# Grating Spacing With Fourier Optics

• nanath
In summary, the spacing for a Fourier transform of an amplitude grating is proportional to f*lambda/b, where b is the spacing between slits in the grating. For a phase grating, the spacing is exp(iT) and the FT is significantly more complicated and qualitatively different than the amplitude case.
nanath
Just ahead of time, no this is not related to homework or coursework in my case. I am a TA for an optics lab and need to know it so I can help the students in my class.

I am trying to find an expression for the x-spacing for a grating imaged using the 4f method, which is a grating that is 1 focal length away from the first lens, which is then 1f away from the Fourier plane, which is 1f away from the second lens which has a focal length equal to the first, which is finally 1f away from the image plane. Specifically, I am interested in knowing what the spacing is for the Fourier transform of a grating when it passes through a Fourier lens. I've heard it is somehow proportional to f*lambda/b, where b is the spacing between slits in the grating, f is the focal length of the lens and lambda is the wavelength of the laser, but I have not been able to find a source that solves such a problem.

I would appreciate it if you could give me either an equation for the spacing or a more direct method for solving it. So far, the only methods I've found end with a k-space solution and don't try to translate the peaks you would see to actual displacements (x-coordinate).

You haven't mentioned if it's an amplitude or a phase grating. For an amplitude grating (Ronchi ruling), the transmission function can be written as T = rect(bx)**comb(ax), where 'b' is the width of a single "slit" and 'a' the spacing between slits. '**' means convolution, not multiplication, so the transmission function is interpreted as a row of slits of width 'b' with a center-to-center spacing of 'a'. If this is illuminated with a plane wave, the FT is sinc(x'/b)*comb(x'/a), where '*' is multiplication and x' = x/(lambda*f) due to the focal length of the lens and illumination wavelength. This demonstrates that decreasing the slit width and/or spacing broadens the diffraction pattern, and illumination with decreasing wavelength and/or focal length also broadens the pattern.

For a phase grating, the transmission function is now exp(iT) (if the phase grating is a square-wave type, otherwise substitute in your own phase function) and the FT is significantly more complicated and qualitatively different than the amplitude case. Even so, decreasing the width and/or spacing of the phase grating broadens the pattern, etc.

Detailed derivations are easily found:

http://ocw.mit.edu/courses/mechanic...se-sinusoidal-and-binary/MIT2_71S09_lec16.pdf

This is an amplitude grating. So just to clarify then, the FT would look, in the spatial domain, like a series of spikes with a sinc function describing their amplitudes from a defined center? Also, the spacing between said spikes would be lambda*f/a?

## 1. What is grating spacing?

Grating spacing refers to the distance between the parallel lines or grooves on a diffraction grating. It is typically measured in micrometers (µm) or nanometers (nm).

## 2. How is grating spacing related to Fourier optics?

Grating spacing is a key factor in Fourier optics, as it affects the diffraction pattern produced by the grating. The spacing determines the angles at which diffraction occurs, which in turn affects the intensity and direction of diffracted light.

## 3. What is the significance of Fourier optics in grating spacing?

Fourier optics is important in understanding the behavior of light as it passes through a diffraction grating. By using Fourier transforms, we can analyze the diffraction pattern and determine the properties of the grating, such as its spacing and orientation.

## 4. How does grating spacing affect the resolution of a diffraction grating?

The grating spacing directly affects the resolution of a diffraction grating. A smaller spacing leads to a larger angular dispersion, which results in a higher resolution and ability to distinguish between closely spaced spectral lines.

## 5. Can grating spacing be adjusted or changed?

Yes, grating spacing can be adjusted or changed by altering the fabrication process or by using specialized techniques such as holographic gratings. However, it is typically a fixed parameter for a given grating and cannot be easily altered once the grating is produced.

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