Module 3 - Summary

Summary Notes Module 3

Amplitude distribution of a Gaussian beam

\[ u(x,y,z) = A_0 \frac{w_0}{w(z)} \exp\left( - j k z \right) \exp\left( - \frac{x^2+y^2}{w^2(z)} \right) \exp\left( - j \frac{k (x^2 + y^2)}{2R(z)} \right) \exp\left( j \varphi(z) \right) \]

\( \exp\left( - j k z \right) \) represents a plane wave

\( \exp\left( - \frac{x^2 + y^2}{w^2(z)} \right) \) represents a Gaussian modulation

\( \exp\left( - j \frac{k (x^2 + y^2)}{2R(z)} \right) \) represents a paraboloidal wavefront

\( \exp\left( j \varphi(z) \right) \) represents a phase gained through propagation

Intensity distribution of a Gaussian beam

The irradiance/intensity distribution of a Gaussian beam follows a Gaussian function.

\( i(x, y, z) = |u(x, y, z)|^2 = \frac{A_0^2}{w_0^2} \frac{w^2(z)}{w_0^2} \exp\left( -2 \frac{x^2 + y^2}{w^2(z)} \right) \)

86% of the intensity is located within the beam waist \( w_0 \).

Beam Parameters

The beam width is given by: \[ w(z) = w_0 \sqrt{1 + \left( \frac{z}{z_0} \right)^2} \]

The beam waist is: \[ w_0 = \sqrt{\frac{\lambda z_0}{\pi}} \]

The Rayleigh range/distance is: \[ z_0 = \frac{\pi w_0^2}{\lambda} \]

The beam curvature is: \[ R(z) = z \left[ 1 + \left( \frac{z_0}{z} \right)^2 \right] \]

The phase retardance is: \[ \varphi(z) = \tan^{-1} \left( \frac{z}{z_0} \right) \]

Relation between intensity distributions at two axial planes

At \( z = 0 \) (waist plane), the beam width coincides with the beam waist, \( w(z = 0) = w_0 \). The intensity at \( z = 0 \) is: \[ i_0(x, y) = i(x, y, z = 0) = |u(x, y, z = 0)|^2 = \frac{A_0^2}{w_0^2} \exp \left( -2 \frac{x^2 + y^2}{w_0^2} \right) \]

The intensity at any axial plane \( z \) is: \[ i(x, y, z) = \left| u(x, y, z) \right|^2 = \frac{A_0^2}{w_0^2} \frac{w^2(z)}{w_0^2} \exp \left( -2 \frac{x^2 + y^2}{w^2(z)} \right) = \frac{1}{M^2} i_0(x, y) \]

where \( M = \frac{w(z)}{w_0} \)

Power of Gaussian Beams. The power is given by: \[ P = \frac{1}{2} i_0 \left( \pi w_0^2 \right) \] which is independent of \( z \).

Energy Conservation Law for Gaussian Beams

\[ \int_0^\infty i(\rho, z) 2\pi \rho d\rho = \int_0^\infty i_0(\rho) 2\pi \rho d\rho \] \[ \int_0^\infty i(\rho, z) 2\pi \rho d\rho = \int_0^\infty \frac{1}{M^2} i_0(x, y) 2\pi \rho d\rho = \int_0^\infty i_0(x', y') 2\pi \rho' d\rho' \]

Where we have defined a new variable \( x' = \frac{x}{M} \) and \( y' = \frac{y}{M} \).

Features of Gaussian Beams

• At any transverse plane (z): \( w(z) > w_0 \)
• If \( z = \pm z_0 \), \( w(z = \pm z_0) = \sqrt{2} w_0 \), which means that at the area of the spot beam \( (\pi w^2(z)) \) it is double \( (2 \pi w_0^2) \) at \( z = \pm z_0 \)
• Depth of focus: \( 2z_0 \), defined as the axial range in which the beam width is “almost” constant
• If \( z = 0 \) (i.e., beam waist plane), \( R(z = 0) = \infty \), so \( \exp\left( - j \frac{k (x^2 + y^2)}{2R(z)} \right) = 1 \). In other words, there is no spherical wavefront term, i.e., the wavefront is a plane wave at the beam waist.
• If \( z = \pm z_0 \) (i.e., Rayleigh range), the radius of curvature is minimum: \( R(z = \pm z_0) = \pm 2z_0 \)
• If \( z \gg z_0 \), the beam width is linearly proportional to the distance \( z \): \[ w(z) = w_0 \frac{z}{z_0} \]

• Beam divergence \( \theta_0 = \frac{w_0}{z_0} = \frac{\lambda}{\pi w_0} \
• If \( z \gg z_0 \), \( R(z) = z \), which means the Gaussian beam acts as a spherical wave confined within the divergence angle \( \theta_0 \).

q-parameter: \( q(z) = z + j z_0 = \frac{1}{R(z)} - j \frac{\lambda}{\pi w^2(z)} \) where \( z \) is the distance of the Gaussian beam to its beam waist, and \( z_0 \) is its Rayleigh range.

Propagation of Gaussian Beams through ABCD Matrix Optics: A Gaussian beam propagating through optical components remains a Gaussian beam. However, its features change on the ABCD matrix of the optical system.

\[ q_2 = z_2 + j z_{02}, \quad q_2 = \frac{C q_1 + D}{A q_1 + B} \] where \( q_2 \) defines a Gaussian beam at \( z_2 \) and Rayleigh range \( z_0^2 \).