously described with reference to  FIG. 8 . The recovered reference from reference recovery  1314  and signals representative of signals received by each duplexer-transducer element  1308  are passed to digital phase amplifier  1316 , which implements a sampled-data equivalent of the phase amplifier circuitry of  FIG. 7  or  FIG. 10 . Once phase-amplified, resolver and imager  1318 , uses conventional or MUSIC methods to identify the targets and resolve images, which are then passed to a display  1320 . 
     A first embodiment of the system of  FIG. 14  is a sonar system for mapping the ocean bottom and for identifying submerged objects. A second embodiment is an ultrasonic imaging device for imaging internal organs of patients. A third embodiment is an over-the-horizon radar system. 
       FIG. 12  illustrates simulated performance of normal radar using a 10 GHz, 64-element phased array with 0.003 radian separation between targets. The separation between elements is d=λ/2. One signal impacts the array normally, while another incidents from 0.003 radians. The resolution is far below the classic Rayleigh Limit, as shown by the angular signal strength distribution. Depicted are two incident waves, and an overall signal. The sensor is unable to distinguish two signals from the overall signal. 
       FIG. 13  illustrates simulated performance of a radar using the same array and angular separation between targets, but incorporating an 8× phase amplifier. The resulting overall signal clearly has a bimodal distribution, indicating presence of two targets instead of one target. It is clear the sensor is able to distinguish two incident signals. The QPA concept presented therefore promises to achieve resolution beyond classic Rayleigh Limit and possibly the Cramer-Rao Bound. 
     Passive Approach 
     In this embodiment, a lens with a refractive index less than 1 but greater than 0, such as may be constructed of an artificial material such as a metamaterial, is added as a covering or coating on the sensor array. With such a material, Refraction angle is away from the normal of the antenna array by the nature of the lens material, and effective phase amplifying is achieved as the incident wavefront arrives at the sensor array behind the lens. 
     A material with a refractive index less than unity is referred to as a phase-advance material since the phase change per unit length for a wave traveling in such a material is less than that if the wave was traveling in free-space. This implementation generally requires such a phase-advance material for microwave or optical lens application. 
     Metamaterials having microwave refractive index less than one have been demonstrated under laboratory conditions. Metamaterials are typically static assemblies of a particular geometry and material that can be tuned to provide desired properties. In optics and electromechanical applications, such as with RF and microwave signals, for example, lenses and gratings are typically constructed of homogenous materials having particular shapes. As utilized in the embodiments disclosed herein, metamaterials depart from this conventional approach in that they can be non-homogenous constructed devices that exhibit passive behavior normally associated with regular materials. In some applications, the metamaterials act like a band-pass filter, except according to the present embodiments, phase can be filtered, and not just frequency. By filtering phase components, significantly greater measurement resolution can be realized with respect to time, angle, and other measured components. 
     Whereas the active approach, described above, can be particularly advantageous for use with digital processing, RADAR, and ultrasound applications, this passive approach is seen by the present inventors to have significant advantages where light applications, such as LIDAR, are also present. One advantage of this passive approach is that it is capable of bypassing stringent requirements seen when dealing with “non-classical” light situations. This passive approach further allows for a more general implementation for various types of signals, including at least those described above. 
     Heisenberg Scaling 
     The phase amplifier achieves Heisenberg resolution scaling, R˜1/Energy or R˜1/N for N received photons per unit time. One way is simply to consider equation (6), which shows R˜1/g. The maximum g value is just given by the mean photon (or phonon) number N, although phase noise limits this gain to a somewhat lower value. This implies R˜1/N. In particle sense, the energy is carried by the particles; therefore, the energy is proportional to the particle number. Consequently, the resolution is proportional to 1/Energy, which is the Heisenberg scaling. 
     Suppose we wish to resolve two coherent-state plane waves whose propagation directions differ by an angle φ. This means that the photon states have mean phase values equal to, say 0 and φ, the variances of which scale as  δφ 2   ∝1/N for N mean photons in the mode. The incident beams have angular Gaussian distribution, whose bandwidths are σ φ . Since the angular separation between two beans is φ, we define the resolution proportional to the ratio of the angular beam width over the angular separation. These phase distributions are distinguishable if R˜σ φ /φ˜(φ√{square root over (N)}) −1  is small enough. After phase amplification, φ→gφ and σ φ →√{square root over (g/N)}, so after post-amplification the resolution is R˜φ −1 (gN) −1/2 . But, again, the maximum g value is just given by the mean photon number N; this implies R˜1/N. Since the photon number has been squeezed g times, the energy; therefore, is reduced g times as well. If we want to improve the Signal-to-Noise-Ratio (SNR), at a fixed number of post-amplification detected photons, we need to increase the transmitted power by a factor of g to achieve a g-fold improvement in resolution. This situation is completely analogous to the classical Heisenberg-like resolution attainable by increasing both power and frequency, except we do not need to propagate shorter wavelength photons to the target. 
     Scaling to Increased Resolution 
     Gain g is a parameter in the QPA sensor, and the resolution enhancement is a factor of g. As described in the previous section, since the PA deamplifies the photon number by the same scale factor g, the maximum allowable gain is given simply by the mean photon number received from the target for fixed SNR, and the maximum gain is the ratio of the pre- to post-amplification photon number. Therefore, the theoretical resolution improvement scales directly with the power transmitted to the target. Practically, one will be limited by the feasibility of attaining high gain amplification. In addition, the effect of phase noise due to atmospheric turbulence must also be considered, since it too will increase with gain (as it would for propagating shorter wavelength). 
     While the forgoing has been particularly shown and described with reference to particular embodiments thereof, it will be understood by those skilled in the art that various other changes in the form and details may be made without departing from the spirit and hereof. It is to be understood that various changes may be made in adapting the description to different embodiments without departing from the broader concepts disclosed herein and comprehended by the claims that follow.