This page titled 9.4: Gaussian Light Beams is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by John F. Cochran and Bretislav Heinrich.

\[\text{E}_{\text{x}}(\text{x}, \text{y}, \text{z})=\frac{\text{E}_{0}}{\sqrt{1+\left(\text{z} / \text{z}_{\text{R}}\right)^{2}}} \exp \left(\frac{i \text{k}_{0}}{2 \tilde{\text{q}}}\left(\text{x}^{2}+\text{y}^{2}\right)\right) \exp \left(i\left[\text{k}_{0} \text{z}-\psi\right]\right), \label{9.27}\]

The radius of curvature is infinite at z=0 corresponding to a plane wave-front. For z ≫ zR the radius of curvature approaches the distance z.

Gaussianbeamformula

\[\text{E}_{\text{x}}(\text{x}, \text{y}, \text{z})=\frac{\text{E}_{0}}{\sqrt{1+\left(\text{z} / \text{z}_{\text{R}}\right)^{2}}} \exp \left(\frac{i \text{k}_{0}}{2 \tilde{\text{q}}}\left[\text{x}^{2}+\text{y}^{2}\right]\right) \exp \left(i\left[\text{k}_{0} \text{z}-\psi\right]\right), \label{9.46}\]

\[\text {Exponent}=-\frac{\text{w}_{0}^{2}}{4}\left(1+\frac{2 i \text{z}}{\text{w}_{0}^{2} \text{k}_{0}}\right)\left[\text{p}^{2}-\frac{4 i \text{px}}{\text{w}_{0}^{2}\left(1+\frac{2 i \text{z}}{\text{w}_{0}^{2} \text{k}_{0}}\right)}\right]. \label{9.40}\]

\[\text {Exponent}=-\frac{\text{w}_{0}^{2}}{4}\left(\text{p}^{2}-\frac{4 i \text{px}}{\text{w}_{0}^{2}}+\frac{2 i \text{p}^{2} \text{z}}{\text{w}_{0}^{2} \text{k}_{0}}\right), \nonumber \]

Gaussianbeamsoftware

\[\text{A}(\text{p}, \text{q})=\frac{\text{E}_{0}}{4 \pi} \text{w}_{0}^{2} \exp \left(-\frac{\text{w}_{0}^{2}}{4}\left[\text{p}^{2}+\text{q}^{2}\right]\right). \nonumber\]

Gaussianbeamprofile

Now using the approximation Equation (\ref{9.25}) in Equation (\ref{9.21}) investigate the beam profile at some arbitrary value of z:

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The instrument has 4 measuring speeds from 0.1 to 1 s / measuring cycle. PM fiber can be in the fast Align measurement mode in real time. For very accurate measurement results, the measurement time can be extended. PER, axis angle, Power level and power quotient are automatically measured and displayed in parallel on the OLED.

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\[\text{E}_{\text{x}}(\text{x}, \text{y}, \text{z})=\frac{\text{E}_{0} \text{w}_{0}^{2}}{4\left(\frac{\text{w}_{0}^{2}}{4}+\frac{i \text{z}}{2 \text{k}_{0}}\right)} \exp \left(\frac{i \text{k}_{0}\left[\text{x}^{2}+\text{y}^{2}\right]}{2 \tilde{\text{q}}}\right) \exp \left(i \text{k}_{0} \text{z}\right), \label{9.43}\]

\[\exp \left(\frac{-\text{k}_{0} \text{z}_{\text{R}}\left(\text{x}^{2}+\text{y}^{2}\right)}{2\left(\text{z}^{2}+\text{z}_{\text{R}}^{2}\right)}\right)=\exp \left(-\frac{\left(\text{x}^{2}+\text{y}^{2}\right)}{\text{w}^{2}}\right), \label{9.33}\]

\[\frac{1}{w_{0}^{2}}\left[x^{2}+i p w_{0}^{2} x\right]=\frac{1}{w_{0}^{2}}\left[x+\frac{i p w_{0}^{2}}{2}\right]^{2}+\frac{p^{2} w_{0}^{2}}{4}. \label{9.37}\]

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Gaussianbeamcalculator

\[\frac{1}{\tilde{\text{q}}}=\frac{1}{\text{z}-\text{z}_{\text{R}}}=\frac{\text{z}+\text{i} \text{z}_{\text{R}}}{\text{z}^{2}+\text{z}_{\text{R}}^{2}}. \label{9.31}\]

The ERM-202 is a dual channel polarization extinction ratio (PER) meter specifically designed to simultaneously measure the PER and power ratio of a device with two polarization maintaining (PM) outputs, such as a Y-branch fiber gyro IOC, PM coupler (PMC), or polarization beam splitter (PBS), as well as evaluate the performance (output DOP) of depolarizers.

