Laser sources play a critical role in numerical simulations across various fields, including photonics, material processing, biomedical optics, and optical communication. In Finite Element Analysis (FEA) and Finite-Difference Time-Domain (FDTD) simulations, different types of laser sources are modeled to study their interaction with matter, wave propagation, and nonlinear effects. The choice of laser source depends on factors such as wavelength, temporal characteristics, coherence properties, and beam profile. This article provides an in-depth discussion of various laser sources used in numerical simulations and their mathematical representations.
1. Continuous Wave (CW) Laser Sources
Continuous wave (CW) lasers emit a constant optical power over time, making them ideal for steady-state simulations in waveguide optics, interferometry, and thermal studies. The electric field of a CW laser source is given by:
$$
E(t) = E_0 e^{i(\omega t + \phi)},
$$
where:
- $E_0$ is the amplitude,
- $\omega = 2\pi f$ is the angular frequency,
- $\phi$ is the phase.
In FDTD simulations, a CW laser is often represented as a time-harmonic source with a fixed wavelength. CW sources are widely used in photonic crystal simulations to study band structures and resonance effects.
2. Pulsed Laser Sources
Pulsed lasers deliver energy in short bursts, characterized by pulse duration ($\tau_p$) and peak power. They are commonly used in ultrafast optics, nonlinear photonics, and laser ablation studies. The electric field of a Gaussian-shaped pulsed laser is modeled as:
$$
E(t) = E_0 e^{-\left(\frac{t}{\tau_p}\right)^2} e^{i(\omega t + \phi)},
$$
where $\tau_p$ is the pulse width.
Pulsed sources enable the study of nonlinear effects such as self-focusing, two-photon absorption, and supercontinuum generation in simulations.
2.1 Femtosecond and Picosecond Pulses
Ultrashort laser pulses, typically in the femtosecond ($10^{-15}$ s) and picosecond ($10^{-12}$ s) ranges, exhibit high peak intensities and short temporal coherence. The Fourier transform of a femtosecond pulse reveals its broad spectral bandwidth:
$$
\Delta \omega \approx \frac{0.44}{\tau_p}.
$$
FEM-based simulations involving Kerr nonlinearities, Raman scattering, and mode-locking often require ultrashort pulse modeling.
3. Modulated Laser Sources
Modulated laser sources are essential in optical communication and microwave photonics simulations. These sources are modulated in amplitude, frequency, or phase to encode information. A common modulation format is amplitude modulation (AM), expressed as:
$$
E(t) = E_0 [1 + m \cos(\omega_m t)] e^{i\omega t},
$$
where:
- $m$ is the modulation index,
- $\omega_m$ is the modulation frequency.
Phase-modulated lasers are used in coherent communication systems, where the phase of the electric field varies as:
$$
E(t) = E_0 e^{i[\omega t + \beta \sin(\omega_m t)]}.
$$
4. Supercontinuum Sources
Supercontinuum generation occurs when a high-intensity pulsed laser propagates through a nonlinear medium, producing a broadband optical spectrum. The evolution of a supercontinuum spectrum is governed by the generalized nonlinear Schrödinger equation (GNLSE):
$$
\frac{\partial A}{\partial z} + \frac{\alpha}{2} A - \sum_{n \geq 2} \frac{i\beta_n}{n!} \frac{\partial^n A}{\partial t^n} = i\gamma |A|^2 A.
$$
Here:
- $A$ is the slowly varying envelope of the pulse,
- $\beta_n$ are the dispersion coefficients,
- $\gamma$ is the nonlinear coefficient.
Supercontinuum sources are used in simulations of optical coherence tomography (OCT) and spectroscopy.
5. Gaussian Beam Sources
Gaussian beam sources are widely used in FEA and FDTD simulations to model laser beams with diffraction-limited profiles. The spatial distribution of a Gaussian beam is given by:
$$
E(r, z) = E_0 \frac{w_0}{w(z)} e^{- \left( \frac{r^2}{w(z)^2} \right)} e^{i(kz - \eta(z))}.
$$
where:
- $w(z) = w_0 \sqrt{1 + \left(\frac{z}{z_R}\right)^2}$ is the beam waist variation,
- $z_R = \frac{\pi w_0^2}{\lambda}$ is the Rayleigh range.
Gaussian beam sources are essential for simulating free-space optical propagation, fiber coupling, and focusing properties.
6. Plane Wave Sources
Plane wave sources are used for simulating wave propagation in periodic structures such as photonic crystals and metamaterials. A plane wave is mathematically represented as:
$$
E(\mathbf{r}, t) = E_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}.
$$
Plane wave sources are particularly useful in computing photonic band structures using Bloch’s theorem.
7. Random and Noise Sources
In some simulations, stochastic laser sources are required to model the effects of quantum noise, spontaneous emission, or turbulent wavefronts. A laser field with additive white Gaussian noise can be expressed as:
$$
E(t) = E_0 e^{i(\omega t + \phi)} + n(t),
$$
where $n(t)$ is a random noise function. These sources are critical in simulating laser speckle, optical coherence, and phase noise effects.
Conclusion
Laser sources in numerical simulations vary widely depending on their temporal, spectral, and spatial characteristics. CW lasers are used for steady-state optical analysis, pulsed lasers enable ultrafast phenomena modeling, modulated sources facilitate communication system simulations, and supercontinuum lasers expand spectral bandwidth for broad applications. FEA and FDTD simulations leverage these different laser models to study photonic devices, wave propagation, and nonlinear interactions. Choosing the appropriate laser source is crucial for achieving accurate and realistic simulation results.
Setting up laser pulse in any numerical simulation can be tricky and challenging. If you need any help, then you can connect with me.
Discussions? let's talk here
Check out YouTube channel, published research
you can contact us (bkacademy.in@gmail.com)
Interested to Learn Engineering modelling Check our Courses 🙂
--
All trademarks and brand names mentioned are the property of their respective owners.
