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# Mode Matching

Definition: the precise spatial matching of the electric field distributions of laser beams and resonator modes or waveguide modes

German: Modenanpassung

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In many situations, it is necessary to spatially match a laser beam precisely to another beam or a mode in order to obtain some kind of efficient coupling. Examples are:

The necessary matching of modes means not only creating a good spatial overlap of the intensity profiles, but also matching the optical phase profiles. In other words, the wavefronts of the beams will then be matched. If the complex amplitude profiles of two beams (having the same optical frequency) are well matched in a certain plane, they will remain well matched during further propagation.

Mode matching can be achieved by using suitable relay optics (typically some combination of curved mirrors or lenses), provided that the beam quality of the initial beam is close to diffraction-limited.

Mathematically, the quality of mode matching can be quantified with an overlap integral. The following formula, involving the square of such an overlap integral, calculates the coupling efficiency concerning optical powers:

$$\eta = \frac{{{{\left| {\int {E_1^*{E_2}\;{\rm{d}}A} } \right|}^2}}}{{\int {{{\left| {{E_1}} \right|}^2}{\rm{d}}A\;\int {{{\left| {{E_2}} \right|}^2}{\rm{d}}A} } }}$$

where <$E_1$> and <$E_2$> are the complex electric fields in a plane, referring e.g. to a laser beam and the field of a resonator or waveguide mode, and the integration spans the whole beam cross-section. That quantity is preserved during propagation in free space.

A similar overlap integral can be used for calculating complex mode amplitudes.

If the beam from a frequency-tunable single-frequency laser hits a symmetric Fabry–Pérot interferometer and the laser frequency is tuned over the whole free spectral range of the resonator, the transmitted light can be used to analyze the degree of mode matching. For perfect matching to a cavity mode (typically the fundamental Gaussian mode), complete transmission of the resonator can be observed when the resonance condition is met, whereas other resonances (corresponding to other resonator modes) can not be excited.

## Questions and Comments from Users

2021-02-15

How does one account for polarization in this equation?

Not at all. The calculation is based on scalar fields, i.e., neglecting polarization. If you want to calculate the overlap often arbitrary polarized field with a polarized mode, you first have to decompose the field into its polarization components.

2021-09-17

How to couple light into a waveguide which does not have a precise mode description but only a rough mode width; how to use the formula?

You can at least normally assume that the wavefronts of the modes are flat. As a first guess, you may then assume a Gaussian amplitude profile.

2021-10-28

Do the two fields E1 and E2 have to be normalized to the maximum?

No – the formula contains the normalization. You can easily see, for example, that the result won't change if you double <$E_1$>.

2021-11-23

What if we do not have the phase information, but measured the intensity profiles with a camera? I guess one could work with the square root of the intensity, but that would only obtain an upper limit, wouldn't it?

You need the full phase information to do any calculations on mode matching.By taking the square root of the intensity profile, you are effectively assuming a flat phase profile. That might serve as an upper limit for the possible mode matching efficiency, if the ideal output field has flat phase fronts.

2022-04-22

Is there any relation between the “overlap INTEGRAL” as described on this page and the “overlap FACTOR” (<$\eta_\textrm{m}$>) on your page concerning optical heterodyne detection?

Yes, it is the same. The heterodyne signal is proportional to the amplitude of the input signal; its signal-to-noise ratio is proportional to the square of that. That's compatible with the formula for the <$\eta$> value shown in this article.

2022-09-12

Does the formula take into account the imaginary part of the electric field? Is there any problem if I only consider the real part of the electric field?

The formula works with complex electric fields, as indicated. Here, you cannot simply ignore their imaginary parts. For example, a wavefront curvature as seen in the complex phase is relevant for mode matching.

2022-10-06

I am a bit confused by your answer on 2022-09-12. When we describe waves using the complex representation it is always understood that this is for algebraic convenience and only the real part has physical significance. Your answer seems to imply that there is information in the complex part, and that the complex representation is actually necessary. Is this a contradiction?

An interesting and important question. The physical electric field is the real part of the product of that complex amplitude and the oscillating factor <$\exp(i \omega t)$>. Written out, I mean <$E(t) = Re(E \cdot \exp(i \omega t))$>. You see from this that the imaginary part of that complex amplitude alone does matter! You also see that from the denominator in the formula: the modulus squared appears, and that also depends on the imaginary part.

2022-10-13

Is there a good book on the topic? Especially I would be interested in mode overlap integrals for LG- and HG- higher-order modes for different waist sizes or misalignment.

I would simply numerically calculate that myself without searching for literature. The equations are reasonably simple.

2023-09-21

I have a question regarding the angle under which the beams arrive. For example in a star coupler, does this angle affect the coupling? Or is the narrower field, resulting from one port being under an angle, sufficient for estimating the efficiency?

No, the angle is important because any tilt is associated with a linearly varying phase change. You cannot consider the intensity profile only.

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