12.3 Diffraction Grating, Resolving Power & Polarization of Light

Diffraction Grating, Resolving Power & Polarization of Light

In the Intermediate Physics Part I syllabus and the UET ECAT entry test, diffraction and polarization are critical topics. In the ECAT examination, questions frequently require computing the grating element, determining the highest observable diffraction order, calculating telescope resolving limits, and applying Brewster's and Malus's laws. A thorough understanding of wave optics and polarization mechanics is essential for rapid problem-solving.

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1. The Phenomenon of Diffraction

Diffraction is the bending of waves around the edges of an obstacle or the spreading of waves beyond an aperture into the geometrical shadow region. Diffraction is a fundamental property of all wave motions (sound, surface waves, light, electrons).

Condition for Observable Diffraction

Diffraction is significant only when the physical dimension of the obstacle or aperture $a$ is comparable to the wavelength $\lambda$ of the incident wave:

$a \approx \lambda$

• Sound Waves: Audible sound has wavelengths ranging from $1.7\text{ cm}$ to $17\text{ m}$. Because common doorways and windows ($1\text{ m}$ to $2\text{ m}$) are comparable to these wavelengths, sound diffracts easily around corners, allowing people to hear around obstacles.
• Light Waves: Visible light has microscopic wavelengths ($400\text{ nm}$ to $700\text{ nm}$). Everyday obstacles are millions of times larger than $\lambda_{\text{light}}$; therefore, light casts sharp geometrical shadows, exhibiting rectilinear propagation under macroscopic conditions. To observe optical diffraction, microscopic slits, fine wires, or pinholes must be used.

Fresnel vs. Fraunhofer Diffraction

Optical diffraction is divided into two distinct classes depending on source and observation geometry:

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2. Fraunhofer Diffraction at a Single Slit

Consider a narrow rectangular slit of width $a$ illuminated normally by monochromatic plane waves of wavelength $\lambda$. By Huygens' principle, every point across the open aperture of width $a$ acts as a secondary emitter of coherent spherical wavelets.

Derivation of Diffraction Minima (Dark Fringes)

To find the condition for destructive interference at an angle $\theta$:

1. Divide the slit width $a$ into two equal halves, each of width $\frac{a}{2}$.
2. For every wavelet originating in the upper half, there exists a corresponding wavelet in the lower half separated by distance $\frac{a}{2}$.
3. The path difference between corresponding pairs is $\Delta x = \frac{a}{2}\sin\theta$.
4. Destructive interference occurs if this path difference equals half a wavelength: $\frac{a}{2}\sin\theta = \frac{\lambda}{2} \implies a\sin\theta = \lambda$.

Generalizing by dividing the slit into $2m$ equal zones, destructive interference (intensity minima) occurs at angles satisfying:

$a\sin\theta = m\lambda \quad (m = \pm 1, \pm 2, \pm 3, \dots)$

ECAT Conceptual Trap: In Young's double-slit interference, $d\sin\theta = m\lambda$ defines bright fringes (maxima). In single-slit diffraction, $a\sin\theta = m\lambda$ defines dark fringes (minima). Note that $m = 0$ corresponds to the central maximum, not a minimum.

Width of the Central Diffraction Maximum

The first diffraction minimum occurs at angle $\theta_1$ where:

$\sin\theta_1 = \frac{\lambda}{a} \implies \theta_1 \approx \frac{\lambda}{a} \quad \text{(for small } \theta\text{)}$

The central maximum extends from $-\theta_1$ to $+\theta_1$. Thus:

• Angular Width of Central Maximum:

$2\theta_1 = 2\frac{\lambda}{a}$

• Linear Width on Screen at Distance $L$:

$W = 2y_1 = 2L\tan\theta_1 \approx 2\frac{\lambda L}{a}$

The linear width of the central maximum is twice as wide as any of the secondary maxima ($W_{\text{secondary}} = \frac{\lambda L}{a}$) and contains roughly $90\%$ of the diffracted light energy (for a circular aperture, the central Airy disc holds about $84\%$).

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3. The Diffraction Grating

A diffraction grating is an optical device consisting of a large number of equidistant, closely spaced parallel slits ruled on a transparent or reflective glass plate. Modern transmission gratings typically contain $300$ to $1{,}000$ lines per millimeter ($3{,}000$ to $10{,}000$ lines per centimeter).

Grating Element ($d$)

The distance between the centers of two consecutive transparent slits is called the grating element $d$:

$d = a + b = \frac{L}{N}$

where $a$ is the transparent slit width, $b$ is the opaque ruling width, $L$ is the total ruled length (typically $1\text{ cm}$), and $N$ is the total number of lines ruled across that length.

