Unit 3: Optics: Diffraction and Polarization

Diffraction

Diffraction is the phenomenon where light waves bend around obstacles or spread out after passing through small apertures. This occurs when the wavelength of light is comparable to the size of the obstacle or aperture. Diffraction is a clear demonstration of the wave nature of light.

Fraunhofer Diffraction

Fraunhofer diffraction occurs when both the light source and the observation screen are effectively at infinite distances from the diffracting obstacle or aperture. This typically means using lenses to ensure that the incident light is a plane wavefront and the diffracted light is collected at a distant point (or in the focal plane of a lens).

Single Slit Diffraction

When monochromatic light passes through a single narrow slit, a diffraction pattern consisting of a broad central bright maximum flanked by weaker, narrower secondary maxima and minima is observed.

• Central Maximum: The brightest and widest part of the pattern, occurring at θ = 0.
• Minima (Dark Fringes): Occur when the path difference between waves from the two edges of the slit is an integral multiple of the wavelength.

  Condition for Minima: a sinθ = nλ, where n = ±1, ±2, ±3,...
  Here, 'a' is the slit width, 'θ' is the angle of diffraction, and 'λ' is the wavelength of light.

• Secondary Maxima (Bright Fringes): Occur approximately halfway between the minima.

  Condition for Secondary Maxima: a sinθ = (2n+1)λ/2, where n = ±1, ±2, ±3,...

• Width of Central Maximum: The angular width is 2θ₁, where θ₁ is the angle of the first minimum (sinθ₁ = λ/a). The linear width on a screen at distance D is 2λD/a.

Double Slit Diffraction

The diffraction pattern from a double slit is a combination of interference from two coherent sources and diffraction from each individual slit. The overall intensity distribution is modulated by the single-slit diffraction envelope.

• Interference Maxima: Occur due to constructive interference from the two slits.

  Condition for Interference Maxima: d sinθ = nλ, where n = 0, ±1, ±2, ...
  Here, 'd' is the distance between the centers of the two slits.

• Interference Minima: Occur due to destructive interference from the two slits.

  Condition for Interference Minima: d sinθ = (2n+1)λ/2, where n = 0, ±1, ±2, ...

• Missing Orders: If a maximum of the interference pattern coincides with a minimum of the single-slit diffraction pattern, that interference maximum will be absent. This occurs when d/a is an integer.

Multiple Slits and Diffraction Grating

A diffraction grating consists of a large number of equally spaced parallel slits. It produces highly resolved and sharp interference maxima compared to a double slit.

• Grating Element (d): The distance between the centers of two adjacent slits. If 'a' is the slit width and 'b' is the opaque space between slits, then d = a + b.
• Principal Maxima: The main bright fringes produced by a diffraction grating.

  Condition for Principal Maxima: d sinθ = nλ, where n = 0, ±1, ±2, ...
  Here, 'n' is the order of the spectrum.

• Resolving Power (R): The ability of a grating to separate two closely spaced wavelengths.

  R = nN, where 'n' is the order of the spectrum and 'N' is the total number of lines (slits) on the grating.

• Absent Spectra: Similar to double-slit, if the principal maxima of the grating coincide with the minima of the single-slit diffraction pattern for each individual slit, those orders of spectra will be absent.

Fresnel Diffraction

Fresnel diffraction occurs when either the source or the screen (or both) are at finite distances from the diffracting object or aperture. The wavefronts are spherical, and the exact calculation involves considering the interference of waves from various parts of the wavefront.

Half-Period Zones (Fresnel Zones)

To analyze Fresnel diffraction, the concept of half-period zones is used. The wavefront is divided into concentric circular zones such that the path difference from the edge of one zone to the observation point and the edge of the next zone to the observation point is λ/2. This means waves from adjacent zones arrive at the observation point with a phase difference of π.

• Radius of n^th zone (r_n): For a plane wavefront incident on an aperture, and observation point P at distance 'b', the radius is approximately:

  r_n ≈ √[nλb]

  For a spherical wavefront originating from S at distance 'a' and observed at P at distance 'b', the radius is:

  r_n = √[nλab / (a+b)]

• Area of each zone: The area of each Fresnel zone is approximately equal (A_n ≈ πbλ).
• Amplitude Contribution: The amplitude of light contributed by each successive zone at the observation point gradually decreases due to increasing obliquity and distance.
• Resultant Amplitude: Due to the π phase difference, contributions from successive zones tend to cancel. The resultant amplitude from a large unobstructed wavefront is approximately half the amplitude of the first zone.

Zone Plate

A zone plate is a device that functions like a lens by selectively blocking or transmitting alternate Fresnel half-period zones. It consists of a series of concentric transparent and opaque rings.

• Principle: By blocking alternate zones (e.g., all even-numbered zones), the remaining odd-numbered zones contribute constructively to form a bright spot at a specific point.
• Action: It has multiple focal lengths (both real and virtual), determined by the radii of the zones and the wavelength.

  Focal length of a zone plate: f_n = r₁² / nλ, where 'n' is the order of the focus and r₁ is the radius of the first zone.

• Comparison with Convex Lens: Unlike a lens which uses refraction, a zone plate uses diffraction to focus light. It exhibits chromatic aberration (focal length depends on λ) more severely than a conventional lens.

