Does a diffraction grating with a shape form fourier image

[steph17]

Main Question or Discussion Point

i just wanted to get this cleared that a beam falling on a diffraction grating with a shape gives the fourier images of the grating object which can be reobtained by placing a biconvex lens that would converge the rays and form a focussed fourier image at its focal length and the image of the object at other points.
please correct me if i have misunderstood any phenomenon above
and is there any relation between the grating lines and the fourier image? how can i calculate the transfer function of the lens?

 

[blue_leaf77]

FT relation between object and its image holds true only in far-field region and paraxial rays. The far-field image will give you the FT of the grating object if the used grating upstream was illuminated with plane wave, otherwise the far-field image will be the FT of the field just after the grating (i.e. product between incoming beam and grating transmission function). For the effect of placing a lens, I suggest this file: http://users.ece.utexas.edu/~becker/FOch5-6.pdf

is there any relation between the grating lines and the fourier image?

To obtain this relation you have to calculate the FT of the grating lines arrangement.

 

[steph17]

FT relation between object and its image holds true only in far-field region and paraxial rays. The far-field image will give you the FT of the grating object if the used grating upstream was illuminated with plane wave, otherwise the far-field image will be the FT of the field just after the grating (i.e. product between incoming beam and grating transmission function). For the effect of placing a lens, I suggest this file: http://users.ece.utexas.edu/~becker/FOch5-6.pdf


To obtain this relation you have to calculate the FT of the grating lines arrangement.

any particular good book/reference where i can find how to mathematically? thanks for the help

 

[blue_leaf77]

Fundamentals of Photonics by Saleh and Teich

 

[Andy Resnick]

i just wanted to get this cleared that a beam falling on a diffraction grating with a shape gives the fourier images of the grating object which can be reobtained by placing a biconvex lens that would converge the rays and form a focussed fourier image at its focal length and the image of the object at other points.
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I'm a little confused by your question- do you mean that the diffraction grating was cut into a particular shape, like a circle or paper doll? Alternatively, by 'shape of the grating' do you mean the groove profile?

 

[steph17]

I'm a little confused by your question- do you mean that the diffraction grating was cut into a particular shape, like a circle or paper doll? Alternatively, by 'shape of the grating' do you mean the groove profile?

groove profile, like a cartoon character on the squares, thats all.

 

[Andy Resnick]

groove profile, like a cartoon character on the squares, thats all.

Thanks, that helps me understand. For example, you may have a laser pointer attachment that projects a square or cartoon character rather than the 'raw beam', right? That attachment is a 2-D phase grating, but if you want to suppress the undiffracted beam as well as undesired diffraction orders, then the grating is nonperiodic.

In any case, the far-field diffraction pattern is the Fourier Transform of the grating transmission. Shining the diffracted beam onto a positive lens (it doesn't have to be biconvex, but it does have to be a positive lens) simply moves the far-field diffraction pattern from infinity to a user-defined plane that is closer and also converts the angular diffraction pattern into a linear diffraction pattern with a conversion factor that involves the focal length of the lens.

Does this help?

 

[steph17]

Thanks, that helps me understand. For example, you may have a laser pointer attachment that projects a square or cartoon character rather than the 'raw beam', right? That attachment is a 2-D phase grating, but if you want to suppress the undiffracted beam as well as undesired diffraction orders, then the grating is nonperiodic.

In any case, the far-field diffraction pattern is the Fourier Transform of the grating transmission. Shining the diffracted beam onto a positive lens (it doesn't have to be biconvex, but it does have to be a positive lens) simply moves the far-field diffraction pattern from infinity to a user-defined plane that is closer and also converts the angular diffraction pattern into a linear diffraction pattern with a conversion factor that involves the focal length of the lens.

Does this help?

yes, thanks