• If a grating has $5{,}000\text{ lines/cm}$:

$d = \frac{1\text{ cm}}{5000} = \frac{10^{-2}\text{ m}}{5000} = 2.0 \times 10^{-6}\text{ m} = 2.0\ \mu\text{m}$

The Grating Equation

When a parallel beam of monochromatic light strikes a diffraction grating at normal incidence, waves emerging from adjacent slits at an angle $\theta$ have an optical path difference of $\Delta x = d\sin\theta$. Principal intensity maxima (sharp bright spectral lines) occur when this path difference equals an integer number of wavelengths:

$d\sin\theta = m\lambda \quad (m = 0, \pm 1, \pm 2, \pm 3, \dots)$

where $m$ is the order of the spectrum:

• $m = 0$: Central undeviated zero-order maximum (all wavelengths superpose; white light produces a white central image).
• $m = 1$: First-order diffraction spectrum.
• $m = 2$: Second-order diffraction spectrum.

Dispersion of White Light by a Grating vs. a Prism

Because $\sin\theta = \frac{m\lambda}{d}$, the diffraction angle is directly proportional to wavelength:

• Red light ($\lambda \approx 700\text{ nm}$) diffracts through a larger angle than blue light ($\lambda \approx 400\text{ nm}$).
• In a prism, shorter wavelengths (blue) bend more due to higher refractive index ($n_{\text{blue}} > n_{\text{red}}$).
• Thus, the spectral dispersion of a diffraction grating is reversed relative to the spectral dispersion of a triangular glass prism.

Maximum Observable Order ($m_{\max}$)

Because the maximum possible physical value for $\sin\theta$ is $1$ (corresponding to $\theta = 90^\circ$):

$\sin\theta = \frac{m\lambda}{d} \le 1 \implies m \le \frac{d}{\lambda}$

The theoretical maximum observable order $m_{\max}$ is the greatest integer less than or equal to the ratio $\frac{d}{\lambda}$:

$m_{\max} = \left\lfloor \frac{d}{\lambda} \right\rfloor$

ECAT Speed Rule: The total number of observable bright lines on a screen is $N_{\text{total}} = 2m_{\max} + 1$ (accounting for positive orders $+m$, negative orders $-m$, and the central zero order $m = 0$). If $d/\lambda$ is an exact integer $n$, the $n$th order would emerge at $\theta = 90^\circ$, grazing along the grating surface, so only orders up to $n - 1$ actually reach a screen.

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4. X-Ray Diffraction and Bragg's Law

X-rays have extremely short wavelengths ($\lambda \approx 0.01\text{ nm}$ to $1\text{ nm}$, that is $10^{-11}$ to $10^{-9}\text{ m}$), which are thousands of times smaller than rulings on mechanical diffraction gratings. However, interatomic lattice spacings in crystalline solids ($d \approx 0.2\text{ nm}$ to $0.5\text{ nm}$) match X-ray wavelengths, making natural crystals ideal three-dimensional diffraction gratings.

In 1913, W. H. Bragg and W. L. Bragg derived the condition for constructive interference of X-rays reflected from adjacent parallel crystal lattice planes separated by distance $d$.

Let $\theta$ be the glancing angle (the angle between the incident X-ray beam and the crystal plane, NOT the surface normal). The path difference between rays reflected from two consecutive planes is:

$\Delta x = 2d\sin\theta$

For constructive interference, this path difference must equal an integral number of wavelengths, yielding Bragg's Law:

$2d\sin\theta = n\lambda \quad (n = 1, 2, 3, \dots)$

Bragg reflection forms the foundation of modern X-ray crystallography, utilized to determine the atomic lattice structures of metals, semiconductors, and biomolecules (including the double-helix structure of DNA).

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5. Resolving Power & Rayleigh's Criterion

Definition of Resolving Power

The resolving power of an optical instrument (such as a telescope, microscope, or human eye) is its ability to distinguish and separate two closely spaced points or adjacent spectral lines as separate entities.

Rayleigh's Criterion for Resolution

Due to wave diffraction, the image of a point source formed by a circular aperture is not a point, but a bright central circular disc (the Airy disc) surrounded by faint concentric dark and bright diffraction rings.

Lord Rayleigh established that two point objects are just resolved when the central maximum of the diffraction pattern of one object falls exactly on the first minimum of the diffraction pattern of the second object.

For a circular aperture of diameter $D$:

$\theta_{\min} = 1.22\frac{\lambda}{D}$

where $\theta_{\min}$ is the minimum angular separation (limit of resolution) in radians.

