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Photon Flux and Bunching Noise from Measurement of the Shot Noise Variance
Abstract
We report the experimental observation of photon-bunching noise through shot noise measurements made on a pseudo-thermal state of light using balanced detection. A full theory describing the measurement is developed, and, in agreement with the theory, it is found that the shot noise variance in the balanced signal reproduces the time series of the flux of the primary incoherent beam. Moreover, when the average power of the pseudo-thermal light is varied, the balanced detection is seen to track this change. A comparison of direct detection and balanced detection of the thermal field shows that the balanced detection performs at least as well as the direct detection and under some conditions appears to outperform the direct detection. There is not necessarily a contradiction with quantum field theory, which predicts that at best the performance of the balanced detection should be equal to the direct detection because the direct detection process is subject to nonlinearity that has not been excluded by measurements (even though any tests we performed suggest that such effects are small). This is the first time that the bunching noise effect of high occupation number chaotic light via the shot noise of the field has successfully been measured, to the point of using it to infer the flux of the field. The findings may be relevant to radio receiver design, specifically from the viewpoint of sensitivity improvement. © 2018. The American Astronomical Society. All rights reserved..
... The mean value of the added electronic noise was calibrated to zero to reduce the effect of electronic noise. [30][31][32][33][34] Note that the electronic noise level was adjusted to result in an order of O(10 0 ) with respect to the Poisson noise level for both breast volumes. Charge collection coefficient (i.e., conversion rate of optical photons to e-h pairs by the photodiode) was set to 1 to simplify the quantification of the electronic noise order. ...
Background
In our previous study, we proposed a convolutional neural network (CNN)‐based model observer for signal known statistically (SKS) and background known statistically (BKS) tasks to assess the detection performance of breast tomosynthesis systems by varying acquisition angles and the number of projections at a constant dose level. Despite demonstrating the significant potential of the CNN‐based model observer in approximating the ideal observer (IO) performance, further research is required to extend its applicability to clinically relevant tasks and validate its robustness across diverse imaging scenarios.
Purpose
Exploring the generalizability of the CNN‐based model observer is essential for advancing its practical utility in diagnostic imaging. In this work, we explored the generalizability of a CNN‐based model observer for SKS and BKS detection tasks in breast tomosynthesis images with two different volume glandular fractions (i.e., VGFs: 30% and 50%), and two different sizes of spiculated signals (i.e., 1 and 2 mm). These efforts aim to provide deeper insights into the factors that influence network optimization for consistent and robust detection performance.
Methods
Five different network architectures were used to verify whether optimizing the match between the receptive field (RF) size and signal size would enhance the detection performance; the networks were designed in terms of theoretical receptive field (TRF) size. The detection performance of the CNN‐based model observer was compared to that of the Hotelling observer (HO) under various training and testing schemes to observe the key factors in optimizing the network to enhance its generalizability.
Results
Throughout the study, we demonstrated that each network focuses more on discriminating the presence of similarly sized signals over achieving robustness to noise variations during training. The CNN‐based model observer showed better detection performance compared to the HO, except when the trained and tested datasets incorporated differently sized signals. Networks trained on datasets involving signals of both sizes resulted in better generalizability compared to those trained on mixed‐VGF datasets (i.e., datasets comprising both VGFs). Contrary to our assumption, the match between the TRF size and signal size did not improve the detection performance. This led to exploring the effective receptive field (ERF) size of the network as a descriptive metric of network generalizability, using pixelwise gradient activation mapping (pGrad‐CAM). We showed that a relationship between the ERF size and signal size exists, thus presenting its clear relevance to the detection performance of the CNN‐based model observer.
Conclusions
Networks trained on datasets sharing similarly sized signals exhibited optimal detection performance, but showed limited generalizability when applied to datasets with signals of different sizes, while those trained on datasets involving signals of both sizes resulted in better generalizability. This work suggests that task‐based exploration is crucial for designing CNN‐based model observers that can perform and generalize consistently, providing valuable guidance for developing robust networks for varying imaging scenarios.
... To reduce the effect of electronic noise, the mean value of the added electronic noise was calibrated to zero. [44][45][46][47][48] The Hanning-weighted ramp filter was then applied to the projection data. 49 The filtered projection data were voxel-driven backprojected with linear interpolation to reconstruct a breast tomosynthesis volume of size 64×64×16 (i.e., 6.4×6.4×6.4 mm 3 ).Note that the reconstructed voxel size was 0.1×0.1×0.4 mm 3 in our work, thus resulting in a slice thickness of 0.4 mm. ...
