Understanding Optical Computing: Fourier Transform and Grating Experiment, Exercises of Advanced Physics

An insight into the concept of optical computing, its history, and the experimental process of generating a fourier transform using a grating. Optical computing is a numerical processing technique that uses light instead of electricity. How a complicated object is divided into plane waves using a fourier transform, and how these plane waves can be recombined to form an image of the object. The document also discusses the use of lasers, lenses, and detectors in optical computing, and provides a simple experiment to demonstrate the generation of a fourier transform.

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Modern day wants fast computing. Present day computer using electronic components for
computing is fast but may not meet future computing requirements. So scientists thing about a
computer which works with speed of light. This type of computer is called optical computer.
This computer uses Fourier transform technique to transform data into optical form for
executions. Our this experiment is one part of optical computing Fourier part, i.e. studying the
transformation of data to optical form.
Optical Computing:-. The concept of optical computing arose in late 1960s when NASA
faced a problem. NASA obtained pictures of moon in form of small strips. These strips were
combined by other methods which degraded the quality of pictures by introducing lines at point
of junction. A new technique were developed which removed the lines from the picture. This
technique was called optical computing. The lines can be removed by producing Fourier
transform of picture and plane waves generated by the unwanted lines can be stopped which
removes the lines form picture.
Now this type of technique is used in future computer known as optical computer.
Optical computers:-:-. Common computer to us works on electricity or in others words
electronics and make use of transistors and integrated circuits based on electronics. These
computers are much faster than old time vacuum tube computers but still high speed is required
in modern day. So it is dreamed to make a computer which works at speed of light known as
optical computer. We consider optical computers that encode data using images and compute by
transforming such image.
Optical computing is numerical processing through light. Optical computing started with the
design and implementation of optical systems to arbitrarily modify the complex valued spatial
frequencies of an image.
Hardware for optical computing:-
1. Lasers:- Lasers are almost monochromatic light sources. LED can also be used in optical
computing but only in those systems where noise is tolerated. Lasers are also used
because of avoiding chromatic dispersion of light during refracting from optical
components.
2. Modulators:- In optical computing image which is spatial function of spatial
frequencies, is encoded in light waves. In fact light is modulated in this process.
Modulation can be done by reflection and also by transmission. Modulators can be a
photographic film and electro-optic, magneto-optic, and acousto-optic devices
3. Detectors:- Detectors are used to measure intensity of light signals. In fact we cannot
measure phase and amplitude of a wave but we can observe it amplitude Mod square i.e.
its intensity. In optical computing, detectors also have some role.
4. Lenses:- Optical computing make use of light from image. This reflected or transmitted
light from image make a Fourier transform at infinity. So lenses i.e. convex lenses are
used for to make Fourier transform at some finite distance.
Following is a simple experiment which demonstrates some part of optical computing i.e.
generation a Fourier transform.
Fourier Transform:- A mathematical tool used to solve partial differential equation. But it
has an important application in image processing. When light strikes a complicated body, then
simple plane waves are generated which have different orientations. These plane waves carry
information of complicated object. So we can say that a complicated object is divided into a
number of simple plane waves. This phenomenon is represented by a simple Fourier transform
equation.
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Modern day wants fast computing. Present day computer using electronic components for computing is fast but may not meet future computing requirements. So scientists thing about a computer which works with speed of light. This type of computer is called optical computer. This computer uses Fourier transform technique to transform data into optical form for executions. Our this experiment is one part of optical computing Fourier part, i.e. studying the transformation of data to optical form.

Optical Computing:-. The concept of optical computing arose in late 1960s when NASA

faced a problem. NASA obtained pictures of moon in form of small strips. These strips were combined by other methods which degraded the quality of pictures by introducing lines at point of junction. A new technique were developed which removed the lines from the picture. This technique was called optical computing. The lines can be removed by producing Fourier transform of picture and plane waves generated by the unwanted lines can be stopped which removes the lines form picture.

Now this type of technique is used in future computer known as optical computer.

Optical computers:-:-. Common computer to us works on electricity or in others words

electronics and make use of transistors and integrated circuits based on electronics. These computers are much faster than old time vacuum tube computers but still high speed is required in modern day. So it is dreamed to make a computer which works at speed of light known as optical computer. We consider optical computers that encode data using images and compute by transforming such image.

Optical computing is numerical processing through light. Optical computing started with the design and implementation of optical systems to arbitrarily modify the complex valued spatial frequencies of an image.

