Unit 5: Fresnel Diffraction and Holography

Fresnel's Assumptions

In Fresnel diffraction, the source of light and the screen are kept at finite distances from the diffracting obstacle. To explain the diffraction phenomena under these conditions, Augustin-Jean Fresnel made several key assumptions about how light propagates and interferes.

The Key Assumptions

• Division of Wavefront: A wavefront can be resolved or divided into a large number of small elements or zones called Fresnel's zones. The effect at any point on the screen is the resultant of the individual effects of these zones.
• Secondary Wavelets: Every point on a wavefront acts as a source of secondary disturbance, emitting secondary wavelets that spread out in all directions.
• Mutual Interference: The secondary wavelets originating from different parts of the wavefront interfere with each other to produce the resultant intensity at any point on the screen.
• Obliquity Factor: The effect of a secondary wavelet at a point is not uniform in all directions. It is maximum in the forward direction and decreases as the angle of obliquity increases. Mathematically, the amplitude of the secondary wavelet at a point is proportional to (1 + cos(theta)), where theta is the angle between the normal to the wavefront and the direction of propagation to the point. This factor is zero in the backward direction (where theta = 180 degrees), explaining why light does not propagate backward.

Definition: Obliquity Factor
The obliquity factor is given by the relation: K(theta) = (1 + cos(theta)) / 2. This factor accounts for the directional variation of wave amplitude, reaching its maximum value of 1 when theta = 0, and falling to 0 when theta = pi.

Fresnel's Half-Period Zones for Plane Wave

To determine the resultant intensity of light at a point due to an unobstructed plane wavefront, Fresnel introduced the method of dividing the wavefront into concentric circular rings called half-period zones.

Construction of Half-Period Zones

Let ABCD represent a plane wavefront of monochromatic light of wavelength lambda travelling in the forward direction. We want to find the resultant effect of this wavefront at a point P situated at a distance b from the pole O of the wavefront.

1. With P as center and radii equal to b + lambda/2, b + 2*(lambda/2), b + 3*(lambda/2), ... b + n*(lambda/2), draw concentric spheres.
2. These spheres intersect the plane wavefront in a series of concentric circles with O as the common center.
3. The annular regions between successive circles are called Fresnel's half-period zones. The circular area of radius OM_1 is the 1st zone, the annular region between M_1 and M_2 is the 2nd zone, and the region between M_(n-1) and M_n is the nth zone.

Mathematical Derivations

Let us calculate the radius, area, and mean distance of the nth zone.

1. Radius of the nth Zone (r_n)

From the right-angled triangle OM_nP:

(M_nP)^2 = (OM_n)^2 + (OP)^2

Since M_nP = b + n*lambda/2 and OP = b:

(b + n*lambda/2)^2 = r_n^2 + b^2
b^2 + n*b*lambda + (n^2 * lambda^2)/4 = r_n^2 + b^2

Since wavelength lambda is very small, the term containing lambda^2 can be neglected:

r_n^2 = n * b * lambda
r_n = sqrt(n * b * lambda)

Thus, the radii of the half-period zones are proportional to the square roots of the natural numbers (1, 2, 3, ...).

2. Area of the nth Zone (A_n)

The area of the nth zone is the difference between the area of the nth circle and the (n-1)th circle:

A_n = pi * [r_n^2 - r_(n-1)^2]
A_n = pi * [n * b * lambda - (n - 1) * b * lambda]
A_n = pi * b * lambda

Important Observation: The area of each half-period zone is approximately equal and independent of the order of the zone n (as long as n is not very large and the approximation holds).

3. Average Distance of the nth Zone (d_n)

The average distance of the nth zone from the point P is:

d_n = b + (2n - 1) * lambda / 4

Resultant Amplitude at P

The phase difference between secondary wavelets arriving at P from any two consecutive zones is pi (corresponding to a path difference of lambda/2). Therefore, if the amplitudes due to the 1st, 2nd, 3rd... nth zones are denoted by R_1, R_2, R_3... R_n, they will have alternating signs due to this phase difference:

R = R_1 - R_2 + R_3 - R_4 + ... + (-1)^(n-1) * R_n

The amplitude R_n decreases continuously as n increases due to two factors:

• The average distance from P increases, which reduces the wave amplitude.
• The obliquity factor (1 + cos(theta)) decreases as the angle increases.

