Gaussian Beams from the Paraxial Wave Equation

Derivation

Gaussian Beams from the Paraxial Wave Equation

D-210 Home PU-206 Threads light · waves Depends on Optical Wave and Helmholtz Equations from Maxwell, Paraxial Ray-Transfer (ABCD) Matrices

Statement

Starting from the paraxial Helmholtz equation for the slowly varying envelope of a monochromatic scalar field, we derive the fundamental (TEM\(_{00}\)) Gaussian mode: its transverse profile, the complex beam parameter \(q(z)\), the waist \(w_0\), the Rayleigh range \(z_R\), the spot size \(w(z)\), the wavefront radius \(R(z)\), and the Gouy phase \(\psi(z)\). We then show that \(q\) transforms under free-space and thin-lens propagation exactly as a ray coordinate does through an ABCD matrix, giving the ABCD law \(q' = (Aq+B)/(Cq+D)\).

Why it matters

Every real laser beam is a finite-aperture bundle, not a plane wave and not a geometric ray. The fundamental Gaussian mode is the exact self-consistent solution of the paraxial wave equation that a stable spherical-mirror cavity selects, so it is the working description of laser light for beam delivery, focusing, coupling into fibres and cavities, and Gaussian optics design.

The complex beam parameter compresses the entire transverse structure — spot size and wavefront curvature — into a single complex number that propagates by the same ABCD matrices used for rays. This unifies ray optics (PU thread ray-transfer-abcd-matrices) and wave optics into one algebra, so a system already characterised by its ray matrix immediately tells you what it does to a Gaussian beam.

Assumptions

Monochromatic scalar field.If dropped, a single scalar \(U\) cannot capture polarization coupling or the frequency spread of a pulse; each frequency component would need its own envelope and the beam parameter would become frequency-dependent (spatio-temporal coupling).

Paraxial propagation: energy travels within a small angle of the \(z\)-axis, \(\theta \sim w_0/z_R \ll 1\).If dropped, the second axial derivative of the envelope is no longer negligible and the transverse profile is not exactly Gaussian; tightly focused beams need vector diffraction (Richards–Wolf), not this equation.

Slowly varying envelope: \(U=\psi(x,y,z)\,e^{-ikz}\) with \(|\partial_z^2\psi|\ll|2k\,\partial_z\psi|\) and \(|\partial_z^2\psi|\ll|\nabla_\perp^2\psi|\).If dropped, we cannot drop \(\partial_z^2\psi\) and reduce the Helmholtz equation to first order in \(z\); the problem stays a full elliptic boundary-value problem with no simple marching solution.

Homogeneous, isotropic, source-free linear medium with real uniform index (here vacuum, \(k=2\pi/\lambda\)).If dropped, \(k\) varies in space, gain or loss makes \(k\) complex, or nonlinearity couples intensity to phase (self-focusing); the linear separable Gaussian ansatz then fails.

Fundamental transverse mode only (no higher Hermite/Laguerre–Gauss content).If dropped, the field is a superposition of modes with different Gouy phases, and the simple single-\(q\) description must be replaced by a modal sum.

Derivation

\[ \nabla^2 U + k^2 U = 0, \qquad U(\mathbf r)=\psi(x,y,z)\,e^{-ikz}. \]

Start from the scalar Helmholtz equation (from scalar-wave-equation-from-maxwell with \(U\propto e^{i\omega t}\)) and factor out the fast plane-wave carrier along \(z\). A

\[ \nabla_\perp^2\psi + \frac{\partial^2\psi}{\partial z^2} - 2ik\frac{\partial\psi}{\partial z} - k^2\psi + k^2\psi = 0. \]

Insert the ansatz and expand \(\nabla^2\). Here \(\nabla_\perp^2=\partial_x^2+\partial_y^2\); the derivatives of \(e^{-ikz}\) generate the \(-2ik\,\partial_z\psi\) and \(-k^2\psi\) terms, the latter cancelling the \(+k^2\psi\). A

\[ \nabla_\perp^2\psi - 2ik\frac{\partial\psi}{\partial z} = 0. \]