The PER Meter ERM-202 is characterized by its large PER dynamic range of 50 dB, an angular resolution of 0.06 ° and its brilliant and high-contrast OLED display.

The PER Meter ERM-202 is a dual channel instrument for measuring the Polarization Extinction Ratio (Polarization Extinction Ratio PER). The second input can be PER and measuring the optical power level on components with two PM outputs in parallel. The PER Meter ERM-202 is therefore very well suited for integrated optical chips (IOC) with Y-coupler as used in fiber gyroscopes, PM coupler (PMC) and polarization beam splitter (PBC / S).

\[\text{I}=\frac{\sqrt{\pi}}{\sqrt{\frac{\text{w}_{0}^{2}}{4}+\frac{i z}{2 \text{k}_{0}}}} \exp \left(\frac{i \text{k}_{0} \text{x}^{2}}{2\left[\text{z}-i \text{k}_{0} \text{w}_{0}^{2} / 2\right]}\right). \label{9.42}\]

\[\text{A}(\text{p}, \text{q})=\frac{\text{E}_{0}}{4 \pi^{2}} \int \int_{-\infty}^{\infty} \text{d} \text{xdy} \exp \left(-\left[\frac{\text{x}^{2}}{\text{w}_{0}^{2}}+i \text{px}\right]\right) \exp \left(-\left[\frac{\text{y}^{2}}{\text{w}_{0}^{2}}+i \text{qy}\right]\right). \label{9.35}\]

\[\text{E}_{\text{x}}(\text{x}, \text{y}, \text{z}, \text{t})=\int \int_{-\infty}^{\infty} \operatorname{dpdq} \text{A}(\text{p}, \text{q}) \exp (i[\text{px}+\text{qy}+\text{kz}-\omega \text{t}]), \label{9.21}\]

It is impossible to generate an unbounded plane wave, of course. Nevertheless, the concept of unbounded plane waves is a very useful one because a finite beam of radiation can be described as the superposition of plane waves having different amplitudes and phases and propagating in slightly different directions, see Figure (9.3.8). To simplify matters let us assume that the amplitude function, A(p,q), is symmetric in p,q: ie. A(-p,-q)= A(p,q). This simplification allows one to construct a beam in which the electric field is polarized along a particular direction in the plane- along the x-direction, say. Eqn.(\ref{9.21}) illustrates how such a beam could be constructed:

Gaussianbeamintensity formula

\[\begin{aligned} \text { Exponent } &=-\frac{\text{w}_{0}^{2}}{4}\left(1+\frac{2 i \text{z}}{\text{w}_{0}^{2} \text{k}_{0}}\right) \\ & \cdot\left(\left[\text{p}-\frac{2 i \text{x}}{\left(\text{w}_{0}^{2}+\frac{2 i \text{z}}{\text{k}_{0}}\right)}\right]^{2}+\frac{4 \text{x}^{2}}{\left(\text{w}_{0}^{2}+\frac{2 \text{i} z}{\text{k}_{0}}\right)^{2}}\right), \end{aligned}\]

\[\text{I}=\int_{-\infty}^{\infty} \text{dp} \exp \left(-\frac{\text{w}_{0}^{2} \text{p}^{2}}{4}+i \text{px}-i \frac{\text{p}^{2} \text{z}}{2 \text{k}_{0}}\right). \label{9.39}\]

Notice that the amplitude function (\ref{9.24}) becomes very small if p2 or q2 is greater than \(4 / \text{w}_{0}^{2}\): : this means that the waves in the bundle describing the radiation beam that have transverse components p,q much larger than ±2/w0 can be neglected. In a typical case the laser beam radius is ∼1 mm so that the amplitude A(p,q) becomes small for | p |, | q | larger than 2 × 103 m−1 . But at optical frequencies λ ∼ (1/2)×10−6 m so that k0 ∼ 2\(\pi\)/λ ∼ 4\(\pi\) ×106 m−1 . Thus the important values of the transverse components p,q of the plane waves that make up the beam are very small compared with the total wavevector k0. The longitudinal component of the wave-vector, the z-component k, is given by

