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# No Beat Note for Orthogonal Modes

Posted on 2009-07-29 as a part of the Photonics Spotlight (available as e-mail newsletter!)

Abstract: It is explained that a beat note does not always occur when two modes with different optical frequencies are superimposed. The orthogonality of modes is an important issue.

Ref.: encyclopedia article on beat note

Imagine a situation where two different transverse modes with different optical frequencies and equal polarization directions oscillate simultaneously in a laser. For example, it could be a TEM00 mode and a TEM10 mode. Both frequencies are contained in the emitted laser beam. Most people would take it for granted that a photodetector, hit by that beam, would indicate a beat note with the difference of the two optical frequencies, assuming that the detection bandwidth is large enough to resolve the beat frequency. It may be astonishing to learn that this is not necessarily the case.

A crucial insight is to realize that the electric field distributions associated with different modes of an optical resonator are orthogonal. This means that their overlap integral is zero. (This is the complex field distribution of one mode times the complex conjugate of the field of the other mode, integrated over the full beam cross-section.) When calculating the total power of the superimposed beams, one sees that beat term would be proportional to that overlap integral, and is thus zero. In other words, the recorded total optical power is simply the sum of the optical powers of both modes, with no oscillating contribution in addition.

So there may be no beat note at all. Note, however, that orthogonality may easily be lost if the superimposed beam is truncated at an edge of the active area of the photodetector. This situation is of course easily obtained by somewhat misaligning the detector. It is even sufficient that the detector's responsivity is not uniform over the active area (e.g. when a part of it has been burned with a focused laser beam). In such cases, the beat note is recovered.

It is also instructive to consider two separate beams with different frequencies, both having e.g. Gaussian profiles. Their overlap integral can be very small, even if their intensity profiles overlap well on the detector, if the beams come from different directions. In that case, the relative optical phase varies across the beam profile, and the contributions to the overlap integral may cancel each other. Assuming two diffraction-limited beams, it can be shown that the overlap integral becomes small if the difference of angles exceeds the beam divergence angle. The larger the beam areas, the more critical is the angular adjustment required for recording a beat note.

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