Drop \(\partial_z^2\psi\) by the slowly varying envelope assumption (\(|\partial_z^2\psi|\ll|2k\,\partial_z\psi|\)). This is the paraxial Helmholtz (paraxial wave) equation — a Schrödinger-type equation with \(z\) as "time". B

\[ \psi(x,y,z)=A(z)\,\exp\!\left[-\frac{ik\,(x^2+y^2)}{2\,q(z)}\right], \qquad r^2=x^2+y^2. \]

Try a self-similar Gaussian/spherical ansatz: a complex amplitude \(A(z)\) times a quadratic-in-\(r\) phase controlled by one complex function \(q(z)\). A complex \(q\) simultaneously encodes wavefront curvature (real part of \(1/q\)) and beam width (imaginary part). C

\[ \nabla_\perp^2\psi=\left[-\frac{2ik}{q}+\left(-\frac{ik r}{q}\right)^2\right]\psi,\qquad \frac{\partial\psi}{\partial z}=\left[\frac{A'}{A}+\frac{ik r^2}{2q^2}q'\right]\psi. \]

Differentiate the ansatz. The transverse Laplacian in 2D gives \(-2ik/q\) from the two \(\partial^2/\partial x^2,\partial^2/\partial y^2\) "diagonal" terms plus the squared first-derivative term \(-(k r/q)^2\). C

\[ \left(-\frac{2ik}{q}-\frac{k^2 r^2}{q^2}\right) - 2ik\left(\frac{A'}{A}+\frac{ik r^2}{2q^2}q'\right)=0. \]

Substitute Step 5 into the paraxial equation (Step 3) and divide out the common \(\psi\ne 0\). B

\[ \underbrace{-\frac{k^2 r^2}{q^2}\big(1-q'\big)}_{\text{coeff. of }r^2}\;+\;\underbrace{\left(-\frac{2ik}{q}-2ik\frac{A'}{A}\right)}_{r\text{-independent}}=0. \]

Group by powers of \(r\). Because the equation must hold for all \(r\), each bracket vanishes separately (polynomial identity in \(r\)). C

\[ \frac{dq}{dz}=1 \quad\Longrightarrow\quad q(z)=q_0+z. \]

The \(r^2\) bracket gives \(q'=1\): the beam parameter simply increments by the propagation distance. This is the core propagation law of the beam parameter in free space. B

\[ \frac{A'}{A}=-\frac{1}{q}\quad\Longrightarrow\quad A(z)=A_0\,\frac{q_0}{q(z)}. \]

The \(r\)-independent bracket gives \(A'/A=-1/q=-1/(q_0+z)\); integrate to get amplitude scaling. The factor \(q_0/q\) both conserves power and carries the Gouy phase (Step 13). B

\[ \frac{1}{q(z)}=\frac{1}{R(z)}-\frac{i\,\lambda}{\pi\,w^2(z)} = \frac{1}{R(z)} - \frac{2i}{k\,w^2(z)}. \]

Definition. Split \(1/q\) into real and imaginary parts. Substituting back into \(\exp[-ikr^2/(2q)]\), the real part \(1/R\) gives a phase \(e^{-ikr^2/2R}\) (a spherical wavefront of radius \(R\)) and the imaginary part gives a real Gaussian envelope \(e^{-r^2/w^2}\). This defines \(R\) and \(w\). A

\[ \text{Choose waist at }z=0:\ R\to\infty,\ w=w_0 \;\Rightarrow\; \frac{1}{q_0}=-\frac{2i}{k w_0^2} \;\Rightarrow\; q_0=\frac{ik w_0^2}{2}\equiv i z_R,\quad z_R\equiv\frac{\pi w_0^2}{\lambda}. \]

At the waist the wavefront is flat (\(R\to\infty\)) and the spot is minimal (\(w=w_0\)), so \(q_0\) is purely imaginary. This defines the Rayleigh range \(z_R\). Hence \(q(z)=z+iz_R\). C

\[ \frac{1}{q}=\frac{1}{z+iz_R}=\frac{z}{z^2+z_R^2}-\frac{i\,z_R}{z^2+z_R^2}. \]