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The beam radius at the output mirror, the position of the minimum beam radius, is usually \(\text{w}_{0} \cong 1 \text{mm}\) for a typical gas laser operating in the visible. For a wavelength of λ = 5 × 10−7 meters the Rayleigh range for such a laser is zR= 6.28 meters. Therefore the beam diameter will have expanded by only \(\sqrt{2}\) = 1.41 at a distance of 6.28 meters from the laser output mirror.

The PER Meter ERM-202 is an optimal solution for aligning PM fiber pigtails especially in PM fiber pigtail assembly on optical modulators, PM waveguide structures, laser diodes, fiber optic PM components such as PM coupler and beam splitter, PM fiber plug assembly, characterization of depolarizers, alignment of PM fibers before splicing, production of PM fiber coils and fiber gyroscopes.

202291 — The amount of light in a photo is controlled by the camera's aperture, and the aperture is itself controlled by what is known as f-stops.

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When Equation (\ref{9.31}) is introduced into the expression for the electric field, Equation (\ref{9.27}), the imaginary part of 1/\(\tilde{\text{q}}\) gives rise to a Gaussian spatial variation

\[\text {Exponent}=-\left(\frac{\text{w}_{0}^{2}}{4}+\frac{i \text{z}}{2 \text{k}_{0}}\right)\left[\text{p}-\frac{2 i \text{x}}{\left(\text{w}_{0}^{2}+\frac{2 \text{i} x}{\text{k}_{0}}\right)}\right]^{2} \label{9.41}\]

See Section(9.4.2) for the details of the calculation. The variable \(\tilde{\text{q}}\) is called the complex radius of curvature of the beam. This nomenclature stems from the description of a spherical wave-front, Figure (9.4.9) as will be explained in the next paragraph. The length zR is called the Rayleigh range.

where R is the radius of curvature. A comparison of this expression with Equation (\ref{9.27}) shows why \(\tilde{\text{q}}\) is called the complex radius of curvature. One can separate the reciprocal of the complex radius of curvature into its real and imaginary parts:

\[\text{I}=\exp \left(\frac{-\text{x}^{2}}{\left(\text{w}_{0}^{2}+\frac{2 \text{i} \text{z}}{\text{k}_{0}}\right)}\right) \int_{-\infty}^{\infty} \text{du} \exp \left(-\left[\frac{\text{w}_{0}^{2}}{4}+\frac{i \text{z}}{2 \text{k}_{0}}\right] \text{u}^{2}\right), \nonumber \]

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\[\begin{align} \mathrm{E}_{\mathrm{x}}(\mathrm{x}, \mathrm{y}, \mathrm{z})=& \frac{\mathrm{E}_{0} \mathrm{w}_{0}^{2}}{4 \pi} \int \int_{-\infty}^{\infty} \operatorname{dpdq} \exp \left(-\frac{\mathrm{w}_{0}^{2}}{4}\left[\mathrm{p}^{2}+\mathrm{q}^{2}\right]\right). \label{9.26} \\ & \cdot \exp (i[\mathrm{px}+\mathrm{qy}]) \exp \left(-i \frac{\left[\mathrm{p}^{2}+\mathrm{q}^{2}\right] \mathrm{z}}{2 \mathrm{k}_{0}}\right) \exp \left(i \mathrm{k}_{0} \mathrm{z}\right) \nonumber \end{align}\]

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Eqn.(\ref{9.21}) is an example of a Fourier Integral. The amplitude function A(p,q) can be chosen to give the required beam profile in the x-y plane at some plane z=constant; it is convenient to choose this plane to be at z=0. The beam profile at any other position z can be obtained using the integral (\ref{9.21}). As an example of how this works let us treat a specific case for which the mathematics can be easily worked out. Suppose that at z=0 the beam cross-section can be described as a plane wave whose amplitude falls off exponentially along x and y:

\[\text{I}=\int_{-\infty}^{\infty} \text{d} \text{x} \exp \left(-\left[\frac{\text{x}^{2}}{\text{w}_{0}^{2}}+i \text{px}\right]\right). \label{9.36}\]

For each channel of the ERM-202 there is an analogue output which provides the PER measurement signal as voltage level provides. This allows the design of a control loop to automate the alignment of PM fibers.

