2.1. Light Propagation in Free Space

2. Groundwork

2.1. Light Propagation in Free Space

Here we review the propagation of light in free space and Fourier optics
Review the solutions of wave propagation in free space
Helmholtz equation describes the propagation of field emitted by a source s 1
U is the scalar field, a function of position r and frequency , is the vacuum wavenumber (or propagation constant),
To solve Eq. 1, need to specify whether the propagation takes place in 1D, 2D, or 3D, and take advantage of any symmetries of the problem
The fundamental equation associated with Eq. 1, which provides the Green’s function of the problem, is obtained by replacing the source term with an impulse function, i.e. Dirac delta function
Fundamental equation is then solved in the frequency domain. Below, we derive the well known solutions of plane and spherical waves, for the 1D and 3D propagation, respectively.

2.1.1. 1D Propagation: Plane Waves.

For 1D, fundamental equation is

g is Green’s function, and is the 1D delta-function, which in 1D describes a planar source of infinite size placed at the origin.
Taking the Fourier transform of Eq. 2 gives an algebraic equation

using the differentiation theorem of Fourier transforms,

Thus, the frequency domain solution is simply

To find Green’s function, we Fourier transform back to the spatial domain.

Note that is shifted by. Thus, use the Fourier transform of and the shift

Combining Eqs. 4 and 5, we obtain

Equation 6 gives the solution to wave propagation, as emitted by the infinite planar source at the origin. Ignoring the prefactor , phase shift, the solution through 1D space, , is

is a superposition of two counter-propagating waves, of wavenumbers and . If we prefer to work with the complex analytic signal associated with , we suppress the negative frequency component

In this case, the complex analytic solution is

We arrived at the representation of the plane wave solution of the wave equation, and established that it can be regarded as the complex analytic signal associated with the real field in Eq. 7.
Since the wave equation contains second order derivatives in space, both , and their linear combinations are all valid solutions. Green’s function in Eq. 7 is such a linear combination. Thus, the amplitude of the plane wave is constant, but its phase is linearly increasing with the propagation distance.
Finally, we note that, if the propagation direction is not parallel to one of the axes, the plane wave equation takes the general form

is the direction of propagation, . The phase delay at a point described by position vector is =, is the component of k parallel to r.

2.1.2. 3D Propagation: Spherical Waves.

Find the Green’s function associated with the vacuum propagation, or the response of free-space to a point source. The fundamental equation becomes

represents a 3D delta function. Fourier transform Eq. 10 and use the relationship

Equation 3 readily yields the solution in the representation

g depends on the modulus of k and not its orientation; the propagation is isotropic.
Eq. 12 looks similar to its analog in 1D (Eq. 4) except that the x component of the wave vector is now replaced by the modulus of the wave vector, k.
In order to obtain the spatial domain solution, , we Fourier transform Eq. 12 back to spatial domain.
For propagation in isotropic media, the problem is spherically symmetric and the Fourier transform of Eq. 12 can be written in spherical coordinates as a 1D integral

Expanding the first factor under the integrand and expressing in exponential form,

The integral in Eq. 14 is a 1D Fourier transform of a function similar to Eq. 4. Similarly, we obtain

This is the spherically symmetric wave propagation from a point source, i.e. a spherical wave
The complex analytic signal associated with this solution is (up to an unimportant constant, 1/2i)

Knowing the Green’s function associated with free space (Eq. 16) we can easily calculate the response to an arbitrary source via a convolution,

Equation 17 is the essence of the Huygens principle, which establishes that, upon propagation, points set in oscillation by the field become new sources, and the emerging field is the summation of spherical wavelets emitted by all these point sources. Each point reached by the field becomes a secondary source, which emits a new spherical wavelet and so on.
The integral in Eq. 17 is difficult to evaluate. In the following section, we study the propagation of fields at large distances from the source, i.e. in the far zone (we will use the phrase “far zone” instead of “far field” to avoid confusions with the actual complex field).

2.2. Fresnel Approximation of Wave Propagation.

Next we describe useful approximations of the spherical wave.
When the propagation distance along one axis is far greater than along the other two axes, the spherical wave can be first approximated by

Using the fact that the amplitude attenuation is a slow function of x and y, when , such that .
However, the phase term is significantly more sensitive to x and y variations, such that the next order approximation is needed.
Expanding the radial distance in Taylor series, we obtain

The spherical wavelet is now approximated by

Equation 20 is the Fresnelapproximation of the wave propagation.
The region of distance z where this approximation holds is called the Fresnel zone. The transverse (x-y) dependence is a quadratic phase term.

