Photocurrent and shot noise
In this blog post, I discuss the intricacies of measuring photocurrents in photodetectors and the accompanying shot noise, especially under non-ideal conditions. Photodetectors are crucial in numerous applications, ranging from scientific instrumentation to commercial optical devices. Understanding the nuances of how these devices operate at a quantum level offers invaluable insights, particularly for graduate students and professionals in quantum physics and mathematics.
Photocurrent in ideal and non-ideal photodetectors
The fundamental operation of a photodetector involves converting light into an electrical signal. Ideally, each photon incident on the detector generates an electron, leading to a photocurrent. This ideal behavior is described by the equation:
\mathbf{i}_T = q_e \frac{\mathbf{N}(T)}{T}
where \mathbf{i}_T represents the photon current over a time T, q_e is the electron charge, and \mathbf{N}(T) is the number of photons detected in time T. By averaging this over time, the current no longer depends on T, simplifying to:
\langle \mathbf{i}_T \rangle = q_e \frac{\langle \mathbf{N}(T) \rangle}{T} = q_e \frac{\Phi}{\hbar \omega_\lambda}
\Phi denotes the photon flux, and \hbar \omega_\lambda is the energy per photon at wavelength \lambda.
Accounting for quantum efficiency
In reality, detectors are not 100% efficient. Quantum efficiency \eta, which may be less than one, reflects the detector’s effectiveness in converting photons into charge carriers. This efficiency modifies the current as follows:
\langle \mathbf{i}_T \rangle = \eta q_e \frac{\Phi}{\hbar \omega_\lambda}
Even with non-ideal efficiency, the relationship between the input light and the measured current remains linear, scaled by \eta.
Shot noise and its implications
Shot noise represents the quantum uncertainty inherent in the photocurrent due to the discrete nature of charge and photon arrival statistics. It’s crucial for understanding the limits of detector sensitivity and noise characteristics. The standard deviation of the current, indicative of shot noise, is given by:
\Delta \mathbf{i} = q_e \sqrt{\frac{\Phi}{\hbar \omega_\lambda T}}
This relationship shows that shot noise increases with the square root of the measurement bandwidth.
Frequency considerations
Considering a typical electrical circuit’s filtering characteristics, related to the measurement bandwidth \Delta f, shot noise can be described in the frequency domain as:
\Delta \mathbf{i} = \sqrt{2 q_e \langle \mathbf{i} \rangle \Delta f}
This formula, essential for designing and interpreting experimental setups, highlights how the noise scales with both the average current and the bandwidth.
Practical insights and power spectral density
The power spectral density (PSD) of the shot noise is a crucial concept, providing a measure of noise power per unit frequency. For shot noise, PSD is constant across frequencies, characteristic of white noise:
S_i(f) = 2 q_e \langle \mathbf{i} \rangle
This constant PSD implies that the total noise power between any two frequencies is simply the product of the PSD and the frequency interval, underscoring the pervasive nature of shot noise across all frequencies.
Conclusion
The study of photocurrent and shot noise in photodetectors bridges quantum physics and practical electronics, offering deep insights into both fundamental physics and engineering applications. Understanding these phenomena allows for the design of better instruments and the improvement of experimental techniques in both academic and industrial settings.
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