The Laser Formulas: Mathematics Behind the Beam

Lasers are one of the most versatile technologies in the world today, used in everything from medical treatments to entertainment. Yet, beneath their captivating beams and cutting-edge applications lies a deep well of mathematics that explains how lasers function. In this blog, we will explore some of the key formulas and mathematical concepts behind lasers, giving a clearer understanding of the science behind the beam.

What is a Laser?

Before diving into the math, it’s important to first understand what a laser is. Laser ليزر stands for Light Amplification by Stimulated Emission of Radiation. It works by emitting light that is coherent, meaning the light waves are uniform in wavelength, direction, and phase. Unlike conventional light sources such as bulbs, which emit light in all directions and at varying wavelengths, lasers focus light in a narrow beam of a single wavelength.

Lasers rely on the stimulated emission of photons, where excited electrons within atoms or molecules return to their lower energy states, releasing energy in the form of photons. These photons stimulate the emission of additional photons, resulting in light amplification.

Understanding how this process works mathematically can provide valuable insights into the operation of lasers. Below are some key mathematical formulas that help describe laser physics.

1. The Einstein Coefficients and the Rate of Stimulated Emission

One of the most fundamental equations in laser physics is the relationship between the stimulated emission rate and the Einstein coefficients. These coefficients describe the probability of an atom or molecule transitioning between different energy states due to the interaction with light.

Einstein Coefficients Formula:

$dN2dt=−B21⋅N2⋅ρ(ν)\frac{dN_2}{dt} = -B_{21} \cdot N_2 \cdot ho(u)$

Where:

$N_2$ is the number of particles in an excited state.
$B_{21}$ is the Einstein coefficient for stimulated emission.
$ρ (ν)$ is the spectral energy density of the incident radiation at frequency $ν$ .

The equation represents the rate at which electrons in the excited state ( $N_2$ ) transition to the lower energy state due to the stimulated emission caused by the incident radiation. This rate is crucial in determining how much energy can be amplified in the lasing medium, which will eventually result in the laser output.

2. Population Inversion and the Threshold Condition

For a laser to work, population inversion must be achieved, meaning there are more particles in the excited state than in the lower energy state. This is the key to the amplification of light, as stimulated emission dominates over absorption.

The population inversion condition can be mathematically expressed as:

$N_2 > N_1$

Where:

$N_2$ is the number of particles in the excited state.
$N_1$ is the number of particles in the lower energy state.

The threshold condition is a critical concept in laser operation. It refers to the minimum amount of energy required to achieve population inversion and start the laser process. If the energy input is insufficient, spontaneous emission will prevail, and the system will not lase.

3. The Rate of Photon Emission and the Output Power

Once population inversion is achieved, the next step is the amplification of light. The rate of photon emission is directly related to the power output of the laser. The output power of a laser is the result of the stimulated emission of photons, which are released as the excited atoms transition to their lower energy states.

The power output $P$ of a laser can be determined by the following formula:

$\cdot u \cdot \frac{dN}{dt}$

Where:

$P$ is the output power of the laser.
$h$ is Planck’s constant ( $J\cdotps6.626 \times 10^{-34} \, \text{J·s}$ ).
$ν$ is the frequency of the emitted light.
$dNdt\frac{dN}{dt}$ is the rate of stimulated emission (how quickly the population of excited atoms is decreasing due to photon emission).

This formula shows that the output power of a laser is directly proportional to the frequency of the emitted light and the rate of photon emission. The higher the frequency and the faster the rate of emission, the more powerful the laser will be.

4. The Heisenberg Uncertainty Principle and Laser Spectral Width

The Heisenberg Uncertainty Principle plays a significant role in determining the spectral width of a laser. This principle states that the product of the uncertainty in energy ( $ΔE\Delta E$ ) and the uncertainty in time ( $Δt\Delta t$ ) must be greater than or equal to a constant value:

$ΔE⋅Δt≥ℏ2\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$

Where:

$ΔE\Delta E$ is the uncertainty in energy.
$Δt\Delta t$ is the uncertainty in time.
$ℏ\hbar$ is the reduced Planck's constant.

This principle implies that there is a fundamental limit to how narrow the frequency (and thus the spectral linewidth) of a laser can be. A narrow linewidth means the laser is very monochromatic, emitting light at a very specific wavelength. However, this comes at the cost of uncertainty in the energy of the photons, which is related to the coherence time of the laser.

5. Laser Beam Divergence and the Diffraction Limit

The divergence of a laser beam refers to how much the beam spreads out as it travels away from its source. A perfect laser beam would maintain its narrow focus indefinitely, but in practice, the beam will diverge due to the diffraction limit. The angle of divergence $θ\theta$ of a laser beam can be expressed as:

$θ≈λπ⋅w0\theta \approx \frac{\lambda}{\pi \cdot w_0}$

Where:

$λ\lambda$ is the wavelength of the laser light.
$w_0$ is the radius of the laser beam at its narrowest point (the beam waist).

This formula shows that the divergence of the laser beam is inversely proportional to the beam's waist size and directly proportional to the wavelength. Shorter wavelengths and smaller beam waists result in a more focused laser beam, which is highly desirable for applications like cutting, medical treatments, and communication systems.

6. The Rate of Spontaneous Emission and its Role in Lasers

While stimulated emission is the dominant process in lasers, spontaneous emission still plays an important role, especially at low population inversion. The rate of spontaneous emission can be described using the following formula:

$dN1dt=A21⋅N2\frac{dN_1}{dt} = A_{21} \cdot N_2$

Where:

$A_{21}$ is the Einstein coefficient for spontaneous emission.
$N_2$ is the number of particles in the excited state.

This formula shows that spontaneous emission decreases as population inversion increases, which is a key condition for laser operation. However, at very low population inversion, spontaneous emission may dominate and prevent the laser from operating effectively.

Conclusion: The Beauty of Laser Mathematics

The mathematics behind lasers is an elegant and powerful tool for understanding how these remarkable devices work. From the fundamentals of stimulated emission and population inversion to the complexities of beam divergence and spectral width, laser science combines physics, engineering, and mathematics to create one of the most important technological advancements of our time.