Below are the 2D patterns used in this activity:
Square
Circular annulus
square annulus
two slits
two dots
Circular annulus
square annulus
two slits
two dotsBelow are the respective FT of the 2D patterns shown above:
FT of Square
The FT of a square aperture is a Bessel function intensity symmetric with x and y axes.
FT of circular annulus
The FT of a circular annulus is a Bessel function intensity that is radially symmetric with origin.
FT of square annulus
This is similar with that of the square pattern only that there are other far-side intensities and smaller amplitude intensity at the origin.
FT of two slits along y direction
The FT is at the opposite direction of the aperture. That is in this example, the slit is along the y direction but the FT plot is along the x-direction. Also, this FT is similar to the square annulus aperture missing the slit along the x-direction.
FT of two dots
The FT of the two dots along the x-axis is just like a french fries that is oriented along the y-axis. Also, it can be observed that the vertical lines are not purely black and white stripes. It looks like the edges are grayscaling. I suspect that these are somewhat sinusoids in y-axis direction.
FT of SquareThe FT of a square aperture is a Bessel function intensity symmetric with x and y axes.
FT of circular annulusThe FT of a circular annulus is a Bessel function intensity that is radially symmetric with origin.
FT of square annulusThis is similar with that of the square pattern only that there are other far-side intensities and smaller amplitude intensity at the origin.
FT of two slits along y directionThe FT is at the opposite direction of the aperture. That is in this example, the slit is along the y direction but the FT plot is along the x-direction. Also, this FT is similar to the square annulus aperture missing the slit along the x-direction.
FT of two dotsThe FT of the two dots along the x-axis is just like a french fries that is oriented along the y-axis. Also, it can be observed that the vertical lines are not purely black and white stripes. It looks like the edges are grayscaling. I suspect that these are somewhat sinusoids in y-axis direction.
Sinusoidal waves pattern
sine pattern
FT of sine pattern f = 4
The FT of the sine pattern oriented horizontally with freq = 4 is represented by two dots along the y axis.
FT of sine pattern f = 6
FT of sine pattern f = 8
As the frequency of the pattern is increased, the separation between the two dots also increased. We can say that the FT is symmetric with the x-axis since our sine pattern is oriented along the x direction. Also, we can say that both x and y axes are frequency units.
sine pattern with bias y + 1
FT of sine pattern with bias y + 1
Observing the FT of the sine pattern above with see that the original sine wave has frequency = 8 which is represented by the previous result. There is an additional dot at the center of the plot representing 0 frequency. Thus we say that the FT has a property of linear combination since the FT plot shows the FT of sine with f = 8 and FT of the bias with obviously with f = 0 since it does not repeat.
rotated sine pattern theta = 30
FT of rotated sine pattern theta = 30
When the vertically oriented sine pattern is tilted by theta = 30, the spot concentration in the FT plot also tilts with respect to the x axis and the spot spreads along the y direction.
sine product f = 4
FT of the above pattern
We can see that the FT follows the superposition since the FT displays that of two sine patterns oriented in opposite direction with same f = 4.
sine product pattern f = 4, f = 6
FT of the above pattern
This is just the same as the previous pattern the difference is just there are two distinct frequencies comprising the mat pattern
sine product plus two rotated sines theta = 30 & 60 f = 4
FT of the above pattern
The above example is a combination of added sine patterns of the same freq along x and y direction and two rotated sines with theta = 30 and 60.
FT of sine pattern f = 4The FT of the sine pattern oriented horizontally with freq = 4 is represented by two dots along the y axis.
FT of sine pattern f = 6
FT of sine pattern f = 8As the frequency of the pattern is increased, the separation between the two dots also increased. We can say that the FT is symmetric with the x-axis since our sine pattern is oriented along the x direction. Also, we can say that both x and y axes are frequency units.
sine pattern with bias y + 1
FT of sine pattern with bias y + 1Observing the FT of the sine pattern above with see that the original sine wave has frequency = 8 which is represented by the previous result. There is an additional dot at the center of the plot representing 0 frequency. Thus we say that the FT has a property of linear combination since the FT plot shows the FT of sine with f = 8 and FT of the bias with obviously with f = 0 since it does not repeat.
rotated sine pattern theta = 30
FT of rotated sine pattern theta = 30When the vertically oriented sine pattern is tilted by theta = 30, the spot concentration in the FT plot also tilts with respect to the x axis and the spot spreads along the y direction.
sine product f = 4
FT of the above pattern
We can see that the FT follows the superposition since the FT displays that of two sine patterns oriented in opposite direction with same f = 4.
sine product pattern f = 4, f = 6
FT of the above pattern
This is just the same as the previous pattern the difference is just there are two distinct frequencies comprising the mat pattern
sine product plus two rotated sines theta = 30 & 60 f = 4
FT of the above patternThe above example is a combination of added sine patterns of the same freq along x and y direction and two rotated sines with theta = 30 and 60.
I give myself 10 points for finishing this activity. Its fun!
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