Hardware for optical computing:-

1. Lasers:- Lasers are almost monochromatic light sources. LED can also be used in optical computing but only in those systems where noise is tolerated. Lasers are also used because of avoiding chromatic dispersion of light during refracting from optical components. 2. Modulators:- In optical computing image which is spatial function of spatial frequencies, is encoded in light waves. In fact light is modulated in this process. Modulation can be done by reflection and also by transmission. Modulators can be a photographic film and electro-optic, magneto-optic, and acousto-optic devices 3. Detectors:- Detectors are used to measure intensity of light signals. In fact we cannot measure phase and amplitude of a wave but we can observe it amplitude Mod square i.e. its intensity. In optical computing, detectors also have some role. 4. Lenses:- Optical computing make use of light from image. This reflected or transmitted light from image make a Fourier transform at infinity. So lenses i.e. convex lenses are used for to make Fourier transform at some finite distance. Following is a simple experiment which demonstrates some part of optical computing i.e. generation a Fourier transform.

Fourier Transform:- A mathematical tool used to solve partial differential equation. But it

has an important application in image processing. When light strikes a complicated body, then simple plane waves are generated which have different orientations. These plane waves carry information of complicated object. So we can say that a complicated object is divided into a number of simple plane waves. This phenomenon is represented by a simple Fourier transform equation.

( ) ∬ ( ) (^ )

And there is an inverse Fourier transform equation which is

( ) ∬ ( ) (^ )

Meaning of above two equations is given as follows. If f(x,y) is a complicated object and some illumination falls on it, then the object is changed into a number of plane wave of different orientations and weighting factor calculated by Fourier transform equation. The plane waves generated can be recombined which give image of object. This is given by inverse Fourier

transform equation in which (^ )^ is plane wave and ( ) is weighting factor. And

integral means that all plane waves, from are added. fx and fy represent Fourier frequencies in two directions on Fourier plane. Fourier frequencies are related to spatial frequencies as inverse of them. While spatial frequencies are number of line pairs per inch. Actually a complicated object is divided into line pairs, pair of dark and white line.

Apparatus:- Convex lenses of different focal length, laser source, stands, meter rod, screen,

polarizer, detector.

Procedure:-

Step 1 Alignment:- For making Fourier transform, first of all laser should be aligned. Its alignment be such that its beam is parallel to board on which t is fixed and also should be parallel to lines drawn if form of holes on board. Put a screen in front of laser and mark center of the beam.

Step 2 Collimation:- After making alignment collimation of the beam is needed. For collimation of beam, two lenses are used, one with short focal length and other with long focal length. In our case first lens of focal length f 1 =10mm was used which focus beam from laser on its focal length and behind that point it diverges. Second lens of focal length f2=160mm was placed at distance. As second lens is at distance equal to focal length of lens from the point of divergence so this lens will focus refract the beam such that all lines of beam become parallel to lens' axis. This beam after coming out from lens 2 is called collimated beam. Diameter of

collimated beam is given by ⁄. So in our experiment diameter of collimated beam was

1.6cm.

and only partial beam was allowed to make Fourier transform. But the partial beam was also intense so it saturated the current. Then we took help of polarizer. Polarizer is very efficient in reducing the intensity of beam. As intensity coming out from polarizer is dependent on angle of polarization of incident light and angle of polarization angle of polarizer, so depending on angle we can get beam of any intensity.

Polarizer helped us because the laser we used was source of plane polarized light. For zeroth and first order we adjusted large angle (=80 degree) and only then we were able to measure the intensity.

For second and third orders we changer angle to relatively smaller angles gradually and measured the intensities of those orders. The changing of angles is necessary because for one orientation of polarizer which we adjusted for zeroth order, higher orders were eliminated from F.T. screen.

Using of polarizer does not destroy the original information of intensities if Fourier frequencies. As we know that ( ), where θ is angle between plane of polarization of light and polarization axis of polarizer. Io is incident intensity on polarizer or we can say that original intensity of Fourier frequency. I is intensity measured by detector. So using this equation we can recover the original intensity of Fourier frequencies.

Observations:-

  1. Changing the grating also change the Fourier transform. For grating with small number of lines per unit length separation between the Fourier frequencies were smaller and for large number of grating lines per unit of length that separation was larger. Also it was observed the line in which Fourier transform was formed was perpendicular to lines of grating.
  2. Using two gratings at one time demonstrates the convolution theorem. For single grating one F.T. was formed. Let name this F.T. as FT1. Similarly F.T. for second grating when there is no other grating is FT2. Now let we use both gratings at a time. If grating 1 is used first then it will generate FT1 on screen and the grating 2 is also placed with grating 1 then the Fourier screen will give a pattern which is obtained by placing FT2 at each Fourier frequency of FT1. This is convolution theorem demonstrated in experiment.

Intensities of different orders of Fourier transform observed are

Fig. Fourier transform of two gratings placed perpendicular to each other demonstrating convolution theorem.

Order Intensity Order 1 Intensity

0th 6.3 - 1st 6

1st 6 - 2nd 3.

2nd 3.1 - 3rd 1.

3rd 1.6 - 4 th^ 1.

4th 1.1 - 5 th^ 0.

5th 0.9 - 6 th^ 0.

6th 0.5 - 7 th^ 0.

7th 0.