Because the decrease is gradual and continuous, we can write the amplitude of any intermediate zone as the average of its neighbors:

R_2 = (R_1 + R_3) / 2
R_4 = (R_3 + R_5) / 2

Rewriting the series for the resultant amplitude:

R = R_1/2 + (R_1/2 - R_2 + R_3/2) + (R_3/2 - R_4 + R_5/2) + ...

Since the terms in parentheses are approximately zero, the entire series simplifies to:

R = R_1 / 2

The resultant intensity at P is proportional to R^2, which is (R_1)^2 / 4. This means the intensity due to the entire unobstructed wavefront is only one-quarter of the intensity due to the first half-period zone alone.

Explanation of Rectilinear Propagation of Light

Historically, the wave theory of light was challenged because light appears to travel in straight lines and casts sharp shadows, unlike sound waves which easily bend around obstacles. Fresnel used his concept of half-period zones to successfully resolve this conflict and explain the rectilinear propagation of light.

Step-by-Step Explanation

1. Consider a plane wavefront propagating towards a screen. The resultant amplitude at a point P on the screen is R = R_1 / 2, where R_1 is the amplitude contributed by the first half-period zone.
2. The radius of the first half-period zone is given by r_1 = sqrt(b * lambda). Let us calculate this for a typical physical setup:
   • Let b = 10 cm = 0.1 m
   • Let lambda = 5000 Angstroms = 5 * 10^(-7) m (green light)
   • r_1 = sqrt(0.1 * 5 * 10^(-7)) = sqrt(5 * 10^(-8)) = 2.24 * 10^(-4) m = 0.22 mm
3. The radius of the first zone is less than a quarter of a millimeter. The zones of higher orders are even narrower concentric rings surrounding the pole O.
4. If an obstacle of size slightly larger than a fraction of a millimeter is placed in the path of the light near O, it covers the first few zones completely.
5. Because the contribution of higher-order zones is negligible due to the large obliquity factor and high distance, the resultant amplitude at point P drops to virtually zero immediately behind the obstacle.
6. Thus, the light is effectively blocked by small obstacles, casting a sharp shadow. This localized transmission explains why light appears to travel in straight lines (rectilinear propagation).

Common Student Mistake

Common Mistake: Thinking that the rectilinear propagation of light is absolute. In reality, if an obstacle is comparable in size to the extremely small radius of the first few Fresnel zones (e.g., a microscopic wire or tiny aperture), light will bend significantly around it, exhibiting clear wave behavior (diffraction).

Theory of a Zone Plate

A zone plate is an specially constructed optical screen designed to obstruct light from alternate half-period zones. When placed in the path of a wavefront, it blocks either all even-numbered zones or all odd-numbered zones, leading to a dramatic increase in the intensity of light at a specific point on the axis.

Construction

To construct a zone plate, concentric circles are drawn on white paper with radii proportional to the square roots of natural numbers (1, 2, 3, ...). The alternate zones (either odd or even) are painted black. A highly reduced photograph of this pattern is taken on a glass plate. The transparent portions of the plate allow light to pass through, while the blackened portions block it.

• Positive Zone Plate: The central (1st) zone is transparent, the 2nd is dark, the 3rd is transparent, and so on. It blocks even zones.
• Negative Zone Plate: The central (1st) zone is dark, the 2nd is transparent, the 3rd is dark, and so on. It blocks odd zones.

Working and Focal Length Derivation

Let an object point source of monochromatic light O be situated at a distance u on the axis of a zone plate, and let the screen be at a distance v where an image P is formed.

Let the radius of the nth zone be r_n. The path of light travelling from O to P through the outer boundary of the nth zone is OM_n + M_nP.

