Interference, Diffraction, and Polarization

28.1 Why Wave Optics Is Needed

The ray model of light (geometric optics) treats light as straight-line rays that reflect and refract at boundaries. It works beautifully for mirrors, lenses, and prisms — systems where wavelength is much smaller than any relevant dimension. But it completely fails to predict some of the most striking behaviors of light.

produce a sharp-edged bright strip on the far side. In reality, if the slit is narrow enough (comparable to the wavelength), the light fans out broadly beyond the slit, and a complex pattern of bright and dark bands appears. Geometric optics predicts that

and darkness: interference fringes.

These phenomena — diffraction and interference — demand the wave model. They are not marginal effects: they occur whenever light interacts with structures on the scale of its wavelength (hundreds of nanometers), which includes almost all fine structures in Physics: Foundations of Physical Science nature and technology. The biological optics of butterfly wings and peacock feathers, the spectroscopy used to identify atoms and molecules, the CD and DVD, the anti- reflection coating on camera lenses, the resolution limit of every telescope and microscope, the operation of interferometric gravitational wave detectors — all of these depend on wave optics.

A system moves from geometric to wave optics when the wavelength

500 nm), this means structures smaller than about 0.1 mm begin to require wave treatment.

28.2 Review of Superposition in Waves

Wave optics is built on the superposition principle established in Chapter 24: when two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements at every point. This is exact for linear media (which includes all optical materials for normal light intensities).

Two key concepts govern the outcome of superposition: phase and coherence. Phase describes where in its oscillation cycle a wave is at a given point. Two waves are in phase if their oscillations align (crest with crest, trough with trough). They are out of phase if one’s crest aligns with the other’s trough.

Coherence is the property of two waves that enables stable interference. Two waves are coherent if they maintain a fixed phase relationship over time. A laser produces highly coherent light; an incandescent bulb produces incoherent light (random phase relationships between different atomic emissions).

Incoherent sources can produce interference, but only over extremely short path-length differences; for the interference patterns studied here, coherent sources (lasers, or light from the same narrow source passed through both slits) are required.

For stable interference patterns, the sources must be coherent: they must have the same frequency and a fixed (not random) phase relationship. In practice, this means using the same source for both interfering beams.

28.3 Constructive and Destructive Interference

When two coherent waves of the same frequency and amplitude meet, the nature of their interference depends entirely on the path-length difference δ (delta) between the waves: the difference in distance each wave traveled from its source to the observation point.

Physics: Foundations of Physical Science Constructive interference occurs when the waves arrive in phase: crest aligns with crest, trough with trough. This happens when the path-length difference is a whole number of wavelengths. The waves reinforce, producing amplitude equal to the sum of the individual amplitudes (and intensity proportional to the square of amplitude).

Destructive interference occurs when the waves arrive exactly out of phase: crest aligns with trough. This happens when the path-length difference is a half-integer number of wavelengths. The waves cancel, producing zero amplitude (and zero intensity for equal- amplitude waves).

Destructive:

A phase shift of 180° (half wavelength) can also occur at a reflection boundary when light reflects from a more optically dense medium (higher n). This ‘reflection phase shift’ changes which path differences give constructive vs. destructive interference in thin- film problems.

28.4 Interference of Light Waves

Optical interference is interference between light waves. Because the wavelength of visible light is roughly 400–700 nm, interference effects are visible only when path differences are controlled on this nanometer scale. Two broad contexts produce visible

travel different paths, as in thin films).

Thin-film interference is one of the most common and visually beautiful optical phenomena. When light strikes a thin transparent film (soap bubble, oil slick on water, anti-reflection coating on a lens), it partially reflects from the front surface and partially from the back surface.

These two reflected beams travel different path lengths (differing by approximately 2 times the film thickness times the index) and interfere when they reunite.

The interference is constructive for some wavelengths and destructive for others, producing the brilliant colors seen in soap films.

Newton’s rings are a thin-film interference pattern formed when a slightly curved glass lens rests on a flat glass plate. The air gap between them increases outward from the center, creating a set of concentric circular interference rings. Newton observed these rings in the 17th century, but it was Young in the 19th century who correctly explained Physics: Foundations of Physical Science them using wave optics.

1.38) 100 nm. This quarter-wave coating causes destructive interference of reflected green light, greatly reducing glare.

28.5 Young's Double-Slit Experiment

Young's double-slit experiment (1801) provided decisive evidence that light behaves as

coherent light source. On a screen beyond the slits, instead of the two bright strips predicted by particle/ray optics, he observed a regular pattern of alternating bright and dark bands: interference fringes.

The explanation: each slit acts as a coherent source of circular (cylindrical in 3D) wave fronts. These two sets of wave fronts overlap beyond the slits and interfere. At points on the screen where the path-length difference from the two slits is a whole number of wavelengths, constructive interference produces a bright fringe. Where the path-length difference is a half-integer of wavelengths, destructive interference produces a dark fringe.

For slits separated by distance d, with a screen at distance L (where L ≫d), the position y on the screen of the m-th bright fringe (measured from the center) is:

…)

wavelength, larger screen distance, or smaller slit separation all increase the fringe spacing. For visible light, fringe spacings of millimeters are easily produced with millimeter-scale slit separations and meter-scale screen distances.