Rationalise \(q=z+iz_R\). Matching to Step 10 term by term reads off \(R(z)\) and \(w(z)\) below. A

\[ w^2(z)=w_0^2\!\left(1+\frac{z^2}{z_R^2}\right),\quad R(z)=z\!\left(1+\frac{z_R^2}{z^2}\right),\quad \frac{q_0}{q}=\frac{iz_R}{z+iz_R}=\frac{w_0}{w(z)}\,e^{\,i\psi(z)},\ \psi=\arctan\frac{z}{z_R}. \]

Read off \(w(z)\) from \(\mathrm{Im}(1/q)=-2/(kw^2)\) and \(R(z)\) from \(\mathrm{Re}(1/q)=1/R\). Writing the complex amplitude prefactor \(q_0/q\) in polar form extracts its magnitude \(w_0/w\) (power conservation) and its argument \(\psi(z)\), the Gouy phase. B

\[ U(r,z)=U_0\,\frac{w_0}{w(z)}\exp\!\left[-\frac{r^2}{w^2(z)}\right]\exp\!\left[-ik z-\frac{ik r^2}{2R(z)}+i\psi(z)\right]. \]

Assemble \(U=\psi e^{-ikz}\) from \(A(z)\) (Steps 9, 13) and the split of \(1/q\) (Step 10). This is the fundamental Gaussian mode. A

\[ \text{Free space } d:\ q_2=q_1+d,\qquad \text{Thin lens } f:\ \frac{1}{q_2}=\frac{1}{q_1}-\frac{1}{f}. \]

Free-space translation is Step 8. A thin lens adds a phase \(e^{+ikr^2/2f}\), i.e. shifts \(1/R\) by \(-1/f\), hence shifts \(1/q\) by \(-1/f\). Compare with the ray matrices \(\left(\begin{smallmatrix}1&d\\0&1\end{smallmatrix}\right)\) and \(\left(\begin{smallmatrix}1&0\\-1/f&1\end{smallmatrix}\right)\). C

\[ \boxed{\;q_2=\frac{A\,q_1+B}{C\,q_1+D}\;}\qquad\text{for any system with ray matrix } M=\begin{pmatrix}A&B\\C&D\end{pmatrix}. \]

Both elementary maps are Möbius (fractional-linear) transformations of \(q\), and the ABCD matrices compose by multiplication. Because composition of Möbius maps corresponds to matrix multiplication, the ABCD law holds for any cascade. This is the Kogelnik ABCD law. C

Result

\[ q(z)=z+i z_R,\quad \frac{1}{q}=\frac{1}{R}-\frac{i\lambda}{\pi w^2},\quad z_R=\frac{\pi w_0^2}{\lambda},\quad w(z)=w_0\sqrt{1+\tfrac{z^2}{z_R^2}},\quad q_2=\frac{Aq_1+B}{Cq_1+D}. \]

Reading. A single complex number \(q\) carries everything: its imaginary part fixes the beam width through \(w\), its real part fixes the wavefront curvature through \(R\). At the waist \(q=iz_R\) is purely imaginary (flat wavefront, minimum spot \(w_0\)). Moving a distance \(z\) just adds \(z\) to \(q\); a lens subtracts \(1/f\) from \(1/q\); any paraxial system acts on \(q\) through its ray ABCD matrix by the same fractional-linear map that would move a ray. The beam expands hyperbolically, has far-field half-angle \(\theta=\lambda/(\pi w_0)\), and accumulates an extra \(\pi\) of Gouy phase as it passes through focus.

Units check. \(z_R=\pi w_0^2/\lambda\): \([\text{m}^2]/[\text{m}]=[\text{m}]\), a length, as required for \(q=z+iz_R\). \(w(z)=w_0\sqrt{1+(z/z_R)^2}\): argument dimensionless, so \([w]=[\text{m}]\). \(1/q\) has units \([\text{m}^{-1}]\); \(1/R\) is \([\text{m}^{-1}]\) and \(\lambda/(\pi w^2)=[\text{m}]/[\text{m}^2]=[\text{m}^{-1}]\) match. In \(q_2=(Aq_1+B)/(Cq_1+D)\), \(A,D\) are dimensionless, \(B\) is a length, \(C\) is inverse length, so \(Aq+B\) is a length, \(Cq+D\) is dimensionless, and \(q_2\) is a length. Consistent.

