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Complex beam parameter

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Optics" title="Optics">optics</a>, the complex beam parameter is a <a href="Complex_number" title="Complex number">complex number</a> that specifies the properties of a <a href="Gaussian_beam" title="Gaussian beam">Gaussian beam</a> at a particular point z along the axis of the beam. It is usually denoted by q. It can be calculated from the beam's vacuum <a href="Wavelength" title="Wavelength">wavelength</a> λ0, the <a href="Radius_of_curvature_(optics)" title="Radius of curvature (optics)">radius of curvature</a> R of the phase front, the <a href="Index_of_refraction" class="mw-redirect" title="Index of refraction">index of refraction</a> n (n=1 for air), and the beam radius w (defined at 1/e2 intensity), according to:<a href="#cite_note-Yariv-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{q(z)}}={\frac {1}{R(z)}}-{\frac {i\lambda _{0}}{\pi nw(z)^{2}}}}</annotation>

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</math><img src="./862de24ba3f8c5044df65e7608b4dd65adaea72c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.418ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{q(z)}}={\frac {1}{R(z)}}-{\frac {i\lambda _{0}}{\pi nw(z)^{2}}}}" loading="lazy">.</dd></dl>

Alternatively, q can be calculated according to

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<annotation encoding="application/x-tex">{\displaystyle q(z)=z+{\frac {i\pi nw_{0}^{2}}{\lambda _{0}}}=z+z_{\mathrm {R} }i\ ,}</annotation>

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</math><img src="./d4c9bd6717d3cd89e1f7ca64f2b483b2fa2308fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.658ex; height:6.509ex;" alt="{\displaystyle q(z)=z+{\frac {i\pi nw_{0}^{2}}{\lambda _{0}}}=z+z_{\mathrm {R} }i\ ,}" loading="lazy"><a href="#cite_note-Yariv-1">[1]</a></dd></dl>

where z is the location, relative to the location of the <a href="Beam_waist" class="mw-redirect" title="Beam waist">beam waist</a>, at which q is calculated, zR is the <a href="Rayleigh_range" class="mw-redirect" title="Rayleigh range">Rayleigh range</a>, and i is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.

<div class="mw-heading mw-heading2"><h2 id="Beam_propagation">Beam propagation</h2></div>

The complex beam parameter is usually used in <a href="Ray_transfer_matrix_analysis" title="Ray transfer matrix analysis">ray transfer matrix analysis</a>, which allows the calculation of the beam properties at any given point as it propagates through an optical system, if the ray matrix and the initial complex beam parameter is known. This same method can also be used to find the fundamental mode size of a stable <a href="Optical_resonator" class="mw-redirect" title="Optical resonator">optical resonator</a>.

Given the initial beam parameter, qi, one can use the ray transfer matrix of an optical system, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}}</annotation>

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</math><img src="./e3e11c7395f67ee34caefeb9e3ac42f0da94a97c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.186ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}}" loading="lazy">, to find the resulting beam parameter, qf, after the beam has traversed the system:<a href="#cite_note-Yariv-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle q_{f}={\frac {Aq_{i}+B}{Cq_{i}+D}}}</annotation>

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</math><img src="./3c795343bbfb96e00e69dcffd256629ef2f04233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.476ex; height:5.843ex;" alt="{\displaystyle q_{f}={\frac {Aq_{i}+B}{Cq_{i}+D}}}" loading="lazy">.</dd></dl>

It is often convenient to express this equation in terms of the reciprocals of q:<a href="#cite_note-Yariv-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {1 \over q_{f}}={\frac {C+D/q_{i}}{A+B/q_{i}}}}</annotation>

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</math><img src="./af48f803f9e9c1a38e3412639417f5b66be6f9a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.474ex; height:6.509ex;" alt="{\displaystyle {1 \over q_{f}}={\frac {C+D/q_{i}}{A+B/q_{i}}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Free-space_propagation">Free-space propagation</h3></div>

The effect of propagation in free space is just that of adding the travelled axial distance Δz to the complex beam parameter:<a href="#cite_note-Kochkina_eq4.16-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle q_{f}=q_{i}+\Delta z}</annotation>

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</math><img src="./86a8a16b4aa5547e3415bcafdb6c3d4fee29e2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.973ex; height:2.843ex;" alt="{\displaystyle q_{f}=q_{i}+\Delta z}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Interfaces">Interfaces</h3></div>

For simple astigmatic fundamental Gaussian beams,<a href="#cite_note-Kochkina_ch4-3">[3]</a> the q- parameters for the tangential and sagittal planes are independent. This is no longer true if those planes do not coincide with the principal direction of the surface on which the beam impinges; that case is called general astigmatism.<a href="#cite_note-Kochkina_ch4-3">[3]</a> Formulas for an incidence angle θi were derived in Massey and Siegman's 1969 paper.<a href="#cite_note-Siegman1969-4">[4]</a>

