Fundamentals of Diffraction and Image Formation

# Properties of lenses in relation to the Fourier transform

## Introduction

In what follows, we will assume a monochromatic illumination and we will be interested in the amplitude distribution of light in the image focal plane of the lens. Three cases will be examined depending on the location of the object by respect to the lens.

## Case where the object is located right near the lens

Let us consider an plane object of amplitude transmittance to(x,y) located right before a converging lens (see figure IV-4). Image IV-4- Object placed right in front of the lens [zoom...]Info

We assume that the object is uniformly illuminated with a plane wave of amplitude A at normal incidence. In the plane tangent to the lens side we can write:

The field distribution right after the back of the lens thus becomes:

where, to take into account the finite size of the lens, we define the pupil function:

In order to determine the field distribution at the lens focal plane, we apply the Fresnel diffraction formula with z=f (Eq. III-2) :

By letting aside the constant phase factor exp(jkf) and by replacing U'L(x,y) with its value, we notice that the quadratic phase terms inside the integral are compensating one other:

If the object spatial dimensions are smaller than the lens aperture (see figure IV-5), the factor P(x,y) disappears from the equation (since the field is equal to zero where the object is opaque, and therefore  U L (x,y) P(x,y)=UL (x,y) :

IV-3 Info Image IV-5- Object placed right before the lens [zoom...]Info
• We note that the complex amplitude at the focal plane is the Fraunhofer diffraction figure of the incident field in front of the lens; therefore, using a lens allows to observing Fraunhofer diffraction figure at observation distances far smaller than the ones defined in chapter 3 (here, the observation distance is equal to f).

• Apart from some multiplying factors in front of the integral, the previous expression (IV-3) is therefore a simple Fourier transform  : • The quadratic phase term in front of the FT expresses what we call a “phase curvature”.

• Generally, we are interested in If, the intensity distribution called “power angular spectrum” or “spectral energy” of the object:

## Object located before the lens

Let us consider an object characterized by it amplitude transmittance   t0 (x,y) and located at a distance d in front of the lens (see figure IV-6). Image IV-6- Object located at a distance d in front of the lens [zoom...]Info

The previous setup is a particular case of this one (for d=0). We assume that the object is uniformly illuminated with a plane wave of amplitude A at normal incidence.

We define : and . We know that the transfer function of the propagation phenomenon allows calculating starting from (Eq. II-12 in the section : Scalar theory of diffraction) :

The lens will be considered in the paraxial approximation, therefore the rays are only slightly inclined with respect to the optical axis: and .

We make a Taylor expansion of the square root in the previous expressions and we obtain:

IV-4 Info is a constant phase term that can be omitted in what follows. In addition, we already characterized the propagation of UL (x,y) between a plane located right near the lens and the focal plane. It is described in the previous paragraph (Eq. IV-3):

IV-5 Info

By applying the propagation transfer function (Eq. IV-4), we obtain:

In the exponent term, we factorize the term 1/(λf)², simplify by λ, and introduce :

Therefore the expression (IV-5) becomes:

By regrouping the exponential functions we find:

or again:

IV-6 Info

We note that a phase factor precedes the FT of the object. This phase factor vanishes for d=f (if the object is placed in the front focal plane of the lens). In that case, the phase quadratic curvature disappears, leading to an exact Fourier transform.

## Object located after the lens

The lens is illuminated with a monochromatic plane wave of amplitude A and at normal incidence (same illumination conditions as in the previous paragraphs). The object is therefore illuminated with a spherical wave converging towards the focal plane of the lens in the image space (see figure IV-7). Image IV-7- Object located after the lens at a distance d from the back focal plane [zoom...]Info

Basing ourself on figure IV-8, Image IV-8- Simplified schematic view of figure IV-7. Definition of the variables involved in the calculation [zoom...]Info

A simple calculation done in the geometric optics framework with the paraxial approximation shows that:

• the amplitude at the object location is: .

• the finite extension of the illuminated area, intersection between the object plane and the light cone can be represented by the effective pupil function: P(xL,yL)=P(xf/d,yf/d).

• the spherical wave illuminating the object can be written:

Using those conditions, the complex amplitude Uo(x,y) of the field transmitted by the object can be written:

IV-7 Info

By applying the Fresnel diffraction formula (Eq. III-2) to go from the object plane to the focal plane (z=d), we find an expression of the amplitude in that latter plane:

By replacing Uo(x,y) (Eq. IV-7) by its value, we notice that the quadratic phase term inside the integral vanishes.

Except for the quadratic phase term in front of the integral, the amplitude distribution at the focal plane is therefore the FT of the object part which is limited by the projection of the lens aperture at the object plane. This result is the same as the one obtained in the case where the object is located right near the lens. However, here, the experimentalist can easily modify the FT scale.

By increasing d, he will increase the FT dimensions until the object gets very near to the lens (d=f) . This property can be useful in spatial filtering applications.