Scalar theory of diffraction EE 231 Introduction to Optics Scalar theory of diffraction Lesson 12 Andrea Fratalocchi www.primalight.org

The Scalar theory of diffraction The interaction problem Homework 1: calculate the transfer function of a thin lens

The Scalar theory of diffraction The interaction problem Homework 1: calculate the transfer function of a thin lens From the definition of the transfer function

The Scalar theory of diffraction The interaction problem Homework 1: calculate the transfer function of a thin lens From the definition of the transfer function Input wave: Output wave:

The Scalar theory of diffraction The interaction problem Homework 1: calculate the transfer function of a thin lens From the definition of the transfer function In paraxial approximation:

The Scalar theory of diffraction The interaction problem The transfer function of a thin lens depends only on one degree of freedom, the focal length f

The Scalar theory of diffraction The interaction problem Application of lenses Far field observation According to the Far-Field approximation, the Far-Field is observed when

The Scalar theory of diffraction The interaction problem Application of lenses Far field observation At a wavelength of 500 nm, and for an object of area of 1mm^2, we have:

The Scalar theory of diffraction The interaction problem Is there a better way to visualize the Far-Field? Application of lenses Far field observation At a wavelength of 500 nm, and for an object of area of 1mm^2, we have: This distance is quite large, especially in a lab where the characteristic lengths of optical tables are 2-3 meters

The Scalar theory of diffraction The interaction problem Application of lenses Exercise 1: given the input field Vi impinging on a thin lens of focal f Far field observation Calculate the output intensity on the focal plane at z=f

The Scalar theory of diffraction Exercise 1: given the input field Vi impinging on a thin lens of focal f The interaction problem Application of lenses Far field observation Calculate the output intensity on the focal plane at z=f

The Scalar theory of diffraction Exercise 1: given the input field Vi impinging on a thin lens of focal f The interaction problem Application of lenses Far field observation Calculate the output intensity on the focal plane at z=f This simplifies into:

The Scalar theory of diffraction Exercise 1: given the input field Vi impinging on a thin lens of focal f The interaction problem Application of lenses Far field observation Calculate the output intensity on the focal plane at z=f Fourier Transform of the input field!

The Scalar theory of diffraction Exercise 1: given the input field Vi impinging on a thin lens of focal f The interaction problem Application of lenses Far field observation Calculate the output intensity on the focal plane at z=f The lens visualizes a scaled version of the Fourier transform of the input field

The Scalar theory of diffraction The interaction problem Application of lenses Problem: the lens visualizes the Fourier transform in Intensity, but the amplitude of the output field has a spherical curvature. This implies that the Fourier transform is imaged on a sphere. This is not the most desirable scenario when more lenses are employed, as the spherical term propagates as well How to address this issue? Far field observation

The Scalar theory of diffraction The interaction problem Application of lenses Exercise 2: calculate the output field Vo Use this configuration

The Scalar theory of diffraction The interaction problem Application of lenses Exercise 2: calculate the output field Vo Use this configuration Simple proportionality! In optics, the Fourier transform of a field is an experimental observable

The Scalar theory of diffraction This is the most basic setup for the elaboration of optical signal in free space The interaction problem Application of lenses 2. 4-f optical systems Exercise: calculate the output field of the system The diffractive object acts as a filter with transfer function With this setup, we can design the transfer function in real space!

The Scalar theory of diffraction The interaction problem Application of lenses 3. Collimation and focusing of Gaussian beams Exercise 3: calculate the emerging field from the lens. Is the emerging field still Gaussian?

The Scalar theory of diffraction The interaction problem Exercise 3: calculate the emerging field from the lens. Is the emerging field still Gaussian? Application of lenses 3. Collimation and focusing of Gaussian beams

The Scalar theory of diffraction The interaction problem Exercise 3: calculate the emerging field from the lens. Is the emerging field still Gaussian? Application of lenses 3. Collimation and focusing of Gaussian beams The output beam is still Gaussian, but with a different curvature

The Scalar theory of diffraction The interaction problem Exercise 4: calculate the ABCD matrix of the lens Application of lenses 3. Collimation and focusing of Gaussian beams

The Scalar theory of diffraction The interaction problem Exercise 4: calculate the ABCD matrix of the lens Application of lenses 3. Collimation and focusing of Gaussian beams

The Scalar theory of diffraction The interaction problem The Gaussian beam emerging from the lens is collimated, i.e., has in the lens plane the waist plane of minimum spot-size Application of lenses 3a. Collimation of Gaussian beams What are the conditions for this type of interaction?

The Scalar theory of diffraction The Gaussian beam emerging from the lens is collimated, i.e., has in the lens plane the waist plane of minimum spot-size The interaction problem Application of lenses 3a. Collimation of Gaussian beams What are the conditions for this type of interaction?

The Scalar theory of diffraction The emerging Gaussian beam is focused at a point located at z=z' measured from the lens The interaction problem Application of lenses 3b. Focusing of Gaussian beams How to find the new plane of minimum waist?

The Scalar theory of diffraction The emerging Gaussian beam is focused at a point located at z=z' measured from the lens The interaction problem Application of lenses 3b. Focusing of Gaussian beams How to find the new plane of minimum waist? This is an inverse problem of Gaussian beams

The Scalar theory of diffraction The emerging Gaussian beam is focused at a point located at z=z' measured from the lens The interaction problem Application of lenses 3b. Focusing of Gaussian beams How to find the new plane of minimum waist? At z=0, we have By solving an inverse problem, we can calculate the position of the new waist of the emerging beam

The Scalar theory of diffraction The emerging Gaussian beam is focused at a point located at z=z' measured from the lens The interaction problem Application of lenses 3b. Focusing of Gaussian beams How to find the new plane of minimum waist? At z=0, we have By solving an inverse problem, we can calculate the position of the new waist of the emerging beam

The Scalar theory of diffraction The interaction problem Homework 1: the operations of focusing and collimation are often used together in the following setup Lens A focus, while lend B collimates the beam. This is a Beam expander. Write the input output relationship of the system and the magnification factor

The Scalar theory of diffraction References A. Yariv, Optical electronics in modern communications, Chapter 2.