From the geometry of the setup:

OM_n = sqrt(u^2 + r_n^2) = u * (1 + r_n^2 / u^2)^(1/2) approx= u + r_n^2 / (2u)
M_nP = sqrt(v^2 + r_n^2) = v * (1 + r_n^2 / v^2)^(1/2) approx= v + r_n^2 / (2v)

The total path length is:

OM_n + M_nP = u + v + (r_n^2 / 2) * (1/u + 1/v)

For constructive interference at P, the path difference between light travelling through O-O-P and O-M_n-P must equal n * lambda / 2:

[u + v + (r_n^2 / 2) * (1/u + 1/v)] - (u + v) = n * lambda / 2
(r_n^2 / 2) * (1/u + 1/v) = n * lambda / 2
r_n^2 * (1/u + 1/v) = n * lambda

This yields the fundamental equation of the zone plate:

1/u + 1/v = n * lambda / r_n^2

Comparing this with the standard lens formula 1/u + 1/v = 1/f, we can define the focal length of the zone plate f_n as:

f_n = r_n^2 / (n * lambda)

Since r_n^2 / n is constant for a given zone plate, the focal length depends on the wavelength lambda of the light used. Thus, the zone plate acts as a converging lens with a focal length that is inversely proportional to the wavelength of light.

Multiple Foci of a Zone Plate

Unlike a standard convex glass lens, which has a single primary focal point, a zone plate behaves as if it has multiple focal points of varying intensities along its axis.

This occurs because at points closer to the plate along its axis, the same physical zones on the plate can contain an odd number of half-period zones of the smaller path-difference scale.

The general formula for the multiple focal lengths of a zone plate is given by:

f_p = r_n^2 / (p * n * lambda)

where p is an odd integer: p = 1, 3, 5, 7, ...

Comparison: Zone Plate vs. Convex Lens

Fresnel's Integrals

To analyze complex diffraction patterns mathematically without using geometric zone approximations, Fresnel developed an analytical method based on integrals. These integrals represent the sum of the contributions of all secondary wavelets originating from a wavefront.

The coordinates of any point on a wave projection curve (known as the Cornu spiral) are given by the Fresnel Cosine Integral C(u) and the Fresnel Sine Integral S(u):

C(u) = integral from 0 to u of cos(pi * t^2 / 2) dt
S(u) = integral from 0 to u of sin(pi * t^2 / 2) dt

Here, u is a dimensionless parameter proportional to the distance along the wavefront. These integrals cannot be solved in terms of elementary closed-form functions and must be evaluated numerically or using expansion series.

Key Observations

• As u approaches infinity, both integrals converge to a value of 0.5: C(infinity) = 0.5 and S(infinity) = 0.5.
• As u approaches minus infinity, they converge to -0.5: C(-infinity) = -0.5 and S(-infinity) = -0.5.
• Plotting Y = S(u) against X = C(u) produces the Cornu Spiral, a double-spiral curve used to graphically calculate the intensity of diffraction patterns for edges, slits, and wires.

Fresnel Diffraction Pattern of a Straight Edge

A straight edge consists of a sharp, opaque, straight-line barrier (such as a razor blade) placed in the path of a wavefront. The resulting diffraction pattern exhibits a unique distribution of light intensity around the boundary of the geometrical shadow.

Experimental Setup and Geometry

Let a narrow slit source of monochromatic light S be parallel to the straight edge E. A screen is placed at a distance b behind the edge. The line joining S and E meets the screen at point P_0, which marks the boundary of the geometrical shadow.

• The region above P_0 is the illuminated region.
• The region below P_0 is the geometrical shadow region.

Intensity Distribution and Analysis

The intensity distribution is divided into two distinct regions:

1. Inside the Geometrical Shadow (below P_0)

As we move deeper into the geometrical shadow, more and more half-period zones of the upper half of the wavefront are progressively blocked by the edge. The intensity decreases rapidly and monotonically without any oscillations, quickly falling to zero.

At the exact boundary P_0, the entire lower half of the wavefront is blocked. Thus, the amplitude is exactly half of the unobstructed amplitude: R_0 = R_1 / 4. Consequently, the intensity at P_0 is 1/16th of the unobstructed intensity (or 1/4th of the normal wave intensity R_1^2/4).

2. Inside the Illuminated Region (above P_0)

In this region, as we move away from P_0, the edge progressively uncovers or covers successive half-period zones. This leads to alternating constructive and destructive interference, resulting in a series of dark and bright bands (fringes).

• Maxima: Occur when the uncovered path differences correspond to an odd number of half-period zones.
• Minima: Occur when the uncovered path differences correspond to an even number of half-period zones.
• As the distance from the edge increases, the difference in amplitude between successive zones becomes negligible. Consequently, the oscillations die out, and the intensity stabilizes to a uniform, constant value corresponding to the unobstructed wavefront.