For reflection, the ABCD matrices read:<a href="#cite_note-Kochkina_Mrefl-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{t}&={\begin{pmatrix}1&0\\{\frac {2}{R_{I}\cos \theta _{i}}}&1\end{pmatrix}},&M_{t}&={\begin{pmatrix}1&0\\{\frac {2\cos \theta _{i}}{R_{S}}}&1\end{pmatrix}}.\end{aligned}}}</annotation>

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</math><img src="./17a81eca4e1885f5d445c0bb819bf1d17454f45e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.202ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}M_{t}&={\begin{pmatrix}1&0\\{\frac {2}{R_{I}\cos \theta _{i}}}&1\end{pmatrix}},&M_{t}&={\begin{pmatrix}1&0\\{\frac {2\cos \theta _{i}}{R_{S}}}&1\end{pmatrix}}.\end{aligned}}}" loading="lazy"></dd></dl>

The ones for refraction are:<a href="#cite_note-Kochkina_Mrefr-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{t}&={\begin{pmatrix}{\frac {\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}{n_{r}\cos \theta _{i}}}&0\\{\frac {\cos \theta _{i}-{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}{R_{I}\cos \theta _{i}{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}}&{\frac {\cos \theta _{i}}{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}\end{pmatrix}},&M_{t}&={\begin{pmatrix}1&0\\{\frac {\cos \theta _{i}-{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}{R_{S}n_{r}}}&{1 \over n_{r}}\end{pmatrix}}.\end{aligned}}}</annotation>

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</math><img src="./603427f5c99967949325d8a4b29ec42e312a2911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:78.262ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}M_{t}&={\begin{pmatrix}{\frac {\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}{n_{r}\cos \theta _{i}}}&0\\{\frac {\cos \theta _{i}-{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}{R_{I}\cos \theta _{i}{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}}&{\frac {\cos \theta _{i}}{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}\end{pmatrix}},&M_{t}&={\begin{pmatrix}1&0\\{\frac {\cos \theta _{i}-{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}{R_{S}n_{r}}}&{1 \over n_{r}}\end{pmatrix}}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Fundamental_mode_of_an_optical_resonator">Fundamental mode of an optical resonator</h3></div>

To find the complex beam parameter of a stable <a href="Optical_resonator" class="mw-redirect" title="Optical resonator">optical resonator</a>, one needs to find the ray matrix of the cavity. This is done by tracing the path of beam in the cavity. Assuming a starting point, find the matrix that goes through the cavity and return until the beam is in the same position and direction as the starting point. With this matrix and by making qi = qf, a quadratic is formed as:

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<annotation encoding="application/x-tex">{\displaystyle C{q_{f}}^{2}+(D-A)q_{f}-B=0}</annotation>

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</math><img src="./5b3e75eb826c5f622a9426f9a4c2e5e74560d028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.19ex; height:3.343ex;" alt="{\displaystyle C{q_{f}}^{2}+(D-A)q_{f}-B=0}" loading="lazy">.</dd></dl>

Solving this equation gives the beam parameter for the chosen starting position in the cavity, and by propagating, the beam parameter for any other location in the cavity can be found.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFYariv1989" class="citation book cs1">Yariv, Amnon (1989). Quantum Electronics (3rd ed.). Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-60997-8</bdi>.</cite>

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<li id="cite_note-Kochkina_eq4.16-2"><a href="#cite_ref-Kochkina_eq4.16_2-0">^</a> Kochkina, eq. 4.16

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<li id="cite_note-Kochkina_ch4-3">^ <a href="#cite_ref-Kochkina_ch4_3-0">a</a> <a href="#cite_ref-Kochkina_ch4_3-1">b</a> Kochkina, ch. 4

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<li id="cite_note-Siegman1969-4"><a href="#cite_ref-Siegman1969_4-0">^</a> <cite id="CITEREFMasseySiegman1969" class="citation journal cs1">Massey, G. A.; Siegman, A. E. (1969). <a rel="nofollow" class="external text" href="https://www.osapublishing.org/ao/abstract.cfm?uri=ao-8-5-975">"Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces"</a>. Applied Optics. 8 (5). OSA: 975–978. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969ApOpt...8..975M">1969ApOpt...8..975M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FAO.8.000975">10.1364/AO.8.000975</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20072358">20072358</a>.</cite>

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<li id="cite_note-Kochkina_Mrefl-5"><a href="#cite_ref-Kochkina_Mrefl_5-0">^</a> Kochkina, eq. 4.35

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<li id="cite_note-Kochkina_Mrefr-6"><a href="#cite_ref-Kochkina_Mrefr_6-0">^</a> Kochkina, eq. 4.42,43

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</ol></div>

<ul><li><cite id="CITEREFKochkina2013" class="citation thesis cs1">Kochkina, Evgenia (2013). Stigmatic and astigmatic Gaussian beams in fundamental mode: impact of beam model choice on interferometric pathlength signal estimates (PhD). Gottfried Wilhelm Leibniz Universität Hannover.</cite></li></ul></div><div><div class="zim-footer">

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