Exploring the Input/Output Relationship of Fourier Optics

In summary, the conversation discusses the input/output relationship in Fourier Optics, specifically how to determine the complex amplitude at the output (g(x,y)) given a known complex amplitude at the input (f(x,y)). The conversation also mentions a specific integral that can be used to solve for g(x,y), and the difficulty the speaker is having with integrating the two exponentials. They also mention their success in solving the problem and ask for textbook recommendations on the topic.
  • #1
roeb
107
1

Homework Statement


I'm trying to figure out the input/output relationship for Fourier Optics:

If we have a plane at z = 0 with the complex amplitude f(x,y) = U(x,y,0) and the complex amplitude at the output g(x,y) = U(x,y,d) at z = d.

My question is if f(x,y) = 1, how do I explicitly determine what g(x,y) is? My book gives the following integral (among others)

g(x,y) = [tex]H_0 \int_{-\inf}^{\inf} \int_{-\inf}^{\inf} F(v_x, v_y) exp( j \pi \lambda d (v_x^2 + v_y^2)) exp(-j2 \pi (v_x x + v_y y)) dv_x dv_y[/tex]

I guess my main problem is that I'm not even sure what g(x,y) is supposed to look like, I know the Fourier transform of f(x,y) -> F(vx,vy) = delta(v_x) delta(v_y), but I'm not really sure what to do next. How do I integrate the two exponentials?

Does anyone have any textbook suggestions for this? Goodman doesn't seem to go in detail on this topic and I can't really find any other examples.

Thanks,
roeb
 
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  • #2
I was able to solve the problem,

Thanks for looking

-roeb
 

1. What is Fourier Optics?

Fourier Optics is a branch of optics that focuses on the study of the input/output relationship between light waves and optical systems. It uses mathematical concepts from the field of Fourier analysis to analyze and manipulate light waves.

2. How is Fourier Optics used in scientific research?

Fourier Optics is used in a wide range of scientific research, including in fields such as astronomy, microscopy, and medical imaging. It allows scientists to understand how light interacts with optical systems and to optimize these systems for specific applications.

3. What are some common applications of Fourier Optics?

Fourier Optics has many practical applications, such as in the design of optical devices like lenses and mirrors, in image processing and data compression, and in the analysis of diffraction patterns. It is also used in the development of technologies like holography and optical data storage.

4. How does Fourier Optics relate to other branches of optics?

Fourier Optics is closely related to other branches of optics, such as geometrical optics and physical optics. While geometrical optics focuses on the behavior of light as rays, and physical optics considers light as waves, Fourier Optics combines both approaches to study the input/output relationship of light waves in optical systems.

5. What are some future advancements in Fourier Optics?

Some potential future advancements in Fourier Optics include the development of new materials with unique optical properties, the use of advanced computational techniques to analyze and manipulate light waves, and the integration of Fourier Optics with other technologies to create new applications in fields like biotechnology and telecommunications.

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