Limiting cases

At the waist (\(z=0\)): \(w=w_0\), \(R\to\infty\) (flat wavefront), \(\psi=0\); the field is a pure real Gaussian times \(e^{-ikz}\).
Near field (\(|z|\ll z_R\)): \(w\approx w_0\), the beam is essentially collimated; \(R\approx z_R^2/z\to\infty\).
Far field (\(|z|\gg z_R\)): \(w\approx w_0 z/z_R=\theta z\) with divergence \(\theta=\lambda/(\pi w_0)\); \(R\approx z\) (spherical wave from the waist); \(\psi\to\pm\pi/2\).
Plane-wave limit (\(w_0\to\infty\), \(z_R\to\infty\)): \(q\to iz_R\to i\infty\), \(1/q\to 0\), envelope flat — recovers the uniform plane wave.
Ray limit: identify \(q=x/x'\) (position over slope); a real \(q\) is a geometric ray and the ABCD law reduces to the ray-transfer law from ray-transfer-abcd-matrices.
Total Gouy shift: from \(z=-\infty\) to \(+\infty\), \(\Delta\psi=\pi\) (a half-integer wave of extra phase through focus).

Breaks when

Tight focusing / high NA: when \(w_0\lesssim\lambda\) the divergence \(\theta=\lambda/(\pi w_0)\) is no longer small, the paraxial assumption (Step 3) fails, \(\partial_z^2\psi\) cannot be dropped, and the true focal field requires vector (Richards–Wolf) diffraction. The scalar Gaussian underestimates the spot and ignores longitudinal field components.
Nonlinear or inhomogeneous media: intensity-dependent index (Kerr self-focusing, filamentation) or a graded/aberrated index makes the medium non-uniform, so the separable single-\(q\) ansatz (Step 4) is invalid and \(q\) no longer obeys a linear ABCD law.
Hard apertures / truncation: clipping the beam on a finite aperture injects higher-order and diffraction-ring content; the field is no longer a single Gaussian mode and \(w\), \(R\) lose their simple meaning (Fresnel diffraction of the truncated field is needed).
Gain, loss, or strong dispersion: a complex or frequency-dependent \(k\) (amplifier, absorber, or short pulse) breaks the real-\(k\) monochromatic assumption; the beam parameter becomes complex-\(k\) or frequency dependent and pulses acquire spatio-temporal coupling.

Failure modes

Confusing \(1/e\) field radius with \(1/e^2\) intensity radius: \(w\) is where the field falls to \(1/e\) and the intensity to \(1/e^2\). Students halve or double \(w\) by mixing these; the intensity FWHM is \(w\sqrt{2\ln 2}\approx 1.18\,w\), not \(w\).
Sign / factor errors in \(1/q\): writing \(1/q=1/R+i\lambda/\pi w^2\) with the wrong sign flips the beam between converging and diverging, or between a physical (decaying) and unphysical (growing) Gaussian.
Adding \(w\) or \(R\) directly through a system instead of propagating \(q\): only \(q\) obeys the ABCD law. \(R\) and \(w\) combine nonlinearly, so you must go through \(q\).
Forgetting the Gouy phase: omitting \(\psi(z)\) misplaces resonance frequencies of a cavity and the on-axis phase; the extra \(\pi\) through focus is physically real (measured phase anomaly).
Using \(R(z)=z\) everywhere: that is only the far-field limit. Near the waist \(R\to\infty\), and \(R\) has a minimum \(2z_R\) at \(z=z_R\).
Treating the on-axis intensity as constant: peak intensity falls as \(w_0^2/w^2(z)\); the \(w_0/w\) amplitude factor is what conserves total power, not the peak.
Non-normalised ray matrices: the ABCD law assumes \(\det M=n_1/n_2=1\) in a single medium; using an un-normalised matrix corrupts \(q_2\).

