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Physics and simulation of optoelectronic devices. Conference

Signal-to-noise ratio of arbitrarily filtered spontaneous emission

Marko Bosiljevac, Marko Šprem, D.I. Babic

Published Feb 23, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1117/12.2289090

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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.
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.
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 formalism for computing the signal-to-noise ratio (SNR) for laser radar is reviewed and applied to the tasks of target detection, direction finding, and phase-change estimation with squeezed light. The SNR for heterodyne detection of coherent light using a squeezed local oscillator is lower than that obtained using a coherent local oscillator. This is true for target detection, for phase estimation, and for direction finding with a split detector. Squeezing the local oscillator also lowers SNR in balanced homodyne and heterodyne detection of coherent light. Loss places an upper bound on the improvement that squeezing can bring to direct-detection SNR.
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.
A phase-sensitive (PS) heterodyne detector is intrinsically resistant to classical noises and useful in measurement of low-frequency signals below the shot noise. Despite the existence of image band vacuum, we show that the quantum-noise power level of this heterodyne detector sensing a coherent signal is exactly one light quantum per measurement time, i.e., twice the vacuum fluctuation power, which can be further reduced by use of squeezed light. We then report on an experiment on a PS heterodyne detector with a 10 Hz 1.0×10−18 W optical signal (1064 nm wavelength) at its input. The noise floor of the unmodulated coherent light is 2.2(±0.1)×10−19 W/Hz from 2 Hz to 20 Hz, and the signal-to-noise ratio is about 6.6 dB for the measured signal when the resolution bandwidth is 1 Hz. The quantum noise floor is reduced by 1.6(±0.3)dB when squeezed light is used, and the sub-shot-noise power spectral density is 1.6(±0.1)×10−19 W/Hz between 2 Hz and 20 Hz. This work should be an important advance towards squeezing-improved precision measurements of low-frequency signals with heterodyne detectors, including audio-band gravitational-wave detection.
We consider the problem of estimating the spatial separation between two mutually incoherent point light sources using the super-resolution imaging technique based on spatial mode demultiplexing (SPADE) with noisy detectors. We show that in the presence of noise, the resolution of the measurement is limited by the signal-to-noise ratio (SNR) and the minimum resolvable spatial separation has a characteristic dependence of [Formula: see text]. Several detection techniques, including direct photon counting, as well as homodyne and heterodyne detection are considered.