Fresnel Diffraction Pattern of a Slit

A slit is an aperture bounded by two parallel straight edges separated by a narrow width d. The diffraction pattern of a slit is determined by the number of half-period zones that can fit within its width from the perspective of an observation point P on the screen.

Intensity Distribution Analysis

Let P be a point on the screen, and let the width of the slit expose N half-period zones of the wavefront. The intensity at P depends directly on whether N is an even or odd integer.

• Central Point P_0: If the width of the slit is such that it exposes an odd number of zones (e.g., N = 1, 3, 5...) relative to P_0, the secondary wavelets interfere constructively, and P_0 is a bright point (maximum). If it exposes an even number of zones (e.g., N = 2, 4, 6...), they cancel each other out, and P_0 is a dark point (minimum).
• Other Points on the Screen: As we move away from the center, the effective number of zones exposed by the slit changes. This produces a symmetrical pattern of alternating bright and dark fringes.

Effect of Slit Width

• Very Narrow Slit: If the slit is so narrow that it exposes less than one half-period zone, only a broad, smooth central maximum is observed on the screen, with no distinct minimum.
• Wide Slit: If the slit is wide, it exposes a very large number of half-period zones. The fluctuations between odd and even zones become negligible, and the pattern transitions into a sharp, geometric image of the slit with tiny, barely visible diffraction fringes confined closely to the edges.

Fresnel Diffraction Pattern of a Wire

A thin wire of diameter d acts as an obstacle that blocks a narrow strip of the wavefront, leaving the wavefront open on both sides of the wire.

Experimental Setup

Monochromatic light from a narrow slit S passes around a parallel thin wire W and is projected onto a screen. The region directly behind the wire is the geometrical shadow, bounded by two points on the screen.

Intensity Distribution Analysis

The diffraction pattern of a wire is particularly interesting because it consists of two distinct types of fringes:

1. Inside the Geometrical Shadow

Within the geometrical shadow, light waves bend around both the left and right sides of the thin wire. Since these two secondary wave sources originate from the same coherent wavefront, they act as coherent sources and interfere with one another.

• This produces equidistant interference fringes of equal width inside the shadow.
• The center of the geometrical shadow is always bright because the path difference from both edges of the wire to the center is exactly zero.
• If the wire is thick, the light bending from either side cannot overlap significantly due to distance attenuation, and the interference fringes inside the shadow disappear.

2. Outside the Geometrical Shadow

Outside the shadow boundary, on either side of the wire, we observe typical diffraction fringes of unequal width. These are formed by the interference of wavelets from the unobstructed portion of the wavefront on one side with the partially cut-off wavefront from the other side. The intensity eventually stabilizes to a uniform value far from the shadow.

Principle of Holography

Conventional photography records only the intensity of light (amplitude squared) reflecting off an object, discarding all phase information. Because phase information is lost, photographic images appear flat and two-dimensional.

Invented by Dennis Gabor in 1948, holography is a technique that records both the amplitude (intensity) and the phase of the light waves scattered by an object. This complete wave record (called a hologram) allows the original three-dimensional wavefront to be fully reconstructed, creating a realistic 3D image.

Recording and Reconstruction Methods

Holography is a two-step process: Recording (forming the hologram) and Reconstruction (reading the hologram to view the 3D image).

1. Recording Process (Hologram Construction)

To record a hologram, a highly coherent laser beam is split into two separate beams:

• Object Beam: This beam illuminates the object. The light scatters off the object's uneven surface, carrying unique amplitude and phase information. This scattered wave (Object Wave) travels toward the photographic plate.
• Reference Beam: This beam is redirected by a mirror to illuminate the photographic plate directly, without striking the object. It maintains a clean, undisturbed plane or spherical wavefront.

When the Object Beam and the Reference Beam meet at the photographic plate, they interfere. The plate records this complex microscopic interference pattern (a series of high-density fringes). Once developed, this plate is called the hologram. It does not contain a recognizable image of the object, but rather a coded pattern of lines and swirls.

2. Reconstruction Process

To reconstruct the image, the developed hologram is illuminated by a beam identical to the original reference beam (called the reconstruction beam).

