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Plan for today: 1) Fourier Op5cs “Recipe” and free space propaga5on as a convolu5on

Image from Fundamentals of Photonics 2nd Edi5on by Bahaa E. A. Saleh & Malvin Carl Teich


-‐3) if you are given U(x,y,z), can you find f(x,y) and g(x,y) without the Fourier Recipe ?

-‐2) Write down the “Fourier Recipe” one uses to find the output complex amplitude g(x,y) given the input image func5on f(x,y) (i.e. the input complex amplitude) … when do you need this Fourier Recipe? -‐1) What is f(x,y) for the following cases:

a) A plane wave traveling along z

c) A Gaussian beam traveling along z and coming to a focus at z=0 b) A plane wave traveling at an angle theta w.r.t. z-axis

c) The wave in (b) hitting a plate with long slit aligned along the x-axis



-‐3) if you are given U(x,y,z), can you find f(x,y) and g(x,y) without the Fourier Recipe ?

-‐2) Write down the “Fourier Recipe” one uses to find the output complex amplitude g(x,y) given the input image func5on f(x,y) (i.e. the input complex amplitude) … when do you need this Fourier Recipe? -‐1) What is f(x,y) for the following cases:

a) A plane wave traveling along z

c) A Gaussian beam traveling along z and coming to a focus at z=0 b) A plane wave traveling at an angle theta w.r.t. z-axis

c) The wave in (b) hitting a plate with long slit aligned along the x-axis

f(x,y) = U(x,y,z=0) and g(x,y) = U(x,y,z=d) YES !



The Fourier Recipe 1) Find image function f(x,y) = E(x,y) = U(x,y,z=0)




2) Find the spectrum of the image function





3) Multiply spectrum by the transfer function





3) Multiply spectrum by the transfer function free space transfer function





3) Multiply spectrum by the transfer function free space transfer function

4) Take inverse Fourier transform of result


0) A monochroma5c plane wave is propaga5ng thru free space from z=0 to z=d, as shown below. Find f(x,y) at z=d using 2 methods (one method is Fourier Op5cs)

(alpha is a constant)

0a) Use the Fourier Op5cs Recipe to find f(x,y) at z=d. 0b) what is the other easier way to find f(x,y) at z=d?


(alpha is a constant)

0a) Use the Fourier Op5cs Recipe to find f(x,y) at z=d. 0b) what is the other easier way to find f(x,y) at z=d?

recall, the defini8on of the delta func8on is


(alpha is a constant) Step 1: Find E(x,y) = U(x,y,z=0)

Step 2: Take the Fourier transform of image function

Step 3: Multiply spectrum by the transfer function

Step 4: Take inverse Fourier transform of result







0b) what is the other easier way to find f(x,y) at z=d? U(x,y,z) is given, so just plug z=d into it!


BONUS) What happens for e-‐field paXerns with transverse spa5al periods smaller than ? Slit width : w

… like at the edge of a slit

a)  The electric field just doesn’t change that fast b)  Something weird happens near the edge but then

doesn’t show up in images…

BONUS) What happens for e-‐field paXerns with transverse spa5al periods smaller than ?

IF then

is imaginary and that component decays to zero exponentially as it propagates! This is an “evanescent wave.”

Slit width : w At slit

BONUS) What happens for e-‐field paXerns with transverse spa5al periods smaller than ?

IF then

is imaginary and that component decays to zero exponentially as it propagates! This is an “evanescent wave.”

Slit width : w

F=1

F=10

F=100

Fresnel number

Distance from slit : L

At slit

U(x,y,z) z=d z=0

2) Jump to frequency space

3) Mul5ply by transfer func5on

4) Jump back to real space 1) Find image func5on

free space transfer function

U(x,y,z) z=d z=0

I.e. can we define some “system transfer func9on” for an arbitrary Op9cal System? with lenses, mirrors, etc...

System transfer func8on



4) Jump back to real space 1) Find image func5on

Does the Recipe WORK for an arbitrary optical system ???

free space transfer function

Let’s try to find out with the following example …

U(x,y,z)

z=d z=0



4) Jump back to real space

1) Find image func5on

B A

1) Assuming empty space between z=0 and z=d, suppose you insert a thin plate (with transmiXance t(x,y)) somewhere in between the planes z=0 and z=d. Describe in words how you would account for it in the recipe when the plate is inserted at: A) (i.e. at z=d)

B) (i.e. at z=0)

C) (i.e. at z=d/2)

C

For free space propaga8on

U(x,y,z)

z=d z=0





B A


B) (i.e. at z=0)

C) (i.e. at z=d/2)

C


Recipe stays the same, but the output wave becomes

U(x,y,z)

z=d z=0





B A


B) (i.e. at z=0)

C) (i.e. at z=d/2)

C

The new image func5on is:



U(x,y,z)

z=d z=0





B A


B) (i.e. at z=0)

C) (i.e. at z=d/2)

C




The FT of the new image func5on is: the thin plate acts as a convolu8on in freq. space

U(x,y,z)

z=d z=0





B A


B) (i.e. at z=0)

C) (i.e. at z=d/2)

C




The FT of the new image func5on is:

Use recipe to propagate to d/2 and find U(x,y,d/2). Mul5ply this field by t(x,y)

the thin plate acts as a convolu8on in freq. space

U(x,y,z)

z=d z=0





B A


B) (i.e. at z=0)

C) (i.e. at z=d/2)

C




The FT of the new image func5on is:

Use recipe to propagate to d/2 and find U(x,y,d/2). Mul5ply this field by t(x,y) to get new image func5on at C (E(x,y) = U(x,y,d/2) t(x,y). Use recipe to propagate to z=d.

the thin plate acts as a convolu8on in freq. space

these two results are NOT, in general, the same… To see this, consider, for example, when t(x,y) is a delta func8on in x and y. Then in the first case E3 is a spherical wave and in the second case E3

is a delta func8on in x and y

U(x,y,z) z=d z=0






2a) The radical in this expression makes it difficult to evaluate. Simplify it by making the “Fresnel” approxima5on for the transfer func5on of free-‐space.

2b) Under what condi5ons is this approxima5on valid in a mathema5cal and/or physical sense?

U(x,y,z) z=d z=0








Hint:

U(x,y,z) z=d z=0








Hint:

should VANISH for large values of kx or ky.

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