Discussion

The paraxial wave equation \(\nabla_\perp^2\psi=2ik\,\partial_z\psi\) is formally the time-dependent Schrödinger equation of a free 2D particle, with \(z\leftrightarrow t\) and \(\hbar/2m\leftrightarrow 1/2k\). The Gaussian beam is then the exact analogue of a minimum-uncertainty coherent-state wavepacket: it stays Gaussian, spreads with distance, and its "waist" is where the transverse momentum and position uncertainties are minimally correlated. The Rayleigh range is the diffraction analogue of the wavepacket spreading time, and the hyperbolic \(w(z)\) is the ballistic spreading of that packet. This is why the same complex-\(q\) trick reappears throughout wave physics.

The complex beam parameter is powerful precisely because it linearises a nonlinear problem. Spot size and curvature evolve nonlinearly and couple to each other, but their specific combination \(1/q=1/R-i\lambda/\pi w^2\) evolves by a fractional-linear (Möbius) map, and Möbius maps compose exactly as \(2\times 2\) matrices multiply. That algebraic isomorphism is the whole content of the ABCD law: the ray matrix you already computed for geometric optics, unchanged, propagates the wave-optical \(q\). Cavity stability, mode matching into fibres, and telescope design all reduce to iterating or inverting that one map.

The Gouy phase \(\psi(z)=\arctan(z/z_R)\) is a genuinely wave phenomenon with no ray counterpart: a focused beam accumulates \(\pi\) less phase over its length than a plane wave would, an "anomalous" advance concentrated within a few \(z_R\) of focus. Physically it reflects the spread of transverse momenta: components with nonzero \(k_\perp\) travel with reduced axial wavenumber \(k_z=\sqrt{k^2-k_\perp^2}\approx k-k_\perp^2/2k\), and averaging over the mode's momentum distribution yields exactly the Gouy shift. It sets the longitudinal mode spacing in resonators and, generalised, underlies the topological phase of structured light.

The full mode family emerges by the same route: separating the paraxial equation in Cartesian coordinates yields Hermite–Gaussian modes \(H_m(\sqrt2 x/w)H_n(\sqrt2 y/w)\) and in cylindrical coordinates Laguerre–Gaussian modes carrying orbital angular momentum \(\ell\hbar\) per photon. All share the same \(q(z)\), \(w(z)\), \(R(z)\); only their Gouy phase generalises to \((m{+}n{+}1)\psi\) or \((2p{+}|\ell|{+}1)\psi\). Because different transverse orders acquire different Gouy phases, they are non-degenerate in a real cavity — the physical origin of transverse mode structure and of the ability to select TEM\(_{00}\) by aperturing. These sets are complete, so any paraxial field is a superposition with fixed \(q\), reducing free-space propagation to a phase bookkeeping over mode indices.

Common misconceptions. A Gaussian beam is not a ray bundle that focuses to a point — diffraction forbids a spot smaller than \(\sim\lambda\); the waist \(w_0\) and divergence \(\theta\) are reciprocally locked by \(w_0\theta=\lambda/\pi\), a diffraction "conservation law." Nor does the beam "come from" the waist as a point source except in the far field. And \(q\) is not merely a bookkeeping convenience: its imaginary part is a physical, invariant measure (the beam's \(M^2=1\) quality) that no lossless linear system can degrade.

Worked examples

\[ \textbf{Focusing a HeNe beam.}\quad \lambda=633\ \text{nm},\ w_0=0.5\ \text{mm}. \]

Find the Rayleigh range, the far-field divergence half-angle, and the spot size at \(z=2\) m. Symbols first, then numbers. A

\[ z_R=\frac{\pi w_0^2}{\lambda}=\frac{\pi(5\times10^{-4})^2}{6.33\times10^{-7}}=1.24\ \text{m}. \]

Direct evaluation. \((5\times10^{-4})^2=2.5\times10^{-7}\ \text{m}^2\); divide by \(\lambda\) and multiply by \(\pi\). A

\[ \theta=\frac{\lambda}{\pi w_0}=\frac{6.33\times10^{-7}}{\pi(5\times10^{-4})}=4.03\times10^{-4}\ \text{rad}\ (\approx 0.40\ \text{mrad}). \]