Background
Since human observer studies are resource‐intensive, mathematical model observers are frequently used to assess task‐based image quality. The most common implementation of these model observers assume that the signal information is exactly known. However, these tasks cannot thoroughly represent situations where the signal information is not exactly known in terms of size and shape.
Purpose
Considering the limitations of the tasks for which signal information is exactly known, we proposed a convolutional neural network (CNN)‐based model observer for signal known statistically (SKS) and background known statistically (BKS) detection tasks in breast tomosynthesis images.
Methods
A wide parameter search was conducted from six different acquisition angles (i.e., 10°, 20°, 30°, 40°, 50°, and 60°) within the same dose level (i.e., 2.3 mGy) under two separate acquisition schemes: (1) constant total number of projections, and (2) constant angular separation between projections. Two different types of signals: spherical (i.e., SKE tasks) and spiculated (i.e., SKS tasks) were used. The detection performance of the CNN‐based model observer was compared with that of the Hotelling observer (HO) instead of the IO. Pixel‐wise gradient‐weighted class activation mapping (pGrad‐CAM) map was extracted from each reconstructed tomosynthesis image to provide an intuitive understanding of the trained CNN‐based model observer.
Results
The CNN‐based model observer achieved a higher detection performance compared to that of the HO for all tasks. Moreover, the improvement in its detection performance was greater for SKS tasks compared to that for SKE tasks. These results demonstrated that the addition of nonlinearity improved the detection performance owing to the variation of the background and signal. Interestingly, the pGrad‐CAM results effectively localized the class‐specific discriminative region, further supporting the quantitative evaluation results of the CNN‐based model observer. In addition, we verified that the CNN‐based model observer required fewer images to achieve the detection performance of the HO.
Conclusions
In this work, we proposed a CNN‐based model observer for SKS and BKS detection tasks in breast tomosynthesis images. Throughout the study, we demonstrated that the detection performance of the proposed CNN‐based model observer was superior to that of the HO.
A formula for shot-noise is derived in the frequency-domain. The derivation is complete and reasonably rigorous while being appropriate for undergraduate students; it models a sequence of random pulses using Fourier sine and cosine series, and requires some basic statistical concepts. The text here may serve as a pedagogic introduction to the spectral analysis of random processes and may prove useful to introduce students to the logic behind stochastic problems. The concepts of noise power spectral density and equivalent noise bandwidth are introduced.
Using the quantum Cramér–Rao bound from quantum estimation theory, we derive a fundamental quantum limit on the sensitivity of a temperature measurement of a thermal astronomical source. This limit is expressed in terms of the source temperature Ts, input spectral bandwidth , and measurement duration T, subject to a long measurement time assumption . It is valid for any measurement procedure that yields an unbiased estimate of the source temperature. The limit agrees with the sensitivity of direct detection or photon counting, and also with that of the ideal radiometer in the regime for which the Rayleigh–Jeans approximation is valid, where is the center frequency at which the radiometer operates. While valid across the electromagnetic spectrum, the limit is especially relevant for radio astronomy in this regime, since it implies that no ingenious design or technological improvement can beat an ideal radiometer for temperature measurement. In this connection, our result refutes the recent claim of a radio astronomy technique with much-improved sensitivity over the radiometer.
Lieu et al. (2015) have recently claimed that it is possible to substantially
improve the sensitivity of radio astronomical observations. In essence, their
proposal is to make use of the intensity of the photon shot noise as a measure
of the photon arrival rate. Lieu et al. (2015) provide a detailed
quantum-mechanical calculation of a proposed measurement scheme that uses two
detectors and conclude that this scheme avoids the sensitivity degradation that
is associated with photon bunching. If correct, this result could have a
profound impact on radio astronomy. Here I present a detailed analysis of the
sensitivity attainable using shot-noise measurement schemes that use either one
or two detectors, and demonstrate that neither scheme can avoid the photon
bunching penalty. I perform both semiclassical and fully quantum calculations
of the sensitivity, obtaining consistent results, and provide a formal proof of
the equivalence of these two approaches. These direct calculations are
furthermore shown to be consistent with an indirect argument based on a
correlation method that establishes an independent limit to the sensitivity of
shot-noise measurement schemes. Collectively, these results conclusively
demonstrate that the photon bunching sensitivity penalty applies to shot noise
measurement schemes just as it does to ordinary photon counting, in
contradiction to the fundamental claim made by Lieu et al. (2015). The source
of this contradiction is traced to a logical fallacy in their argument.
Although an electromagnetic pulse traveling through a dispersive medium gradually spreads out, it is shown that for a stationary thermal field all multipoint correlation functions of any order remain unchanged as the field propagates, provided that the refractive index is real.
