**Table of Contents:**

**FORMULATION**

Because of the paraxial/Fresnel approximation, a Gaussian beam maintains its profile as it propagates in space, with the only parameter parameter changing being its wavefront radius R(z). We know from paraxial geometrical optics that the effect of the optical elements on the wavefront radius can be described by the ABCD law of ray transfer matrices. We can therefore use this formulation to describe how the wavefront radius can be modified by optical elements.

Recall that the wavefront curvature R(z) and the radius $\omega(z)$ are not independent but are connected by the q-parameter as:

\begin{equation} \frac{1}{q(z)} = \frac{1}{R(z)}-\frac{i\lambda}{\pi\omega^2(z)} \end{equation}

or

\begin{equation} q(z) = z + iz_R \end{equation}

Case 1: For propagation in free space over a distance d, then clearly if $q(z_1)=q_1$, then $q(z_2)=q_2=q_1+d$. We can see this because the effect of the propagation in the distance term would just be to add a distance d. This can be written as:

\begin{equation} q_2 = \frac{Aq_1 + B}{Cq_1 + D} \end{equation}

where

\begin{equation} \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} \end{equation}

Case 2: With propagation through a thin lens, the lens will transform the wavefront radius R without changing the beam radius $\omega$ since the propagation distance is negligible. Since then $\omega_2 = \omega_1$, the q-parameter change is:

\begin{equation} \frac{1}{q_2} – \frac{1}{q_1} = \frac{1}{R_2} – \frac{1}{R_1} \end{equation}

We know from geometrical optics that the thin lens changes R as:

\begin{equation} R_2 = \frac{AR_1+B}{CR_1+D} \end{equation}

where

\begin{equation} \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} r & 0 \\ -1/f & 1 \end{pmatrix} \end{equation}

Conclusion: Looking at the above two cases, we can see that with a given change of the ABCD matrix, we can have the relation for the complex radius of curvature q:

\begin{equation} q_2 = \frac{Aq_1 +B}{Cq_1+D} \end{equation}

which is known as the ABCD law. This ABCD Law relates the complex radius of curvature at the front and back reference planes of a system through a ray matrix. We can interpret the ABCD law as stating that the complex q parameter transforms in the same way as R in geometrical optics. This turns out to be a generally true statement for paraxial system and not just the two presented above. Since Gaussian beams are paraxial waves with spherical wavefronts, they then transform in the same way as wavefronts defined as normals to paraxial rays.

**FOCUSING A GAUSSIAN BEAM**

We can now consider the case of a well-collimated Gaussian beam incident on a thin lens. This in practice means that the incident beam has a waist at the lens with the Rayleigh range $z_R >> f$. The q-parameter of the incident beam at the lens is then:

\begin{equation} \frac{1}{q_1} = \frac{1}{\infty}-i\frac{\lambda}{\pi\omega_i^2} \end{equation}

so then:

\begin{equation} q_1 = i\frac{\pi\omega_i^2}{\lambda} \end{equation}

where

\begin{equation} z_R_i = \frac{\pi\omega_z^2}{\lambda} \end{equation}

The ABCD matrix propagating the beam through the lens to the focus is then:

\begin{equation} \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -1/f & 1 \end{bmatrix} = \begin{bmatrix} 1 – \frac{d}{f} & d \\ -\frac{1}{f} & 1 \end{bmatrix} \end{equation}

so that the ABCD law with substitution of $q_i$ yields:

\begin{align} q_2 & = \frac{(1-d/f)(i\frac{\pi\omega_i^2}{\lambda})+d}{(-\frac{1}{f})(i\frac{\pi\omega_i^2}{\lambda})+1} \\ & = [\frac{1}{R_2}-i\frac{\lambda}{\pi\omega_2^2}]^{-1} \end{align}

where the second definition came from the definition of $q_2$. Equating the real and imaginary parts of both sides:

\begin{equation} R_2(d) = \frac{(\frac{d}{zR_i})^2+(1-\frac{d}{f})^2}{(\frac{d}{zR_i})^2+\frac{1}{f}(1-\frac{d}{f})} \end{equation}

\begin{equation} \omega_2^2(d) = \omega_i^2(1-\frac{d}{f})^2+\omega_i^2(\frac{d}{zR_i})^2\end{equation}

The waist of the focused beam occurs when $R_2(d) = \infty$, so:

\begin{equation} \frac{d}{z_R_i^2}-\frac{1}{f}(1-\frac{d}{f}) = 0 \rightarrow d(\frac{1}{z_R_i^2}+\frac{1}{f^2}) = \frac{1}{f} \end{equation}

so the focused beam occurs at distance d:

\begin{equation} d = \frac{f}{1+\frac{f^2}{z_R_i^2}} \end{equation}

Note here that when the incident beam is well-collimated, i.e. its beam diameter is large so that $z_{R_i} >> f$, then we get the simple geometrical optics result for the ray picture that d = f. In the case of a slightly diverging beam, the actual focal length is then a little bit shorter than the focal length.

We can then find the focused beam waist by substituting in the expression for d into the equation for $\omega_2^2(d)$:

\begin{equation} \omega_f = \frac{\lambda{f}}{\pi\omega_i}\frac{1}{\sqrt{1+f^2/z_R^2}} \end{equation}

In the typical case where $z_R_i >> f$:

\begin{equation} \omega_f = \frac{\lambda{f}}{\pi\omega_i} \end{equation}

Then, if we define the beam diameter parameters $D_i = 2\omega_i$, $D_f = 2\omega_f$, then we have:

\begin{equation} D_f = \frac{4}{\pi}\lambda(\frac{f}{D_i}) \end{equation}

We previously defined the f-number of a lens as the focal length divided by the diameter of the aperture stop.