Fresnel Diffraction Pattern of a Straight Edge, a Slit and a Wire using Half-Period Zone Analysis

Analyzing these patterns involves considering how many half-period zones are exposed or blocked by the obstacle/aperture.

• Straight Edge:
  When light diffracts past a straight edge, the pattern exhibits bright and dark fringes in the region slightly within the geometrical shadow and a gradual decrease in intensity further into the shadow. Outside the shadow, there are oscillatory fringes of decreasing intensity, eventually leading to uniform illumination. This occurs because the number of active Fresnel zones above the edge varies, causing constructive and destructive interference. The brightest fringe occurs just inside the geometric shadow.
• Slit:
  The Fresnel diffraction pattern of a slit depends on its width relative to the Fresnel zones.
  • Narrow Slit: If the slit exposes only a few half-period zones, a broad central bright band with decreasing intensity on either side is observed.
  • Wide Slit: If the slit is wide enough to expose many half-period zones, the pattern near each edge resembles that of a straight edge. In the central region, if the observation point is far, the intensity can be nearly uniform. The overall pattern can be quite complex with multiple bright and dark bands.
• Wire:
  For a thin wire, a bright line is observed in the center of the geometrical shadow. This is a unique feature of Fresnel diffraction and occurs because light waves diffracted around both edges of the wire interfere constructively at the center of the shadow. Outside the shadow, fringes are observed that resemble those produced by a straight edge.

Polarization

Polarization is a property of transverse waves, such as light, that describes the orientation of their oscillations. Unpolarized light consists of waves oscillating in all possible planes perpendicular to the direction of propagation. Polarized light has its oscillations restricted to a specific plane or direction.

Plane Polarized Light: Light in which the electric field vector oscillates in a single plane perpendicular to the direction of wave propagation.

Nicol Prism

The Nicol prism is an optical device made from a calcite (calcium carbonate) crystal, used for producing and analyzing plane polarized light.

• Construction: A calcite crystal is cut diagonally into two halves, which are then polished and cemented together with a layer of Canada balsam.
• Principle: It works on the principle of double refraction (birefringence) and total internal reflection. When unpolarized light enters calcite, it splits into two refracted rays: the Ordinary ray (o-ray) and the Extraordinary ray (e-ray).

For calcite, n_o ≈ 1.658 and n_e ≈ 1.486. Canada balsam has a refractive index n_balsam ≈ 1.55. The angle of the cut is chosen such that the o-ray, upon hitting the Canada balsam layer, experiences total internal reflection and is diverted out of the prism, while the e-ray is transmitted through the balsam and emerges from the prism as plane polarized light.

Production of Plane Polarized Light by Nicol Prism

Unpolarized light enters the Nicol prism. Inside the calcite, it undergoes double refraction, splitting into o-ray and e-ray. The o-ray encounters the Canada balsam layer at an angle greater than its critical angle (since n_o > n_balsam), so it undergoes total internal reflection and is absorbed. The e-ray, having a different refractive index (n_e < n_balsam for light striking the balsam at the specific angle in the prism), passes through the Canada balsam and emerges from the prism. This emergent e-ray is plane polarized.

Analysis of Plane Polarized Light by Nicol Prism

A second Nicol prism, called an analyzer, is used to analyze polarized light. If plane polarized light is incident on the analyzer, its intensity upon emergence depends on the angle between the plane of polarization of the incident light and the principal section of the analyzer. According to Malus' Law, the transmitted intensity I is given by:

I = I₀ cos²θ
where I₀ is the maximum intensity transmitted by the analyzer, and θ is the angle between the plane of polarization of the incident light and the transmission axis of the analyzer.

When the plane of polarization of the incident light is parallel to the principal section of the analyzer (θ = 0°), maximum light is transmitted. When they are perpendicular (θ = 90°), no light is transmitted (extinction). Rotating the analyzer allows determination of the plane of polarization and the degree of polarization.

Half-Wave Plate (λ/2 plate)

A half-wave plate is a birefringent crystal plate of specific thickness that introduces a phase difference of π (or a path difference of λ/2) between the Ordinary and Extraordinary components of light passing through it.

• Construction: It is typically made of quartz or mica, cut with its optic axis parallel to the surface. The thickness 't' is chosen such that:

  t |n_e - n_o| = λ/2

• Action: If linearly polarized light is incident on a half-wave plate such that its plane of polarization makes an angle θ with the optic axis, the emergent light will still be linearly polarized, but its plane of polarization will be rotated by an angle 2θ with respect to the original direction. This property is useful for rotating the plane of polarization of a light beam.

Quarter-Wave Plate (λ/4 plate)

A quarter-wave plate is a birefringent crystal plate that introduces a phase difference of π/2 (or a path difference of λ/4) between the Ordinary and Extraordinary components of light passing through it.

• Construction: Similar to a half-wave plate, it's made of a birefringent material like quartz or mica, with a thickness 't' chosen such that:

  t |n_e - n_o| = λ/4

• Action:
  • If linearly polarized light is incident on a quarter-wave plate at an angle of 45° to its optic axis, the emergent light will be circularly polarized.
  • If linearly polarized light is incident at any other angle (not 0°, 45°, or 90°), the emergent light will be elliptically polarized.
  • A quarter-wave plate can also convert circularly polarized light into linearly polarized light, and elliptically polarized light into linearly polarized light (under specific conditions).