Resolving Power of Telescopes

The resolving power of an astronomical telescope is the reciprocal of its limit of resolution:

$\text{Resolving Power} = \frac{1}{\theta_{\min}} = \frac{D}{1.22\lambda}$

To increase resolving power (resolve finer details of distant stars):

1. Use larger objective apertures $D$ (e.g., modern observatory mirrors with $D > 8\text{ m}$).
2. Observe at shorter wavelengths $\lambda$.

Resolving Power of a Diffraction Grating

For a diffraction grating, resolving power measures the capacity to distinguish two closely spaced spectral wavelengths $\lambda$ and $\lambda + \Delta\lambda$:

$R = \frac{\lambda}{\Delta\lambda} = m N$

where $m$ is the spectral order and $N$ is the total number of illuminated grating rulings.

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6. Polarization of Light

Interference and diffraction demonstrate conclusively that light is a wave phenomenon, but they do not reveal whether light waves are longitudinal or transverse. The phenomenon of polarization provides conclusive experimental proof that light is a transverse electromagnetic wave.

Unpolarized vs. Linearly Polarized Light

• Unpolarized Light: Light emitted by natural thermal sources (sunlight, incandescent filaments) consists of millions of independent atomic wave trains. The electric field vector $\mathbf{E}$ vibrates randomly in all possible planes perpendicular to the direction of propagation.
• Linearly (Plane) Polarized Light: Light in which the electric field vector $\mathbf{E}$ oscillates strictly within a single, fixed plane containing the direction of propagation.

1. Polarization by Selective Absorption (Polaroids) and Malus's Law

Certain materials (such as Polaroid sheets containing oriented long-chain polyvinyl alcohol molecules impregnated with iodine) absorb electric field components aligned with their polymer chains while transmitting components perpendicular to them. The transmission axis defines the vibration plane of the transmitted light.

When unpolarized light of initial intensity $I_{\text{unpol}}$ passes through an ideal polarizer, exactly half of the total intensity is transmitted:

$I_1 = \frac{1}{2}I_{\text{unpol}}$

When this linearly polarized light strikes a second polarizing filter (the analyzer) whose transmission axis is rotated by angle $\theta$ relative to the polarizer, the transmitted electric field amplitude is $E = E_0\cos\theta$. Because intensity is proportional to $E^2$, the transmitted intensity obeys Malus's Law:

$I = I_0\cos^2\theta$

• If $\theta = 0^\circ$ (parallel axes): $I = I_0$ (maximum transmission).
• If $\theta = 90^\circ$ (crossed axes): $I = I_0\cos^2(90^\circ) = 0$ (complete extinction).
• If $\theta = 45^\circ$: $I = I_0\cos^2(45^\circ) = \frac{I_0}{2}$.

2. Polarization by Reflection: Brewster's Law

When unpolarized light reflects from a dielectric boundary (such as air to glass or air to water), the reflected beam is partially polarized. In 1815, Sir David Brewster discovered that at a specific angle of incidence $\theta_p$ (called Brewster's angle or the polarizing angle), the reflected light becomes $100\%$ linearly polarized, with its electric field vibrating strictly parallel to the reflecting surface (perpendicular to the plane of incidence).

Brewster established that at this angle, the reflected ray and refracted ray are mutually perpendicular:

$\theta_p + r = 90^\circ \implies r = 90^\circ - \theta_p$

Applying Snell's law at the boundary between air ($n_1 = 1$) and medium ($n_2 = \mu$):

$\frac{\sin\theta_p}{\sin r} = \mu \implies \frac{\sin\theta_p}{\sin(90^\circ - \theta_p)} = \mu \implies \frac{\sin\theta_p}{\cos\theta_p} = \mu$

$\tan\theta_p = \mu$

This fundamental relation is Brewster's Law.

• For air to water ($n = 1.33 = \frac{4}{3}$): $\tan\theta_p = 1.33 \implies \theta_p \approx 53.1^\circ$.
• For air to crown glass ($n = 1.50$): $\tan\theta_p = 1.50 \implies \theta_p \approx 56.3^\circ$.

3. Double Refraction (Birefringence) & Optical Activity

• Birefringence: Anisotropic crystals (such as calcite $\text{CaCO}_3$ and quartz) possess direction-dependent refractive indices. An incident unpolarized ray splits into two distinct polarized rays:
  1. Ordinary Ray ($O$-ray): Vibrates perpendicular to the optic axis, obeys Snell's law, and travels with constant speed in all directions.
  2. Extraordinary Ray ($E$-ray): Vibrates in the plane of the optic axis, violates Snell's law, and travels with direction-dependent velocity.
• Optical Activity: The property of certain chiral organic substances (such as glucose, sucrose, tartaric acid) and crystals to rotate the plane of vibration of linearly polarized light through an angle $\theta$. The angle of optical rotation is measured using a polarimeter to quantify chemical concentration.

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7. Diffraction and Polarization Summary Table