General Physics II: Optics, Interference, and Diffraction Study Notes

Nature and Propagation of Light

• Wave-Particle Duality of Light:

  • Propagation of light is best described using a wave model.
  • Emission and absorption of light require a particle approach involving photons.
  • Accelerated electric charges emit electromagnetic light waves.
  • In ordinary light sources (such as incandescent light bulbs or flames), individual atoms emit light independently and uncooperatively, producing a continuous spectrum of frequencies.
  • In a laser (Light Amplification by Stimulated Emission of Radiation), atoms are induced to emit light in a cooperative, coherent fashion, producing an extremely narrow, intense, and highly monochromatic (single-frequency) beam.

• Universal Speed of Light in Vacuum ($c$):

  • Electromagnetic radiation in vacuum travels at a constant speed:     $c = 2.99792458 \times 10^8\,\text{m/s}$
  • To three significant figures, the speed of light is:     $c = 3.00 \times 10^8\,\text{m/s}$

• Wave Fronts:

  • A wave front is defined as the locus of all adjacent points at which the phase of vibration of a physical quantity associated with the wave is identical.
  • At any instant, all points on a wave front are at the exact same part of their variation cycle.
  • Examples of wave fronts include:
  • Concentric expanding circular crests and troughs formed by dropping a pebble into a calm pool of water.
  • Concentric spherical surfaces expanding from a point sound source in still air or a point electromagnetic emitter.
  • Wave front diagrams usually show surfaces corresponding to identical phase separated by one wavelength ($\lambda$), such as wave crests, pressure maxima in sound waves, or electric/magnetic field maxima in electromagnetic waves.
  • At large distances from a small source, a section of a spherical wave front becomes nearly planar, forming a plane wave.

• Light Rays:
  • A ray is an imaginary line drawn along the direction of wave propagation, perpendicular to wave fronts at all points.
  • In particle theory, rays represent the physical trajectories of light particles.
  • For spherical wave fronts, rays radiate outward from the central source.
  • For planar wave fronts, rays are parallel to each other and perpendicular to the wave fronts.

Laws of Reflection and Refraction

• Behavior at Smooth Interfaces:
  • When a light wave hits a smooth boundary between two transparent optical materials (such as air/glass or water/glass), it splits into a reflected wave returning into the first medium and a refracted (transmitted) wave entering the second medium.
  • Light beams are modeled as bundles of rays. At a transparent glass window, an outside observer sees a reflected image while an inside observer sees a refracted image.

• Specular vs. Diffuse Reflection:
  • Specular Reflection: Reflection at a definite angle from a very smooth surface (such as polished glass or metal).
  • Diffuse Reflection: Reflection scattered in random directions from a rough surface.

• Index of Refraction ($n$):

  • The index of refraction (refractive index) of a material is the ratio of the speed of light in vacuum ($c$) to the speed of light in the material ($v$):     $n = \frac{c}{v}$
  • In vacuum, $n = 1$. In all transparent matter, light travels slower than in vacuum ($v < c$), so $n > 1$.
  • Wave speed in a medium is inversely proportional to its refractive index:     $v = \frac{c}{n}$

• Tabulated Indexes of Refraction (Table 33.1 for Yellow Sodium Light, $\lambda_0 = 589\,\text{nm}$):

