Lecture 5

Fourier Optics

Class Test I: Mark Distribution

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Marks for Class Test I will be available from your tutors

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Class Test I

…and you were given on the front sheet:

Class Test I

…and you were given on the front sheet:

You know from PC2 and the lecture notes handout that FT{top-hat}:

and

Class Test I

Class Test I

Common errors I

• Inability to multiply complex exponential expressions for sine and cosine.

• Inability to integrate exponential functions – in Fourier analysis this is a major concern.

• Arbitrarily interchanging x, k, t, w, k0 and w0 (and other symbols)

• Not realising that a series of sine functions is a Fourier series (particularly worrisome)

• Not being able to write down correct expressions for complex exponential form of sine and/or cosine.

Common errors II

•

• Not being able to sketch simple filter response diagrams.

• Inability to write down correct form of Fourier integral even though it was an open book test.

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Recap….

• Diffraction and convolution: double slit experiment

• 2D Fourier transforms• Diffraction gratings• Fourier filtering

Outline of Lecture 5

Reciprocal space and

spatial frequencies

Just as we can build up a complex waveform from a variety of sinusoids of different amplitudes and phases, so too can we generate an image from a Fourier integral.

2D Images and 2D Fourier Transforms

Consider an aperture:

?? f(x,y) in this case can be broken down into two functions f(x) and f(y). Sketch those functions.

2D Images and 2D Fourier Transforms

So, for a square aperture we have two sinc functions, one along kx and one along ky

Figures taken from Optics, Hecht (Addison-Wesley, 2nd Ed. 1987)

2D Images and 2D Fourier Transforms

?? Which area of the diffraction pattern is associated with low spatial frequencies? With high spatial frequencies?

2D Images and 2D Fourier Transforms

Aperture function (2 slits)

2 slit pattern

?? What is the effect on the image if we only pass the spatial frequencies within the circle shown?

2D Images and 2D Fourier Transforms

?? What is the effect on the image if we block the spatial frequencies within the circle shown?

Complex images: Fourier transforming and spatial filtering

Niamh

Niamh’s Fourier transform (modulus2)

Complex images: Fourier transforming and spatial filtering

Optical computer

Complex images: Fourier transforming and spatial filtering

The diffraction gratingThe diffraction grating

• An (infinite) diffraction grating has a transmission function whichlooks like:

?? The transmission function above can be represented as the convolution of two functions. Sketch them.

• We saw earlier how the double slit transmission function could be represented as a convolution of two functions. The grating transmission function can be treated similarly.

The diffraction gratingThe diffraction grating

• The ‘train’ of delta functions is known as a Dirac ‘comb’ (or a Shah function).

…whose Fourier transform is another Dirac comb:

where:

The diffraction gratingThe diffraction grating

?? At what value of k is the first zero in G(k) located?

has Fourier transform:

?? Sketch the Fourier transform (i.e. the diffraction pattern) of the transmission function for the infinite diffraction grating.

The diffraction gratingThe diffraction grating

•Now, what happens if we want to consider a real diffraction grating (i.e. one that is not infinite in extent)?

??The slits in the infinite grating above are spaced by an amount L. Imagine that we want to determine the Fourier transform of a grating which is 50L wide. How do we convert the transmission function for the infinite grating into that for a real grating which is 50L wide?

The diffraction gratingThe diffraction grating