A quasithermal, quasmonochromatic lamp is described which serves as a highly degenerate light source with adjustable coherence time between 10-5 sec and 1 sec. This lamp is used for several demonstration experiments concerning the relations between coherence and fluctuations: The intensity interferometer of Hanbury Brown and Twiss is applied to measure the correlations between intensity fluctuations. The double slit experiment of Young serves to stress the role of fluctuations for classical interferometry. Interference patterns from two independent quasithermal lamps are presented.
The shot noise of vacuum states is a kind of quantum noise and is totally random. In this paper a nondeterministic random number generation scheme based on measuring the shot noise of vacuum states is presented and experimentally demonstrated. We use a homodyne detector to measure the shot noise of vacuum states. Considering that the frequency bandwidth of our detector is limited, we derive the optimal sampling rate so that sampling points have the least correlation with each other. We also choose a method to extract random numbers from sampling values, and prove that the influence of classical noise can be avoided with this method so that the detector does not have to be shot-noise limited. The random numbers generated with this scheme have passed ent and diehard tests.
The advent of stable, highly squeezed states of light has generated great
interest in the gravitational wave community as a means for improving the
quantumnoise- limited performance of advanced interferometric detectors. To
confidently measure these squeezed states, it is first necessary to measure the
shot-noise across the frequency band of interest. Technical noise, such as
non-stationary events, beam pointing, and parasitic interference, can corrupt
shot-noise measurements at low Fourier frequencies, below tens of kilo-Hertz.
In this paper we present a qualitative investigation into all of the relevant
noise sources and the methods by which they can be identified and mitigated in
order to achieve quantum noise limited balanced homodyne detection. Using these
techniques, flat shot-noise down to Fourier frequencies below 0.5 Hz is
produced. This enables the direct observation of large magnitudes of squeezing
across the entire audio-band, of particular interest for ground-based
interferometric gravitational wave detectors. 11.6 dB of shot-noise suppression
is directly observed, with more than 10 dB down to 10 Hz.
We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.
We present a technique for measuring the second-order coherence function g(2)(tau) of light using a Hanbury Brown-Twiss intensity interferometer modified for homodyne detection. The experiment was performed entirely in the continuous-variable regime at the sideband frequency of a bright carrier field. We used the setup to characterize g(2)(tau) for thermal and coherent states and investigated its immunity to optical loss. We measured g(2)(tau) of a displaced-squeezed state and found a best antibunching statistic of g(2)(0)=0.11+/-0.18.
It is shown by a quantum-mechanical treatment that the emission times of photoelectrons at different points illuminated by a plane wave of light are partially correlated, and identical results are obtained by a classical theory in which the photocathode is regarded as a squarelaw detector of suitable conversion efficiency. It is argued that the phenomenon exemplifies the wave rather than the particle aspect of light and that it may most easily be interpreted as a correlation between the intensity fluctuations at different points on a wavefront which arise because of interference between different frequency components of the light. From the point of view of the corpuscular picture the interpretation is much less straight-forward but it is shown that the correlation is directly related to the so-called bunching of photons which arises because light quanta are mutually indistinguishable and obey Bose-Einstein statistics. However, it is stressed that the use of the photon concept before the light energy is actually detected is highly misleading since, in an interference experiment, the electromagnetic field behaves in a manner which cannot be explained in terms of classical particles. The quantitative predictions of the theory have been confirmed by laboratory experiments and the phenomenon has been used, in an interferometer, to measure the apparent angular diameter of Sirius: these results, together with further applications to astronomy, will be discussed in detail in later papers. It is shown that the classical and quantum treatments give identical results when applied to find the fluctuations in the photoemission current produced by a single light beam, and the connexion between these fluctuations and the correlation between photons in coherent beams is pointed out. The results given here are in full agreement with those obtained by Kahn from an analysis based on quantum statistics: however, they differ from those derived on thermodynamical grounds by Fellgett and by Clark Jones and the reasons for this discrepancy are discussed.
Displaced Fock states of the electromagnetic field have been synthesized by overlapping the pulsed optical single-photon Fock state |1〉 with coherent states on a high-reflection beam splitter and completely characterized by means of quantum homodyne tomography. The reconstruction reveals nonclassical properties of displaced Fock states, such as negativity of the Wigner function and photon number oscillations. To our knowledge, this is the first time complete tomographic reconstruction has been performed on a highly nonclassical optical state.
- R Lieu
- T W B Kibble
Lieu, R., & Kibble, T. W. B. 2015, in Proc. of the 26th Symp. on Space
Terahertz Technology (Boston, MA: Harvard Univ. Press), Paper W2-3
(arXiv:1509.01868)