  • Solids:
  • Ice ($\text{H}_2\text{O}$): $n = 1.309$
  • Fluorite ($\text{CaF}_2$): $n = 1.434$
  • Polystyrene: $n = 1.49$
  • Rock salt ($\text{NaCl}$): $n = 1.544$
  • Quartz ($\text{SiO}_2$): $n = 1.544$
  • Zircon ($\text{ZrO}_2 \cdot \text{SiO}_2$): $n = 1.923$
  • Diamond ($\text{C}$): $n = 2.417$
  • Fabulite ($\text{SrTiO}_3$): $n = 2.409$
  • Rutile ($\text{TiO}_2$): $n = 2.62$
  • Glasses (typical values):
  • Crown glass: $n = 1.52$
  • Light flint glass: $n = 1.58$
  • Medium flint glass: $n = 1.62$
  • Dense flint glass: $n = 1.66$
  • Lanthanum flint glass: $n = 1.80$
  • Liquids at $20^\circ\text{C}$:
  • Methanol ($\text{CH}_3\text{OH}$): $n = 1.329$
  • Water ($\text{H}_2\text{O}$): $n = 1.333$
  • Ethanol ($\text{C}_2\text{H}_5\text{OH}$): $n = 1.36$
  • Carbon tetrachloride ($\text{CCl}_4$): $n = 1.460$
  • Turpentine: $n = 1.472$
  • Glycerine: $n = 1.473$
  • Benzene: $n = 1.501$
  • Carbon disulfide ($\text{CS}_2$): $n = 1.628$

• The Laws of Reflection and Refraction:
  • 1. Plane of Incidence: The incident ray, reflected ray, refracted ray, and surface normal at the point of incidence all lie in a single plane.
  • 2. Law of Reflection: The angle of reflection $\theta_r$ equals the angle of incidence $\theta_a$ for all wavelengths and materials:     $\theta_r = \theta_a$
  • 3. Law of Refraction (Snell's Law): For monochromatic light crossing an interface between media $a$ and $b$, the ratio of the sines of angles $\theta_a$ and $\theta_b$ measured from the normal equals the inverse ratio of their refractive indexes:     $\frac{\sin(\theta_a)}{\sin(\theta_b)} = \frac{n_b}{n_a}$$n_a \sin(\theta_a) = n_b \sin(\theta_b)$

• Directional Rules for Refraction:
  • Entering a higher index medium ($n_b > n_a$): Light slows down and bends toward the normal ($\theta_b < \theta_a$).
  • Entering a lower index medium ($n_b < n_a$): Light speeds up and bends away from the normal ($\theta_b > \theta_a$).
  • Normal Incidence: A ray striking along the surface normal ($\theta_a = 0^\circ$) passes straight without bending ($\theta_b = 0^\circ$).
  • Apparent Depth Effect: A ruler half-immersed in water appears bent at the air-water surface because light rays coming from underwater bend away from the normal upon emerging into air ($n_{\text{air}} = 1.00 < n_{\text{water}} = 1.33$), placing the virtual image of the ruler closer to the surface than its actual position.

• Wave Characteristics during Refraction:

  • Frequency Invariance: The frequency $f$ of light remains unchanged when crossing between materials because the boundary surface cannot create or destroy wave cycles.
  • Wavelength Variation: Because $v = \lambda f$ and $f$ is constant, the wavelength $\lambda$ in a medium with refractive index $n$ is reduced relative to its vacuum wavelength $\lambda_0$:     $\lambda = \frac{\lambda_0}{n}$
  • Ratio between two media:     $\frac{\lambda_a}{\lambda_b} = \frac{n_b}{n_a}$

• Worked Example 33.1 (Reflection and Refraction Angles):

  • Incident ray in water ($n_a = 1.33$) strikes glass ($n_b = 1.52$) at an angle $\theta_a = 60.0^\circ$.
  • Reflected angle:     $\theta_r = \theta_a = 60.0^\circ$
  • Refracted angle:     $n_a \sin(\theta_a) = n_b \sin(\theta_b)$$\sin(\theta_b) = \frac{n_a}{n_b} \sin(\theta_a) = \frac{1.33}{1.52} \sin(60.0^\circ) = \frac{1.33}{1.52} (0.8660) = 0.758$$\theta_b = \arcsin(0.758) = 49.3^\circ$
  • Evaluation: Because $n_b > n_a$, the refracted ray bends toward the normal ($\theta_b < \theta_a$).