For a given planar (x-y) field distribution at , calculate the resulting propagated field at distance z, by simply convolving U with the Fresnel wavelet,

Equation 21 is theFresnel diffraction equation, which explains the field propagation using Huygens’ concept of secondary point sources, except that now each of the secondary point sources emits Fresnel wavelets
Although a drastic approximation, the Fresnel equation captures the essence of many important phenomena in microscopy. The prefactor is typically neglected, as it contains no information about the x-y field dependence

2.3. Fourier Transform Properties of Free Space.

The Fraunhofer approximation can be made if the observation plane is farther away. For, the quadratic phase terms can be ignored in Eq. 21,

The field distribution in the Fraunhofer regionis obtained from Eq. 21 as

For a particular direction of propagation, , and analogously for ky.We can re-write the Fourier transform in Eq. 23 as

Equation 24 establishes that upon long propagation distances, the free space performs the Fourier transform of a given field
The spatial frequency of the input field is associated with the propagation of a plane wave along the direction of the wavevector .
The Fraunhofer regime is the situation where different plane waves are generated by different spatial frequencies of the input field. If distance z is large enough, the propagation angles corresponding to different spatial frequencies do not mix.
This allows us to solve many problems of practical interest with extreme ease, by invoking various Fourier transform pairs and their properties

Example 1. Diffraction by a sinusoidal grating.

Consider a plane wave incident on a one-dimensional amplitude grating of transmission , with the period of the grating
In the far-zone, we expect the diffraction pattern:

or, changing notations:

Since the Fourier transform of a cosine function is a sum of two delta functions,

This shows that the diffraction by a sinusoidal grating generates two distinct off-axis plane waves in the far zone and another along the optical axis
This result isused in solving inverse problems. Consider a 3D object, whose spatial distribution of refractive index can be expressed as a Fourier transform. If the object is weakly scattering, such that the frequencies do not mix, measuring the angularly scattered light from the object reveals the entire structure of the object. This solution relies on each angular component reporting on a unique spatial frequency

2.4. Fourier Transformation Properties of Lenses.

Lenses have the capability to perform Fourier transforms, much like free space, without the need for large distance of propagation
Consider the biconvex lens in Fig. 8. We want to determine the effect that the lens has on an incident plane wave. This effect can be incorporated via a transmission function of the form

. 28

The problem reduces to evaluating the phase delay produced by the lens as a function of the off-axis distance, or the polar coordinate

and are the phase shifts due to the glass and air portions, is the wavenumber in air, , is the thickness along the optical axis, is the thickness at distance r off axis, and n is the refractive index of the glass.

can be expressed as

and are the segments shown in Fig. 8, calculated using simple geometry
For small angles, ABC becomes a right triangle, the following identity applies

Since , , and ,

can be expressed from Eq. 30,

With this, the phase distribution in Eq. 29 becomes

In Eqs. 33-34, we used the geometrical optics convention whereby surfaces with centers to the left (right) are considered of negative (positive) radius; and .
We recognize that the focal distance associated with a thin lens is given by

Such that Eq. 34 becomes

The lens transmission function, which establishes how the plane wave field right before the lens is transmitted right after the lens, has the form

Eq. 37 shows that the effect of the lens is to transform a plane wave into a parabolic wavefront.
The negative sign denotes a convergent field, the positive sign marks a divergent field.
Comparing Eqs. 27 and 37, the effect of propagation through free space is qualitatively similar to transmission through a thin divergent lens(Fig. 10).

Consider the field propagation through a combination of free space and convergent lens.
Deriving an expression for the output field as a function of input field can be broken down into a Fresnel propagation over distance , followed by a transformation by the lens of focal distance f, and, finally, a propagation over distance .
Earlier we found that the Fresnel propagation can be described as a convolution with the quadratic phase function. Thus the propagation can be written,

These calculations aretedious. A simplification arises when

If the input field is at the front focal plane and the output field is observed at the back focal plane, the two fields are related via a Fourier transform

This result establishes a simple, powerful way to compute analog Fourier transforms

Example 2. Sinusoidal transmission grating.

Revisit Example 1, where we studied the diffraction by a sinusoidal grating.
Now the grating is placed in the focal plane object of the lens and the observation is in the focal plane image
The Fourier transform of the 1D grating transmission function is

A thins lens can generate the diffraction pattern of a grating just as free space can.

2.5Born approximation of light scattering in inhomogeneous media

In many situations, light interacts with inhomogeneous media, in which case the light-matter interaction is scattering. If the wavelength of the field is unchanged, then it’selastic light scattering.
The goal in light scattering experiments is to solve the scattering inverse problem.This problem can be solved analytically if we assume weakly scattering media.
Recall the Helmholtz equation

 is the propagation constant, or wavenumber.
Equation 42a can be re-arrangedto show the inhomogeneous term on the right side, showing the medium acts as a secondary source. Thus, the field satisfies

is the scattering potential associated with the medium.
Equation 43 shows the inhomogeneous portion of the refractive index as a secondary source of (scattered) light.
The fundamental equation that yields Green’s function,, has the form

We solve the equation by Fourier transforming it with respect to spatial variable and arrive at the well known spherical wave solution

The solution for the scattered field is a convolution between the source term and the Green function

This integral can be simplified if we assume that the measurements are in the far-zone, i.e. . We use the following approximation