Far-field half-angle. Equivalently \(\theta=w_0/z_R\). A

\[ w(2\,\text{m})=w_0\sqrt{1+\left(\tfrac{z}{z_R}\right)^2}=0.5\ \text{mm}\sqrt{1+\left(\tfrac{2}{1.24}\right)^2}=0.5\sqrt{1+2.60}\ \text{mm}. \]

Apply \(w(z)\). \((2/1.24)^2=(1.613)^2=2.60\). A

\[ z_R=1.24\ \text{m},\quad \theta=0.40\ \text{mrad},\quad w(2\,\text{m})=0.95\ \text{mm}. \]

Reading. The beam stays near \(w_0\) for over a metre, then spreads: at 2 m it has nearly doubled. A tighter waist would diverge faster — the reciprocal lock \(w_0\theta=\lambda/\pi\) is unavoidable.

Units check. \(z_R\) in metres, \(\theta\) dimensionless (radians), \(w\) in millimetres. All consistent.

\[ \textbf{Lens focusing via the ABCD law.}\quad \text{Waist } w_0=1\ \text{mm at a lens } f=100\ \text{mm},\ \lambda=1064\ \text{nm}. \]

A collimated Nd:YAG beam has its waist at the lens. Find the new waist size and its location. Use \(q\) and the ABCD matrices. B

\[ z_R=\frac{\pi w_0^2}{\lambda}=\frac{\pi(10^{-3})^2}{1.064\times10^{-6}}=2.95\ \text{m},\qquad q_1=iz_R=i\,2.95\ \text{m}. \]

At the input waist \(q\) is purely imaginary. Because \(z_R\gg f\), the beam is essentially collimated at the lens. B

\[ \frac{1}{q_2}=\frac{1}{q_1}-\frac1f=\frac{1}{i\,2.95}-\frac{1}{0.1}=-\frac{i}{2.95}-10=-10-0.339i\ \ [\text{m}^{-1}]. \]

Lens map from Step 15. \(1/(i\,2.95)=-i/2.95=-0.339i\). B

\[ q_2=\frac{1}{-10-0.339i}=\frac{-10+0.339i}{10^2+0.339^2}=\frac{-10+0.339i}{100.115}=-0.0999+3.39\times10^{-3}i\ [\text{m}]. \]

Invert. \(q_2=z'+iz_R'\Rightarrow z'=-0.0999\) m (waist is \(0.0999\) m past the lens, since after propagating \(+0.0999\) m the real part reaches zero) and \(z_R'=3.39\times10^{-3}\) m. C

\[ z_R'=\frac{\pi w_0'^2}{\lambda}\Rightarrow w_0'=\sqrt{\frac{\lambda z_R'}{\pi}}=\sqrt{\frac{(1.064\times10^{-6})(3.39\times10^{-3})}{\pi}}=3.39\times10^{-5}\ \text{m}. \]

Invert the definition of \(z_R\) to get the focused waist. Cross-check: \(w_0'\approx f\lambda/(\pi w_0)=0.1(1.064\times10^{-6})/(\pi\cdot10^{-3})=3.39\times10^{-5}\) m. C

\[ w_0'\approx 34\ \mu\text{m},\qquad \text{focus at } z'\approx 100\ \text{mm past the lens} \ (\approx f). \]

Reading. A collimated beam focuses essentially one focal length behind the lens to a spot \(w_0'\approx f\lambda/\pi w_0\): a larger input beam or shorter \(f\) gives a tighter focus. The tiny \(z_R'=3.4\) mm shows the focused beam is very short — depth of focus \(2z_R'\approx 7\) mm.

Units check. \(1/q\) in \(\text{m}^{-1}\), \(q\) in m, \(w_0'\) in m. \(f\lambda/\pi w_0\): \([\text m][\text m]/[\text m]=[\text m]\). Consistent.