• Worked Example 33.2 (Laser Light in the Human Eye):

  • Red light from a helium-neon laser has wavelength $\lambda_0 = 633\,\text{nm}$ in air/vacuum and $\lambda = 474\,\text{nm}$ in the aqueous humor inside an eyeball.
  • Index of Refraction of aqueous humor:     $n = \frac{\lambda_0}{\lambda} = \frac{633\,\text{nm}}{474\,\text{nm}} = 1.34$
  • Speed of Light in aqueous humor:     $v = \frac{c}{n} = \frac{3.00 \times 10^8\,\text{m/s}}{1.34} = 2.25 \times 10^8\,\text{m/s}$
  • Frequency of Light in aqueous humor:     $f = \frac{v}{\lambda} = \frac{2.25 \times 10^8\,\text{m/s}}{474 \times 10^{-9}\,\text{m}} = 4.74 \times 10^{14}\,\text{Hz}$
  • Frequency in vacuum/air:     $f_0 = \frac{c}{\lambda_0} = \frac{3.00 \times 10^8\,\text{m/s}}{633 \times 10^{-9}\,\text{m}} = 4.74 \times 10^{14}\,\text{Hz}$
  • Evaluation: Frequency remains identical ($f = f_0$) across media.

Total Internal Reflection

• Critical Angle ($\theta_{\text{crit}}$):

  • Total internal reflection occurs when light traveling in a material with higher index $n_a$ meets a material with lower index $n_b$ ($n_a > n_b$).
  • As the angle of incidence $\theta_a$ increases, the refracted angle $\theta_b$ increases toward $90^\circ$.
  • The critical angle $\theta_{\text{crit}}$ is the incident angle at which the refracted ray emerges tangent to the boundary surface ($\theta_b = 90^\circ$, $\sin(\theta_b) = 1$).

• Formula for Critical Angle:   $n_a \sin(\theta_{\text{crit}}) = n_b \sin(90^\circ)$$\sin(\theta_{\text{crit}}) = \frac{n_b}{n_a}$

• Condition for Total Internal Reflection:

  • Total internal reflection takes place if and only if:
  1. $n_a > n_b$ (light moves toward an optically less dense medium).
  2. The angle of incidence $\theta_a \ge \theta_{\text{crit}}$.
  • For $\theta_a \ge \theta_{\text{crit}}$, no light passes into the second material; 100% of the light energy is internally reflected back into the first medium.

Image Formation by Reflection at Plane and Spherical Surfaces

• Object and Image Concepts:

  • Object Point ($P$): Source point from which light rays originate.
  • Image Point ($P'$): Point from which outgoing reflected or refracted rays appear to diverge or at which they actually converge.
  • Real Image: Image formed when outgoing rays physically pass through the image point $P'$.
  • Virtual Image: Image formed when outgoing rays do not pass through $P'$, but their backwards extensions intersect at $P'$.

• Universal Sign Rules for Optics:

  • 1. Object Distance ($s$): $s > 0$ when the object is on the incoming light side of the surface (real object); $s < 0$ otherwise.
  • 2. Image Distance ($s'$): $s' > 0$ when the image is on the outgoing light side of the surface (real image); $s' < 0$ otherwise (virtual image).
  • 3. Radius of Curvature ($R$): $R > 0$ when the center of curvature $C$ is on the outgoing light side of the surface; $R < 0$ otherwise.
  • 4. Lateral Magnification ($m$): $m > 0$ when the image is erect (upright); $m < 0$ when the image is inverted.

• Image Formation by a Plane Mirror:

  • Outgoing reflected rays diverge from a virtual image point $P'$ located behind the mirror.
  • Object-Image Distance Relationship:     $s = -s'$
  • Lateral Magnification:     $m = \frac{y'}{y} = -\frac{s'}{s} = +1$
  • An image formed by a plane mirror is virtual, erect, unmagnified ($y' = y$), and located at an image distance $s' = -s$ behind the mirror surface.

• Geometry and Parameters of Spherical Mirrors:
  • Center of Curvature ($C$): Center of the sphere forming the mirror surface.
  • Vertex ($V$) or Pole ($P$): Center point on the mirror surface itself.
  • Optic Axis / Principal Axis: Line passing through $C$ and $V$.
  • Focal Point ($F$): Point to which incoming parallel rays converge (concave) or from which they appear to diverge (convex).
  • Focal Length ($f$): Distance from vertex $V$ to focal point $F$, given by $f = \frac{R}{2}$.
  • Paraxial Rays: Rays close to and nearly parallel to the optic axis. Rays far from the axis do not converge to a single focal point, causing spherical aberration.

• Equations for Spherical Mirrors:
  • Object-Image Equation:     $\frac{1}{s} + \frac{1}{s'} = \frac{2}{R} = \frac{1}{f}$
  • Focal Length:     $f = \frac{R}{2}$
  • Lateral Magnification:     $m = \frac{y'}{y} = -\frac{s'}{s}$

• Four Principal Rays for Spherical Mirrors:
  • Ray 1: Incident parallel to optic axis $\rightarrow$ reflects through focal point $F$ (concave) or appears to come from $F$ (convex).
  • Ray 2: Incident through focal point $F$ (concave) or directed toward $F$ (convex) $\rightarrow$ reflects parallel to optic axis.
  • Ray 3: Incident along radius through center of curvature $C$ $\rightarrow$ strikes surface normally and reflects along its original path.
  • Ray 4: Incident at vertex $V$ $\rightarrow$ reflects symmetrically around optic axis.

• Summary Table of Image Formation by Concave Mirrors:
  • Object at Infinity ($s = \infty$):
  • Image Position: At focus $F$
  • Image Size: Highly diminished, point-sized
  • Image Nature: Real and inverted
  • Object Beyond Center of Curvature ($s > R$):
  • Image Position: Between $F$ and $C$
  • Image Size: Diminished
  • Image Nature: Real and inverted
  • Object at Center of Curvature ($s = R$):
  • Image Position: At $C$
  • Image Size: Same size ($|m| = 1$)
  • Image Nature: Real and inverted
  • Object Between $C$ and $F$ ($f < s < R$):
  • Image Position: Beyond $C$
  • Image Size: Enlarged ($|m| > 1$)
  • Image Nature: Real and inverted
  • Object at Focus $F$ ($s = f$):
  • Image Position: At infinity ($s' = \infty$)
  • Image Size: Highly enlarged
  • Image Nature: Real and inverted (parallel outgoing rays)
  • Object Between Pole $V$ and Focus $F$ ($s < f$):
  • Image Position: Behind the mirror ($s' < 0$)
  • Image Size: Enlarged ($m > 1$)
  • Image Nature: Virtual and erect

• Applications of Concave Mirrors:
  • Shaving Mirrors and Makeup Mirrors: Positioned near the face ($s < f$) to produce an enlarged, erect virtual image.
  • Dentist Mirrors: Placed near teeth ($s < f$) so the dentist sees an enlarged virtual image of teeth.
  • Torches, Headlights, and Beam Lights: Light bulb placed at focus $F$ ($s = f$) produces a powerful parallel light beam.
  • Solar Furnaces: Large concave reflectors concentrate incoming parallel sunlight onto focus $F$, producing extreme heat.

• Summary Table of Image Formation by Convex Mirrors:
  • Convex Mirrors ($R < 0$, $f < 0$) always produce erect, diminished, virtual images behind the mirror surface.
  • Object at Infinity ($s = \infty$):
  • Image Position: At focus $F$ behind mirror
  • Image Size: Highly diminished, point-sized
  • Image Nature: Virtual and erect
  • Object Between Infinity and Pole $V$:
  • Image Position: Between $V$ and $F$ behind mirror
  • Image Size: Diminished
  • Image Nature: Virtual and erect

• Applications of Convex Mirrors:
  • Rear-View Mirrors for Vehicles: Images produced are erect (right-side up) and diminished, affording drivers a wide field of view.
  • Shop Security Mirrors: Installed in stores to afford shopkeepers a wide-angle view of large shop areas to prevent shoplifting.

Refraction at Spherical and Plane Surfaces

• Refraction at a Spherical Surface:
  • For a spherical interface between medium $a$ ($n_a$) and medium $b$ ($n_b$) with radius of curvature $R$:     $\frac{n_a}{s} + \frac{n_b}{s'} = \frac{n_b - n_a}{R}$
  • Lateral Magnification for spherical refraction:     $m = \frac{y'}{y} = -\frac{n_a s'}{n_b s}$

• Worked Example (Cylindrical Glass Rod with Hemispherical End):

  • A glass rod ($n_b = 1.52$) in air ($n_a = 1.00$) has a hemispherical surface with radius $R = +2.00\,\text{cm}$. An object is placed on the rod axis $s = +8.00\,\text{cm}$ left of vertex.
  • Find Image Distance $s'$, position, and Lateral Magnification $m$:     $\frac{1.00}{8.00\,\text{cm}} + \frac{1.52}{s'} = \frac{1.52 - 1.00}{+2.00\,\text{cm}}$$0.125\,\text{cm}^{-1} + \frac{1.52}{s'} = 0.260\,\text{cm}^{-1}$$\frac{1.52}{s'} = 0.135\,\text{cm}^{-1} \implies s' = +11.3\,\text{cm}$
  • Image is real and located $11.3\,\text{cm}$ inside the glass rod to the right of vertex.
  • Lateral Magnification:     $m = -\frac{(1.00)(11.3\,\text{cm})}{(1.52)(8.00\,\text{cm})} = -0.929$
  • The image is inverted ($m < 0$) and slightly diminished in size.

• Refraction at a Plane Surface ($R = \infty$):

  • Setting $R = \infty$ yields:     $\frac{n_a}{s} + \frac{n_b}{s'} = 0 \implies s' = -\frac{n_b}{n_a} s$
  • Lateral Magnification:     $m = -\frac{n_a s'}{n_b s} = +1$
  • A plane refracting surface always creates an erect virtual image with lateral size identical to the object ($m = 1$).

• Worked Example (Apparent Depth of a Swimming Pool):
  • A swimming pool is $s = 2.00\,\text{m}$ deep. Water has $n_a = 1.33$, air has $n_b = 1.00$. Looking straight down:     $s' = -\frac{1.00}{1.33} (2.00\,\text{m}) = -1.50\,\text{m}$
  • The pool appears to be $1.50\,\text{m}$ deep ($3/4$ of its true depth).

Thin Lenses and Lens Combinations

• Types and Definitions of Thin Lenses:
  • Converging Lens: Thicker at center than at edges; positive focal length ($f > 0$). Types: Meniscus, Planoconvex, Double convex.
  • Diverging Lens: Thicker at edges than at center; negative focal length ($f < 0$). Types: Meniscus, Planoconcave, Double concave.

• Equations for Thin Lenses:
  • Thin Lens Object-Image Equation:     $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$
  • Lensmaker's Equation:     $\frac{1}{f} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$     where $n$ is index of lens material, $R_1$ is radius of curvature of first surface, and $R_2$ is radius of curvature of second surface.
  • Lateral Magnification:     $m = \frac{y'}{y} = -\frac{s'}{s}$

• Three Principal Rays for Thin Lenses:
  • Ray 1: Parallel to optic axis $\rightarrow$ refracts through focal point $F_2$ (converging) or appears to come from $F_2$ (diverging).
  • Ray 2: Passes through lens center $\rightarrow$ continues without deviation.
  • Ray 3: Passes through focal point $F_1$ (converging) or directed toward $F_1$ (diverging) $\rightarrow$ refracts parallel to optic axis.

• Combination of Lenses (Image of an Image):

  • For two lenses in sequence, the image formed by the first lens acts as the object for the second lens.
  • Total magnification is the product of individual magnifications:     $m_{\text{overall}} = m_A \times m_B$

• Worked Example (Two Converging Lenses System):

  • Lens A ($f_A = +8.0\,\text{cm}$) and Lens B ($f_B = +6.0\,\text{cm}$) are separated by $36.0\,\text{cm}$. Object $O$ of height $y = 8.0\,\text{cm}$ is placed $12.0\,\text{cm}$ left of Lens A.
  • Step 1: Image $I$ formed by Lens A:     $\frac{1}{12.0\,\text{cm}} + \frac{1}{s'_A} = \frac{1}{8.0\,\text{cm}} \implies \frac{1}{s'_A} = \frac{1}{8.0} - \frac{1}{12.0} = \frac{1}{24.0\,\text{cm}} \implies s'_A = +24.0\,\text{cm}$$m_A = -\frac{s'_A}{s_A} = -\frac{24.0\,\text{cm}}{12.0\,\text{cm}} = -2.00$     Image $I$ is inverted, real, $16.0\,\text{cm}$ tall, located $24.0\,\text{cm}$ right of Lens A.
  • Step 2: Object distance for Lens B ($s_B$):     $s_B = 36.0\,\text{cm} - 24.0\,\text{cm} = +12.0\,\text{cm}$
  • Step 3: Final Image $I'$ formed by Lens B:     $\frac{1}{12.0\,\text{cm}} + \frac{1}{s'_B} = \frac{1}{6.0\,\text{cm}} \implies \frac{1}{s'_B} = \frac{1}{6.0} - \frac{1}{12.0} = \frac{1}{12.0\,\text{cm}} \implies s'_B = +12.0\,\text{cm}$$m_B = -\frac{s'_B}{s_B} = -\frac{12.0\,\text{cm}}{12.0\,\text{cm}} = -1.00$
  • Step 4: Overall Magnification and Height:     $m_{\text{overall}} = m_A \times m_B = (-2.00) \times (-1.00) = +2.00$$y'' = (+2.00)(8.0\,\text{cm}) = 16.0\,\text{cm}$     Final image $I'$ is real, erect (same orientation as object $O$), $16.0\,\text{cm}$ tall, located $12.0\,\text{cm}$ right of Lens B.

Two-Source Interference of Light

• Interference Definition:

  • Interference refers to any situation in which two or more light waves overlap in space.

• Principle of Superposition:

  • When two or more waves overlap, the resultant displacement at any point and instant is the sum of instantaneous displacements produced by individual waves.

• Coherence:

  • Two monochromatic sources are coherent if they have identical frequency $f$, identical wavelength $\lambda$, and maintain a constant phase relationship.

• Conditions for Interference between In-Phase Coherent Sources:

  • Path Difference: $\Delta r = r_2 - r_1$
  • Constructive Interference (Reinforcement): Waves arrive in phase (crest meets crest).     $r_2 - r_1 = m \lambda \quad (m = 0, \pm 1, \pm 2, \pm 3, \dots)$
  • Destructive Interference (Cancellation): Waves arrive $180^\circ$ (a half-cycle) out of phase (crest meets trough).     $r_2 - r_1 = \left(m + \frac{1}{2}\right) \lambda \quad (m = 0, \pm 1, \pm 2, \pm 3, \dots)$

• Young's Double-Slit Experiment:
  • Setup: Monochromatic light passes through single slit $S_0$ ($\approx 1\,\mu\text{m}$ wide), forming coherent wave fronts that strike two narrow slits $S_1$ and $S_2$ separated by distance $d$.
  • Geometry for distant screen ($R \gg d$): Rays to point $P$ are approximately parallel.
  • Path Difference:     $r_2 - r_1 = d \sin(\theta)$
  • Constructive Interference (Bright Fringes):     $d \sin(\theta) = m \lambda \quad (m = 0, \pm 1, \pm 2, \dots)$
  • Destructive Interference (Dark Fringes):     $d \sin(\theta) = \left(m + \frac{1}{2}\right) \lambda \quad (m = 0, \pm 1, \pm 2, \dots)$
  • Fringe Position $y_m$ on Screen for Small Angles ($\sin(\theta) \approx \tan(\theta) = \frac{y_m}{R}$):     $y_m = R \frac{m \lambda}{d} \quad (m = 0, \pm 1, \pm 2, \dots)$

• Worked Example 35.1 (Two-Slit Wavelength Measurement):

  • Slits separation $d = 0.200\,\text{mm} = 0.200 \times 10^{-3}\,\text{m}$, screen distance $R = 1.00\,\text{m}$, $m = 3$ fringe distance $y_3 = 9.49\,\text{mm} = 9.49 \times 10^{-3}\,\text{m}$.
  • Calculate Wavelength $\lambda$:     $\lambda = \frac{y_3 d}{m R} = \frac{(9.49 \times 10^{-3}\,\text{m})(0.200 \times 10^{-3}\,\text{m})}{(3)(1.00\,\text{m})} = 633 \times 10^{-9}\,\text{m} = 633\,\text{nm}$

• Worked Example 35.2 (Broadcast Pattern of Radio Station Antennas):

  • Two antennas separated by $d = 400\,\text{m}$ oscillate in phase at $f = 1500\,\text{kHz} = 1.5 \times 10^6\,\text{Hz}$.
  • Wavelength:     $\lambda = \frac{c}{f} = \frac{3.00 \times 10^8\,\text{m/s}}{1.5 \times 10^6\,\text{Hz}} = 200\,\text{m}$
  • Intensity Maxima Angles:     $\sin(\theta) = \frac{m \lambda}{d} = \frac{m (200\,\text{m})}{400\,\text{m}} = \frac{m}{2}$
  • $m = 0 \implies \sin(\theta) = 0 \implies \theta = 0^\circ$
  • $m = \pm 1 \implies \sin(\theta) = \pm 0.5 \implies \theta = \pm 30^\circ$
  • $m = \pm 2 \implies \sin(\theta) = \pm 1.0 \implies \theta = \pm 90^\circ$
  • Values of $|m| \ge 3$ yield $\sin(\theta) > 1$, which is unphysical. Maxima directions are $0^\circ$, $\pm 30^\circ$, and $\pm 90^\circ$.

Intensity Distribution in Two-Source Interference

• Wave Number $k$ and Phase Difference $\phi$:

  • Wave number:     $k = \frac{2\pi}{\lambda}$
  • Phase difference $\phi$ from path difference $r_2 - r_1$:     $\phi = \frac{2\pi}{\lambda} (r_2 - r_1) = k (r_2 - r_1)$

• Resultant Electric Field Amplitude and Intensity:

  • Resultant Electric Field Amplitude $E_P$:     $E_P = 2 E \left|\cos\left(\frac{\phi}{2}\right)\right|$
  • Intensity $I$ as a function of $\phi$:     $I = I_0 \cos^2\left(\frac{\phi}{2}\right)$     where $I_0$ is peak intensity at fringe centers ($4 \times$ individual source intensity).

• Quantitative Relationship between Phase Difference and Intensity:

  • $\phi = 0^\circ \implies \cos^2(0^\circ) = 1.00 \implies I = I_0$
  • $\phi = 60^\circ \implies \cos^2(30^\circ) = 0.75 \implies I = 0.75 I_0$
  • $\phi = 90^\circ \implies \cos^2(45^\circ) = 0.50 \implies I = 0.50 I_0$
  • $\phi = 120^\circ \implies \cos^2(60^\circ) = 0.25 \implies I = 0.25 I_0$
  • $\phi = 180^\circ \implies \cos^2(90^\circ) = 0.00 \implies I = 0$

Single-Slit Diffraction

• Concept of Diffraction:
  • Diffraction is the bending or spreading of waves when passing through an aperture or around an obstacle, produced by mutual interference of wavelets from different parts of the same wave front.
  • Geometric optics predicts a sharp shadow matching slit width $a$. In reality, light spreads out vertically/horizontally to create a broad central bright band flanked by secondary bright and dark bands of decreasing intensity.