In Eq. 47, is the scalar product of vectors r and and is the unit vector associated with the direction of propagation.
With this far-zone approximation, Eq. 46 can be re-written as

Thus, far from the scattering medium, the field behaves as a spherical wave which is perturbed by the scattering amplitude, defined as

To solve the integral in Eq. 49, we assume the scattering is weak, which allows us to expand .
The first orderBorn approximation assumes that the field inside the scattering volume is constant and equal to the incident field, assumed to be a plain wave

Plugging this into Eq. 49, we obtain for the scattering amplitude

The right hand side is a 3D Fourier transform.
Within the first Born approximation, measurements of the field scattered at a given angle give access to the Fourier component of the scattering potential F,

q is the difference between the scattered and incident wavevectors, sometimes called scattering wavevector and in quantum mechanics referred to as the momentum transfer.
From Fig. 14 it can be seen that , with the scattering angle.

Due to the reversibility of the Fourier integral, Eq. 52can be inverted to provide the scattering potential,

Equation 53 establishes the solution to the inverse scattering problem.
Equivalently, measuring U at a multitude of scattering angles allows the reconstruction of F from its Fourier components. For far-zone measurements at a fixed distance R, the scattering amplitude and the scattered field differ only by a constant , and, thus, can be used interchangeably.

In order to retrieve the scattering potential experimentally, two conditions must be met:
The measurement has to provide the complex scattered field
The scattered field has to be measured over an infinite range of spatial frequencies q (i.e. the limits of integration in Eq. 43 are to ).

Great progress has been made recently in terms of measuring phase information. Nevertheless, most measurements are intensity-based. It is important to realize that if one only has access to the intensity of the scattered light, , then the autocorrelation of and not F itself is retrieved,

This result is simply the correlation theorem applied to 3D Fourier transforms

We have only experimental access to a limited frequency range, or bandwidth. The spatial frequency coverage (range of momentum transfer) is intrinsically limited.
Specifically, for a given incident wave vector ki, with , the highest possible q is obtained for backscattering, .
Similarly for an incident wave vector in the opposite direction, , the maximum momentum transfer is also .

Altogether, the maximum frequency coverage is , as illustrated in Fig. 16.
As we rotate the incident wavevector from to , the respective backscattering wavevector rotates from to , such that the tip of q describes a sphere of radius .
This is known as the Ewald sphere, or Ewald limiting sphere.
Let us study the effect of this bandwidth limitation in the best case scenario of the entire Ewald’s sphere coverage.
The measured (i.e. truncated in frequency) field, , can be expressed as

Using the definition of the “ball” function, defined as , we can re-write Eq. 55 as

,56

Thus, the scattering potential retrieved by measuring the scattered field U can be obtained viathe 3D Fourier of Eq. 56

is the Fourier transform of the ball function and has the form

Even in the best case scenario, full Ewald sphere coverage, the reconstructed object is a “smooth” version of the original object, with smoothing function , a radially symmetric function of radial coordinate r’.
Covering the entire Ewald sphere requires illuminating the object from all directions and measuring the scattered complex field over the entire solid angle for each illumination direction. In practice, fewer measurements are performed, at the expense of degrading resolution.
Figure 17 depicts the 1D profile of the 3D function.The FWHM of this function, , is given by , or .
We conclude that the best achievable resolution in reconstructing the 3D object is approximately , a factor of 2 below the common half-wavelength limit. This factor of 2 gain is due to illumination from both sides.

2.6. Scattering by single particles.

By particle, we mean a region in space that ha a dielectric permeability , which differs from that of the surrounding medium(Fig. 18)
The field scattered in the far zone has the form of a perturbed spherical wave,

is the scattered vector field at position r, , and defines the scattering amplitude. f is physically similar to that encountered above except that here it also includes polarization information
The differential cross-section associated with the particle is defined as

and are the Poynting vectors along the scattered and initial direction, respectively, with moduli
From Eq. 60, is only defined in the far-zone. Thus the differential cross-section equals the modulus squared of the scattering amplitude,

The unit for is .
Onecase is obtained for backscattering, when ,

is referred to as the backscattering cross section.
The normalized version of defines the phase function,

The phase function p defines the angular probability density function associated with the scattered light (“phase function” was borrowed from nuclear physics and does not refer to the phase of the field).

The denominator of Eq. 63 defines the scattering cross section.

.64

The unit of the scattering cross section is .
If the particle also absorbs light, we can define an absorption cross section, the attenuation due to the combined effect is governed by a total cross section,

For arbitrary particles, deriving expression for the scattering cross sections, and , is difficult. If simplifying assumptions can be made, the problem becomes tractable

2.7. Particles under the Born approximation

When the refractive index of a particle is only slightly different from that of the surrounding medium, its scattering properties can be derivedwithin the framework of the Born approximation.
The scalar scattering amplitude from such a particle is the Fourier transform of the scattering potential of the particle (Eq. 52)