- #1

ecastro

- 254

- 8

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter ecastro
- Start date

In summary, the Angular Spectrum Method and Fourier Transform of a Fresnel Diffraction are equal under the paraxial approximation, with the former being more accurate as it does not rely on additional approximations. However, when using built-in functions like fft2 in MatLab for the Fourier Transform, the scaling on the coordinates may be different and may require coding a custom function for accurate results.

- #1

ecastro

- 254

- 8

Physics news on Phys.org

- #2

blue_leaf77

Science Advisor

Homework Helper

- 2,637

- 786

- #3

ecastro

- 254

- 8

This is my aperture. A slit with a width of 100 pixels.

This is the Fresnel Diffraction simulation using the Angular Spectrum Method; and

This is the Fresnel Diffraction simulation using the Fourier Transform. Although they seem to be the same at the middle of the diffraction pattern, the pattern from the Fourier Transform seems to be from a rectangular aperture. There seems to be some sort of a "scaling" process done. Is this expected?

- #4

blue_leaf77

Science Advisor

Homework Helper

- 2,637

- 786

- #5

ecastro

- 254

- 8

- #6

blue_leaf77

Science Advisor

Homework Helper

- 2,637

- 786

Which ratio is this?ecastro said:With these parameters, the ratio is 4000.

Anyway, I'm not quite sure if what I think is the same as you, when you say using FT, did you actually transform the field aperture profile

If that's not sufficiently fulfilled, then your experimental conditions might still be in the paraxial/Fresnel region, in which case the FT relation should be performed w.r.t. the aperture profile

Furthermore, it seems like you don't perform the FT correctly, it's obvious that the lower picture is not an FT of a rectangular object. Just for an advice, I don't think that numerical program can give accurate result of a FT computation if the transformed object is not bounded, for example the vertical size of your object extends to the end of the pixel array. What about simulating a more realistic rectangular aperture?

- #7

ecastro

- 254

- 8

blue_leaf77 said:Which ratio is this?

It is the ratio between the distance to the observation plane, and the largest size in the object plane.

blue_leaf77 said:Anyway, I'm not quite sure if what I think is the same as you, when you say using FT, did you actually transform the field aperture profileonly? Such FT relation only holds within the far-field region in which the inequality ##z>>2D^2/\lambda## must be true.

If that's not sufficiently fulfilled, then your experimental conditions might still be in the paraxial/Fresnel region, in which case the FT relation should be performed w.r.t. the aperture profilemultipliedby a quadratic phase in the aperture plane ##\exp{\frac{jk}{2z}(\eta^2+\zeta^2)}##. That is, what must be transformed is not only the aperture profile, but there is an additional phase factor. In either case, knowledge of wavelength is necessary.

Here is the equation that I transformed.

##E\left(x, y, z\right) = \frac{e^{ikz}}{i \lambda z} e^{i \frac{\pi}{\lambda z}\left(x^2 + y^2\right)}\mathcal{F}\left[E\left(x', y', 0\right) e^{i\frac{\pi}{\lambda z}\left(x'^2 + y'^2\right)}\right] ##

where ##x## and ##y## are the coordinates for the output plane, and ##x'## and ##y'## for the input plane. And ##\lambda = 700 \times 10^{-9}## meters.

blue_leaf77 said:What about simulating a more realistic rectangular aperture?

This is the square aperture with 200 pixels on each side, or 0.04 meters on each side.

This is the Angular Spectrum result:

And this is the Fourier Transform:

It's somehow the same with the slit. The diffraction pattern is smaller for the Fourier Transform.

- #8

blue_leaf77

Science Advisor

Homework Helper

- 2,637

- 786

In any case, the Fresnel integral can be shown to coincide with the angular spectrum method provided paraxial approximation is valid within the problem considered, I don't see any reason why the two ways should differ even in scaling.

- #9

ecastro

- 254

- 8

blue_leaf77 said:Now what program do you use to calculate those images, MATLAB? If yes did you calculate the FT using the build-in function such as fft? As far as I know, when you use such build-in function, the coordinate scaling in the target plane is already determined by the function, therefore the scaling on the coordinate on both pictures above may be different, can you also check this?

I did use MatLab, and used the built-in function fft2. Does that mean that I need to code my own function for the Fourier Transform to remove the difference? How can I check the scaling on the coordinates?

- #10

blue_leaf77

Science Advisor

Homework Helper

- 2,637

- 786

If you want to stay using fft2 then you need to know and show the coordinates in the target plane. For this purpose I found this link http://www.gaussianwaves.com/2014/0...b-fft-of-basic-signals-sine-and-cosine-waves/ can be hepful especially example no.4.

- #11

ecastro

- 254

- 8

My sampling rate is the same size as the image, and according to MatLab, the fft2 of x can also be computed by:

fft2 = fft(fft(x).').'

I tried this one by putting the sampling rate on both Fourier Transform syntax, but still no changes. The size of the image (500 x 500 pixels) is still the same after the transform. Is it possible that the raw equation is the problem, not the Fourier Transform?

- #12

blue_leaf77

Science Advisor

Homework Helper

- 2,637

- 786

The Angular Spectrum Method (ASM) is a mathematical technique used to analyze and manipulate signals in the form of waves or frequencies. It is commonly used in fields such as optics, acoustics, and signal processing.

The Fourier Transform is a mathematical operation that breaks down a signal or function into its individual frequency components. It is used to analyze and manipulate signals in the frequency domain, rather than the time domain.

The Angular Spectrum Method is often used in conjunction with the Fourier Transform. It uses the Fourier Transform to decompose a signal into its frequency components, and then applies mathematical operations to manipulate the signal in the frequency domain before using the inverse Fourier Transform to convert it back to the time domain.

The Angular Spectrum Method and Fourier Transform have several advantages, including the ability to accurately analyze and manipulate signals in the frequency domain, which can be useful in fields such as optics and acoustics. They also allow for the removal of unwanted noise or frequency components from a signal, and can be used to reconstruct a signal from its frequency components.

One limitation of the Angular Spectrum Method and Fourier Transform is that they assume signals are linear, meaning that they follow a predictable pattern. This means they may not be as effective in analyzing complex, non-linear signals. Additionally, the accuracy of the results can be affected by noise and other external factors.

- Replies
- 1

- Views
- 3K

- Replies
- 3

- Views
- 4K

- Replies
- 1

- Views
- 2K

- Replies
- 17

- Views
- 2K

- Replies
- 1

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Replies
- 2

- Views
- 2K

- Replies
- 2

- Views
- 1K

- Replies
- 2

- Views
- 4K

- Replies
- 1

- Views
- 987

Share: