We derive an exact expression for the continuous-wave signal-to-noise ratio of filtered incoherent light, which applies to arbitrary optical filtering and arbitrary receiver electrical response. We demonstrate a simple method for making accurate measurements of continuous-wave signal-to-noise ratio, and compare these measurements with values computed from independent measurements of optical-filter coherence time and receiver electrical response; our measurements and computed values agree to within ±0.2 dB.
We derive an exact expression for the continuous-wave signal-to-noise ratio of filtered incoherent light, which applies to arbitrary optical filtering and arbitrary receiver electrical response. We demonstrate a simple method for making accurate measurements of continuous-wave signal-to-noise ratio, and compare these measurements with values computed from independent measurements of optical-filter coherence time and receiver electrical response; our measurements and computed values agree to within ±0.2 dB.
The signal-to-noise ratio (SNR) of filtered incoherent light can be approximated from the product of the coherence time of the light and the equivalent (electrical) noise bandwidth of the detector. This approximation holds only for the light with very short coherence time, that is in the case where the optical bandwidth of the light is much larger than the electrical bandwidth. We present here an expression for accurate evaluation of the SNR of the filtered incoherent light, which computes SNR from arbitrary shapes of optical and electrical filter power spectral densities (PSD). The PSDs of the filters can be measured using optical and electrical spectrum analyzers. Using our expression, we show that the SNR reaches unity when the electrical filter bandwidth is becoming larger than the optical filter bandwidth. To prove the theory, we evaluate and directly measure SNR of an incoherent light source filtered with several optical filters with bandwidths larger and commensurate with the bandwidth of the detector. For later we used optical and electrical filters with 3-dB bandwidths of 15 GHz and 10 GHz, respectively. Using our expression to evaluate SNR we obtained results in a good agreement with directly measured SNR. The results also prove that the approximation for evaluating SNR does not provide accurate results. The PSD of the detector with large noise bandwidth is difficult to measure using spectrum analyzer. There- fore, we report here a method for measuring the electrical noise bandwidth of the detector using the heterodyne linewidth measurement technique with tunable laser.
Random fluctuations in the phase and the amplitude of a source field induce random fluctuations in the field intensity at the network output, which in turn give rise to electrical noise in the detector current. The characteristics of this source-induced noise depend on the properties of the source and on the network transfer function. In the present study we consider a stationary optical source with a finite coherence time coupled to an arbitrary, linear, time-independent network. General expressions for the power-spectral density of the source-induced noise are derived in terms of the network transfer function and the second- and the fourth-order correlation functions of the source field in the frequency domain. The incoherent limit and the case of a source that has two uncorrelated polarization modes are considered in detail, and a simple method for the estimation of the source-induced noise is suggested.
Time-frequency (TF) filtering of analog signals has played a crucial role in the development of radio-frequency communications and is currently being recognized as an essential capability for communications, both classical and quantum, in the optical frequency domain. How best to design optical time-frequency (TF) filters to pass a targeted temporal mode (TM), and to reject background (noise) photons in the TF detection window? The solution for 'coherent' TF filtering is known-the quantum pulse gate-whereas the conventional, more common method is implemented by a sequence of incoherent spectral filtering and temporal gating operations. To compare these two methods, we derive a general formalism for two-stage incoherent time-frequency filtering, finding expressions for signal pulse transmission efficiency, and for the ability to discriminate TMs, which allows the blocking of unwanted background light. We derive the tradeoff between efficiency and TM discrimination ability, and find a remarkably concise relation between these two quantities and the time-bandwidth product of the combined filters. We apply the formalism to two examples-rectangular filters or Gaussian filters-both of which have known orthogonal-function decompositions. The formalism can be applied to any state of light occupying the input temporal mode, e.g., 'classical' coherent-state signals or pulsed single-photon states of light. In contrast to the radio-frequency domain, where coherent detection is standard and one can use coherent matched filtering to reject noise, in the optical domain direct detection is optimal in a number of scenarios where the signal flux is extremely small. Our analysis shows how the insertion loss and SNR change when one uses incoherent optical filters to reject background noise, followed by direct detection, e.g. photon counting. We point out implications in classical and quantum optical communications. As an example, we study quantum key distribution, wherein strong rejection of background noise is necessary to maintain a high quality of entanglement, while high signal transmission is needed to ensure a useful key generation rate.
The recently demonstrated concept of a quantum pulse gate enables coherent filtering of an individual well-defined optical mode from a noisy signal. This contrasts with the standard technique of spectral filtering followed by temporal gating, which exhibits an inherent trade-off between the efficiency and the noise rejection level. Here we compare the performance of the two techniques in application to quantum key distribution using entangled photon pairs. It shown that coherent filtering may lead to a nearly 10 dB enhancement of the key rate.
Imaging is the estimation of the spatial distribution of a physical object by measuring the optical radiation it emits, or by making use of an optical wave that interacts with the object, via reflection or transmission, for example, before being measured by a detector. The resolution of an imaging system is limited by the inability to localize the optical field at points of the object. Under otherwise ideal conditions, resolution is limited by diffraction. The sensitivity of an imaging system is limited by the uncertainty in the measurement. Under ideal conditions, this is determined by photon noise, which depends on the statistical fluctuations of the light. In conventional imaging systems, an extended detector, such as a CCD camera or an array detector, measures the spatial distribution of the optical intensity, which is proportional to the photon flux density. In interferometric systems, the spatial distribution of the optical field is inferred from measurements of the optical intensity. With the emergence of coherence theory, imaging systems based on measurements of the second-order coherence function at pairs of points in the detection plane were developed. An example is the imaging of an incoherent object based on the van Cittert-Zernike theorem. Imaging systems based on measurement of intensity correlation, or the photon coincidence rate, at pairs of points, were developed in the 1960s. A classic example of the photon-correlation imaging of an object emitting thermal light is stellar imaging using a Hanbury-Brown-Twiss intensity-correlation interferometer. More recently, two-photon light, which may be generated via spontaneous parametric downconversion in a second-order nonlinear optical crystal, has been used for imaging. This type of two-photon (or biphoton) imaging, which has come to be called quantum imaging, is also based on the measurement of photon coincidence by the use of photon-counting array detectors or by scanning two photon-counting detectors at pairs of points. To compare the resolution and sensitivity of imaging systems based on the aforementioned types of measurements, it is necessary to derive expressions for the measured quantities in terms of the object distribution. The point-spread functions based on such expressions can be used to assess the resolution. One measure of the sensitivity of the imaging process is the signal-to-noise ratio (SNR) of the measured variables. The statistical nature of the light source must be known in order to determine the SNR. The purpose of this chapter is to compare the resolution and sensitivity of photon-correlation imaging systems that make use of thermal light and two-photon light. We will henceforth refer to these two imaging modalities as classical and quantum photon-correlation imaging, or simply classical and quantum imaging, respectively. Clearly, light in other quantum states can also be used for imaging.
We introduce and numerically demonstrate a simple and general concept for direct reconstruction of the temporal phase profile of an optical signal from temporal intensity measurements at the input and output of an arbitrary linear optical filter. The concept is based on exploiting the linearity and causality properties of any physical filter. Very few restrictions need to be imposed on the optical filter response to ensure an unambiguous phase reconstruction. The filter can be specifically designed to minimize the noise influence on the measurement process.