• Fresnel vs. Fraunhofer Diffraction:
  • Fresnel (Near-Field) Diffraction: Source or screen is close to slit; incoming/outgoing rays are non-parallel.
  • Fraunhofer (Far-Field) Diffraction: Source and screen are far from slit; rays arriving at screen point $P$ are parallel. Can be imaged onto a nearby screen using a converging lens.

• Location of Dark Fringes (Minima):

  • Dividing slit width $a$ into equal halves, light from corresponding strips in top and bottom halves cancels when path difference $\frac{a}{2} \sin(\theta) = \frac{\lambda}{2}$.
  • General condition for single-slit dark fringes:     $\sin(\theta) = \frac{m \lambda}{a} \quad (m = \pm 1, \pm 2, \pm 3, \dots)$
  • Note: $m = 0$ corresponds to the central bright maximum, NOT a dark fringe.

• Small-Angle Approximations ($\sin(\theta) \approx \theta$, $\tan(\theta) \approx \theta = \frac{y_m}{x}$):   $\theta = \frac{m \lambda}{a} \quad (m = \pm 1, \pm 2, \pm 3, \dots)$$y_m = x \frac{m \lambda}{a} \quad (m = \pm 1, \pm 2, \pm 3, \dots)$

• Worked Example 36.1 (Slit Width Calculation):

  • Laser light $\lambda = 633\,\text{nm} = 633 \times 10^{-9}\,\text{m}$ passes through a slit onto screen $x = 6.0\,\text{m}$ away. Distance between first minima on either side is $32\,\text{mm}$.
  • Distance to first minimum $y_1 = \frac{32\,\text{mm}}{2} = 16\,\text{mm} = 16 \times 10^{-3}\,\text{m}$.
  • Slit Width $a$:     $a = \frac{x \lambda}{y_1} = \frac{(6.0\,\text{m})(633 \times 10^{-9}\,\text{m})}{16 \times 10^{-3}\,\text{m}} = 2.4 \times 10^{-4}\,\text{m} = 0.24\,\text{mm}$

• Intensity in Single-Slit Diffraction Pattern:   $I = I_0 \left\{ \frac{\sin\left[\frac{\pi a \sin(\theta)}{\lambda}\right]}{\frac{\pi a \sin(\theta)}{\lambda}} \right\}^2$

  • Peak intensity of central maximum ($\theta = 0$): $I = I_0$.
  • Peak intensities of secondary bright maxima:
  • First secondary peak ($m = 1$): $I \approx 0.0472 I_0$
  • Second secondary peak ($m = 2$): $I \approx 0.0165 I_0$
  • Third secondary peak ($m = 3$): $I \approx 0.0083 I_0$

Diffraction Gratings and Spectroscopy

• Structure and Principle:

  • A diffraction grating consists of a large number $N$ of thin parallel slits (rulings or lines), each of width $a$, spaced equal distances $d$ apart.
  • Grating spacing $d$ is the distance between centers of adjacent slits.

• Condition for Intensity Maxima:

  • Intensity maxima occur at angles $\theta$ where path difference between adjacent slits is an integer number of wavelengths:     $d \sin(\theta) = m \lambda \quad (m = 0, \pm 1, \pm 2, \pm 3, \dots)$
  • Line Orders:
  • $m = 0$: Central zero-order line
  • $m = \pm 1$: First-order lines
  • $m = \pm 2$: Second-order lines
  • Gratings with thousands of slits produce extraordinarily sharp, narrow, and intense intensity maxima (peak intensity proportional to $N^2 I_0$).

• Spectroscopy and Astronomical Applications:
  • Spectroscopy (Spectrometry) is the practice of dispersing light into its spectrum to measure wavelengths precisely.
  • Sunlight passing through the Sun's atmosphere undergoes selective absorption at specific wavelengths. A diffraction grating disperses sunlight into a spectrum containing dark absorption lines (Fraunhofer lines).
  • Comparing solar absorption line wavelengths to laboratory gas spectra allows astronomers to deduce the exact chemical composition of the Sun and distant galaxies.