As this beam passes through the fine interference fringes of the hologram, it undergoes diffraction. The diffracted light splits into three components:

1. A zero-order beam that passes straight through without forming an image.
2. A first-order diffracted wave that diverges. When projected backward, these diverging rays appear to meet, forming a virtual, three-dimensional image of the original object behind the hologram. This image exhibits full parallax.
3. A conjugate first-order diffracted wave that converges to form a real image in front of the hologram, which can be captured on a screen or camera sensor without a lens.

Theory of Holography as Interference between Two Plane Waves

To understand the mathematics of holography, we can model the recording and reconstruction processes using two interfering plane waves: a reference wave and an object wave.

1. The Recording Phase

Let the reference wave be a plane wave incident on the holographic plate at an angle. We can write its complex amplitude at the plate as:

R = A_r * exp(i * k * x * sin(theta))

Let the wave scattered from the object (the object wave) be represented by:

O = A_o(x) * exp(i * phi(x))

The total complex amplitude U at the plate is the sum of these two waves:

U = R + O

The photographic plate responds to the intensity I of the incident light:

I = |U|^2 = (R + O) * (R + O)*
I = R * R* + O * O* + R* * O + R * O*

Substituting the terms:

I = |A_r|^2 + |A_o(x)|^2 + R* * O + R * O*

Here, the third and fourth terms, R* * O and R * O*, contain the crucial phase information of the object wave relative to the reference wave, captured in the spatial variation of the intensity pattern.

2. The Reconstruction Phase

After development, the transmittance T of the hologram is proportional to the recorded intensity I:

T = t_0 + beta * I

where t_0 is a constant transmittance and beta is a constant parameter. When we illuminate the hologram with a reconstruction beam equal to the original reference beam R, the transmitted wave amplitude U_reconstruction is:

U_reconstruction = R * T = R * (t_0 + beta * I)
U_reconstruction = R * [t_0 + beta * (|A_r|^2 + |A_o|^2) + beta * R* * O + beta * R * O*]

Expanding this expression yields three distinct terms:

U_reconstruction = R * [t_0 + beta * (|A_r|^2 + |A_o|^2)] + beta * |R|^2 * O + beta * R^2 * O*

Let's analyze these three terms physically:

• First Term: R * [t_0 + beta * (|A_r|^2 + |A_o|^2)]. This corresponds to the undeflected zero-order beam that passes straight through the plate.
• Second Term: beta * |R|^2 * O. Since |R|^2 = A_r^2 is constant, this term is directly proportional to the original object wave O. This reconstructed wave diverges, creating the virtual image of the object. It is a perfect replica of the original wave in both amplitude and phase.
• Third Term: beta * R^2 * O*. This term contains the complex conjugate of the object wave O*. It represents a wave converging to form the real conjugate image of the object.

Point Source Holograms

A point source hologram is the simplest type of hologram, recorded using a single point source as the object wave and a collimated plane wave as the reference wave. Understanding this simple case provides an intuitive look at the physics of holography, as it directly mirrors the structure of a Fresnel Zone Plate.

Recording Mechanism

Consider a point source of light placed at a distance d in front of a photographic plate. This source emits spherical waves towards the plate. Concurrently, a coherent plane reference wave is directed normal to the plate.

Because the phase of the plane wave is constant across the flat plate, while the phase of the spherical wave varies radially from the center (where the point source is closest), the interference of these two waves produces a pattern of concentric rings.

As we move radially outward from the center of the plate, the path difference between the spherical wave and the plane wave increases. This produces alternating bright and dark circles with radii proportional to the square roots of natural numbers: sqrt(n * lambda * d).

Important Observation: The resulting interference pattern on a point source hologram is identical to a Fresnel Zone Plate.

Reconstruction Mechanism

When this point source hologram (which is physically a zone plate) is illuminated with a plane reconstruction beam:

• The zone plate focuses a portion of the light to a real focal point at a distance d in front of the plate. This forms the real image of the point source.
• It also causes a portion of the light to diverge as if it were originating from a virtual focal point at a distance d behind the plate. This forms the virtual image of the point source.

Thus, any complex 3D object can be thought of as a collection of individual point sources. The hologram of a complex object is simply the superposition of countless microscopic point-source holograms (zone plates) overlapping on the same plate.