Photon flux and beam propagation
In this article, I explore the transformation from the classical description of a freely propagating beam to its quantum formalism counterparts, examining intrinsic quantities like electric field amplitude, irradiance, and the implications of quantum mechanics on measuring beam properties.
Classical description
A freely propagating beam with a transverse area S is traditionally described as a plain wave. The electric field in any section of the beam can be expressed as:
\mathbf{E}(\mathbf{r}, t) = i\mathbf{e}_\lambda E_\lambda(0) \left(e^{i(\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)} + e^{-(i\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)} \right)
This formula shows the oscillatory nature of the electric field as it propagates through space and time.
Irradiance, representing the power per unit surface, can be determined as:
\Pi = 2\varepsilon_0 c |E_\lambda(0)|^2
From here, it’s straightforward to compute the density of energy per unit volume and the total power flowing across any section of the beam, yielding:
\begin{aligned} & D = \frac{\Pi}{c} \\ & \Phi = \Pi S \end{aligned}
Quantum perspective
To translate these classical concepts into quantum mechanics, it’s essential to define a quantization volume. This volume, relevant for photon calculations, can be visualized as a portion of the beam over an arbitrary length L = cT, making:
V_\lambda = S c T
From this, I can discuss the quantum flux of photons, linking classical and quantum descriptions:
\Phi_{\text{photon}} = \frac{\Phi}{\hbar \omega_\lambda}
Bridging classical and quantum descriptions
The amplitude of the electric field in the quantum state can be calculated by modifying the classical formula to incorporate quantum state parameters:
\langle \alpha_\lambda | \mathbf{E}(\mathbf{r}, t) | \alpha_\lambda \rangle = i\mathbf{e}_\lambda \sqrt{\frac{\Phi}{2\varepsilon_0 S c}} \left(e^{i(\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)} + e^{-(i\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)} \right)
This expression is crucial as it demonstrates how the classical field amplitude correlates with the photon flux in a quantized description.
Measurement implications and shot noise
The concept of shot noise, which arises from quantum fluctuations, is significant when measuring beam properties with a photodetector. The photon current observed can be expressed by:
i_T = q_e \frac{N(T)}{T}
Analyzing the fluctuations in this current provides insight into the limits of measurement accuracy, crucial for precision experiments.
Conclusion
By examining both classical and quantum descriptions, I aim to provide a thorough understanding of the dynamics of freely propagating beams and their measurement in contemporary physics research. The transition from a classical to a quantum perspective not only broadens our understanding but also enhances our ability to measure and manipulate these beams with greater precision in experimental setups.
For more insights into this topic, you can find the details here.