Gaussianbeamradius

Gaussianbeampdf

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Gaussianbeamdivergence

A time dependence exp (−iωt) is assumed, but this factor will be suppressed in the following. The output beam from a typical gas laser, a He-Ne laser for example, exhibits the spatial variation (\ref{9.22}) at the output mirror with w0 approximately equal to 1 mm. Such a beam profile is called a Gaussian beam profile. The spatial Fourier integral in (\ref{9.21}) can be inverted for z=0 to obtain

\[\text{A}(\text{p}, \text{q})=\frac{1}{4 \pi^{2}} \int \int_{-\infty}^{\infty} \operatorname{dxdy} \text{E}_{\text{x}}(\text{x}, \text{y}, 0) \exp (-i[\text{px}+\text{qy}]). \label{9.23}\]

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\[\text{I}=\exp \left(-\frac{\text{p}^{2} \text{w}_{0}^{2}}{4}\right) \int_{-\infty}^{\infty} \text{du} \exp \left(-\frac{\text{u}^{2}}{\text{w}_{0}^{2}}\right)=\text{w}_{0} \sqrt{\pi} \exp \left(-\frac{\text{p}^{2} \text{w}_{0}^{2}}{4}\right). \label{9.38}\]

\[\begin{aligned} \frac{\text{E}_{0} \text{w}_{0}^{2}}{4\left(\frac{\text{w}_{0}^{2}}{4}+\frac{\text{i} \text{z}}{2 \text{k}_{0}}\right)} &=\frac{\text{E}_{0}}{\left(1+\frac{i \text{z}}{\text{z}_{\text{R}}}\right)} \\ &=\frac{\text{E}_{0}\left[1-i \text{z} / \text{z}_{\text{R}}\right]}{\left(1+\left[\text{z} / \text{z}_{\text{R}}\right]^{2}\right)} \\ &=\frac{\text{E}_{0}}{\sqrt{1+\left(\text{z} / \text{z}_{\text{R}}\right)^{2}}} \exp (-i \psi) \end{aligned}\]

The ERM-202 is a dual channel polarization extinction ratio (PER) meter specifically designed to simultaneously measure the PER and power ratio of a device with two polarization maintaining (PM) outputs, such as a Y-branch fiber gyro IOC, PM coupler (PMC), or polarization beam splitter (PBS), as well as evaluate the performance (output DOP) of depolarizers.

\[\text{A}(\text{p}, \text{q})=\frac{\text{E}_{0} \text{w}_{0}^{2}}{4 \pi} \exp \left(-\frac{\text{w}_{0}^{2}}{4}\left(\text{p}^{2}+\text{q}^{2}\right)\right). \label{9.24}\]

\[\text{E}_{\text{x}}(\text{x}, \text{y}, 0)=\text{E}_{0} \exp \left(\frac{-\left(\text{x}^{2}+\text{y}^{2}\right)}{\text{w}_{0}^{2}}\right). \label{9.22}\]

The ERM-202 is a dual channel polarization extinction ratio (PER) meter specifically designed to simultaneously measure the PER and power ratio of a device with two polarization maintaining (PM) outputs, such as a Y-branch fiber gyro IOC, PM coupler (PMC), or polarization beam splitter (PBS), as well as evaluate the performance (output DOP) of depolarizers.

Interested readers can learn more about Gaussian beams and Gaussian beam optics in the book ”An Introduction to Lasers and Masers” by A.E. Siegman, McGraw-Hill, New York, 1971; chapter 8.

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This means that as one moves along the beam the radius of the beam slowly increases and becomes greater by \(\sqrt{2}\) at z = zR: ie. at one Rayleigh range removed from the minimum beam radius, or beam waist.

\[\text{k}^{2}=\left(\frac{\omega}{\text{c}}\right)^{2}-\text{p}^{2}-\text{q}^{2}=\text{k}_{0}^{2}-\text{p}^{2}-\text{q}^{2}. \nonumber \]