What Plane Waves Make a Gaussian Beam?

Question

When I think of a light beam, what first comes to mind is this:

Black lines are axes (both spatial axes, at a snapshot in time), and blue lines represent the surfaces of constant phase of a plane wave traveling along the horizontal axis.

However, obviously most light beams are not infinite in extent. This means that most light beams are composed of components with nonzero transverse spatial frequency.

My question is, what do those components look like?

Is the below an example?

If I put such a plane wave through a lens, it gets focused to a different spot in the lens's Fourier plane, leading me to believe that this is the right idea.

Are there other ways to make a plane wave with nonzero spatial frequency aside from just "tilting" the beam?

EDITS: To clarify what I meant by "spatial frequency" as well as the axes on my drawings, and to hone in on my exact question.

Answer 1

Answers to your question:

1. Yes, the tilted plane wave you draw is exactly what is meant by a plane wave component with a non-zero spatial frequency. If you look at the 2D function representing the complex amplitude of that wave on the indicated plane you will see it has uniform amplitude but a spatially varying phase. The phase will vary with a periodicity which is related to the title angle. A larger tilt angle corresponds to a large spatial frequencies. For small titles the relationship is approximately linear.

2. I'm not sure I fully understand you question about other ways to make transverse waves with non-zero spatial frequency. Theoretically the tilted plane wave is exactly what is meant by a wave with non-zero spatial frequency, no more, no less. So theoretically there is no other way. Experimentally the situation is different I guess. You can take a large beam and tilt it and that will increase the spatial frequencies in on direction. You can also impinge a plane wave onto a sinusoidal grating, the output of which will now include components at non-zero spatial frequencies. Lens and other sorts of optics can in general change the spatial frequency spectrum of an optical beam. This is the general discussion of the field of Fourier optics for which I can recommend the books "Fundamentals of Photonics" by Saleh and Teich and "Introduction to Fourier Optics" by Goodman.

I don't understand how the edited title relates to the body of the question so I can't really answer that except to say that a Gaussian beam is a superposition of many beams with differing spatial frequencies the same way a Gaussian function is a superposition of its Fourier components.

Answer 2

I assume your first drawing is for a specific time t, as in a photograph, and it is showing spatial variation. Spatial frequency is a measure of how often a particular feature of the wave repeats per unit distance. A plane wave in the positive x-direction such as in your first drawing has a kx term, e.g. in a sin function, where $k = 2\pi/L$ where L is the wavelength and its reciprocal is the spatial frequency.

More generally, for a wave not in the x-direction, kx is replaced by the dot product of the vector k with the position vector r. The magnitude of k again leads to the spatial frequency using the above equation.

Answer 3

To answer the title:

What Plane Waves Make a Gaussian Beam?

It is the wave packet solution:

one can construct a wave packet solution as a sum of traveling waves:

Almost a gaussian .