Does a diffraction grating with a shape form fourier image

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?

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blue_leaf77
Homework Helper
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.

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
Homework Helper
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?

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?

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