Problems

A beam has \(w_0=0.2\) mm at \(\lambda=800\) nm. Compute \(z_R\) and the spot size at \(z=0.5\) m.
Solution
\(z_R=\pi w_0^2/\lambda=\pi(2\times10^{-4})^2/(8\times10^{-7})=\pi(4\times10^{-8})/(8\times10^{-7})=0.157\) m. At \(z=0.5\) m: \(z/z_R=0.5/0.157=3.18\), so \(w=0.2\,\sqrt{1+3.18^2}=0.2\sqrt{1+10.1}=0.2\times3.33=0.67\) mm.
Show that the wavefront radius \(R(z)=z(1+z_R^2/z^2)\) is minimum at \(z=z_R\) and find \(R_\text{min}\).
Solution
\(R=z+z_R^2/z\). Then \(dR/dz=1-z_R^2/z^2=0\Rightarrow z=z_R\). Second derivative \(2z_R^2/z^3>0\), a minimum. \(R_\text{min}=z_R+z_R^2/z_R=2z_R\). So the most sharply curved wavefront occurs one Rayleigh range from the waist, with radius \(2z_R\).
A Gaussian beam with \(q_1=0.5i\) m (waist, \(z_R=0.5\) m) propagates 0.5 m in free space. Find \(w\) and \(R\) at the new plane. (\(\lambda=1\ \mu\)m.)
Solution
Free space: \(q_2=q_1+d=0.5i+0.5=0.5+0.5i\) m. Then \(1/q_2=(0.5-0.5i)/(0.25+0.25)=(0.5-0.5i)/0.5=1-1\,i\ \text{m}^{-1}\). So \(1/R=\mathrm{Re}=1\Rightarrow R=1\) m. \(\mathrm{Im}(1/q)=-\lambda/\pi w^2=-1\Rightarrow w^2=\lambda/\pi=10^{-6}/\pi=3.18\times10^{-7}\), \(w=5.64\times10^{-4}\) m \(=0.56\) mm. Check: at \(z=z_R\), \(w=w_0\sqrt2\); waist \(w_0=\sqrt{\lambda z_R/\pi}=\sqrt{10^{-6}\cdot0.5/\pi}=3.99\times10^{-4}\) m, and \(w_0\sqrt2=5.64\times10^{-4}\) m. Consistent.
Derive the far-field divergence \(\theta=\lambda/\pi w_0\) from \(w(z)\), and state the beam-parameter product \(w_0\theta\).
Solution
For \(z\gg z_R\), \(w(z)=w_0\sqrt{1+z^2/z_R^2}\to w_0\,z/z_R\). The half-angle is \(\theta=\lim_{z\to\infty}w(z)/z=w_0/z_R\). Substituting \(z_R=\pi w_0^2/\lambda\): \(\theta=w_0/(\pi w_0^2/\lambda)=\lambda/(\pi w_0)\). The product \(w_0\theta=\lambda/\pi\), a diffraction invariant independent of \(w_0\) — squeezing the waist proportionally widens the divergence.
A beam of waist \(w_0=1\) mm (\(\lambda=633\) nm) passes through a telescope of angular magnification \(M=5\) (ray matrix \(\mathrm{diag}(1/M,\,M)=\mathrm{diag}(0.2,5)\), \(B=C=0\)). Find the output waist assuming input and output planes are at the respective waists.
Solution
At a waist \(q_1=iz_R\), \(z_R=\pi w_0^2/\lambda=\pi(10^{-3})^2/6.33\times10^{-7}=4.96\) m, so \(q_1=4.96\,i\). ABCD law with \(A=0.2,B=0,C=0,D=5\): \(q_2=(Aq_1+B)/(Cq_1+D)=0.2\,q_1/5=0.04\,q_1=0.04\times4.96\,i=0.198\,i\). Purely imaginary, so output plane is a waist with \(z_R'=0.198\) m. Then \(w_0'=\sqrt{\lambda z_R'/\pi}=\sqrt{6.33\times10^{-7}\times0.198/\pi}=\sqrt{3.99\times10^{-8}}=2.0\times10^{-4}\) m \(=0.2\) mm. So the waist scales by \(1/M\): a \(5\times\) angular telescope shrinks the beam \(5\times\) (and expands its divergence \(5\times\)), as expected since \(q\to q/M^2\) rescales \(z_R\) by \(1/M^2\) and \(w_0\propto\sqrt{z_R}\) by \(1/M\).