Example — Double-slit fringe spacing

(6.0 × 10 ⁷ × 2.0) / (5.0 × 10 ⁴) = 2.4 × 10 ³ m = 2.4 mm.

Young's 1801 experiment was met with fierce opposition from supporters of Newton's particle theory of light. Physicists such as Henry Brougham attacked Young's work vigorously. Young responded with characteristic calm precision.

It was not until Fresnel's mathematical wave theory (1818) and Foucault's direct Physics: Foundations of Physical Science measurement showing light travels slower in water than in air (1850) that the wave theory became universally accepted.

28.6 Diffraction and Wave Spreading

Diffraction is the bending and spreading of a wave as it passes through an opening or

a, strong diffraction and complex interference patterns appear.

Diffraction is not an anomaly or edge effect — it is fundamental to wave propagation.

Every lens, telescope, and microscope is limited by diffraction. The minimum resolvable detail in any optical instrument is determined not by the quality of the glass but by the diffraction of light at the aperture. This is the diffraction limit (or Rayleigh limit), and it defines the fundamental resolution limit of all wave-based imaging systems.

Rayleigh criterion for resolution: two point objects are just barely resolved when the central diffraction maximum of one falls on the first diffraction minimum of the other.

For a circular aperture of diameter D, the minimum angular separation of two resolvable points is:

(Rayleigh criterion for circular aperture) The factor 1.22 comes from the mathematics of diffraction from a circular opening. This

550 nm in

2.8 × 10 ⁷ radians

0.06 arcseconds — enough to read a

newspaper headline from 2 km away.

28.7 Diffraction Through Openings and Around Obstacles

different parts of the slit interfere with each other on the far side. The result is a central bright maximum flanked by alternating dark and bright bands of decreasing intensity.

Physics: Foundations of Physical Science ±2, …)

wide diffraction patterns; wide slits produce narrow patterns. This inverse relationship is a fundamental feature of diffraction: concentrating a wave in one dimension (narrow slit) causes it to spread in the perpendicular dimension.

Babinet’s Principle: the diffraction pattern from an opaque obstacle has the same intensity distribution as the pattern from an aperture of the same shape and size (except at the exact forward direction). This is why the shadow of a circular disk has a bright spot at its very center (Poisson’s spot or Arago spot) — the diffraction from the circular edge constructively interferes on-axis. This was experimentally predicted as a consequence of Fresnel’s wave theory and confirmed by Arago in 1818, converting many remaining wave-theory skeptics.

28.8 Diffraction Gratings

A diffraction grating is an optical element with a large number of equally spaced, parallel slits or grooves (typically thousands per millimeter). Each slit acts as a coherent source; the combined interference of waves from all slits produces extremely sharp, bright maxima at specific angles.

The grating equation gives the angles at which constructive interference produces bright maxima:

Different wavelengths satisfy the grating equation at different angles, so a grating disperses white light into a complete spectrum much more cleanly and brightly than a

appear at larger angles.

Diffraction gratings are the key element in spectroscopes and spectrometers — instruments that measure the wavelengths of light emitted or absorbed by atoms and molecules. Every element has a unique emission spectrum (set of specific wavelengths), like a fingerprint.

Grating spectroscopy allows astronomers to determine the composition, temperature, and velocity of stars and galaxies from their light; allows chemists to identify compounds; and allows physicists to verify quantum mechanical Physics: Foundations of Physical Science predictions about atomic energy levels.

Example — Grating calculation

1.67 × 10 ⁶ m. For λ = 550 nm, first-order maximum: sinθ₁ = λ/d = 5.50 × 10 ⁷ / 1.67

× 10 ⁶ = 0.329. θ₁ = arcsin(0.329) = 19.2°.

28.9 Polarization of Light

Polarization is a property of transverse waves in which the direction of oscillation is restricted.

For a transverse wave, the oscillation can occur in any direction perpendicular to propagation. Unpolarized light (like sunlight or incandescent light) has its electric field vectors oscillating in all perpendicular directions with equal probability and random phase. Polarized light has its electric field oscillating preferentially in one direction.

axis). Circular polarization: the electric field rotates uniformly around the propagation direction as the wave travels. Elliptical polarization is the general case, of which linear and circular are special cases.

Three mechanisms produce polarized light: selective absorption (polarizing filters), reflection at Brewster’s angle, and scattering. In everyday life, reflected light from non- metallic surfaces (water, glass, road, snow) is partially polarized parallel to the surface.

Polarizing sunglasses exploit this by transmitting only vertically polarized light, blocking the horizontally polarized reflected glare.

28.10 Polarizing Filters

A polarizing filter (polarizer) transmits only the component of the electric field along its transmission axis and absorbs the perpendicular component. When unpolarized light of intensity I₀ passes through an ideal polarizer, the transmitted intensity is exactly I₀/2, because the filter passes only one of the two equally probable oscillation directions.

When polarized light of intensity I passes through a second polarizer (called an

incoming light, the transmitted intensity is given by Malus’s Law:

(Malus’s Law) ₀ When two polarizers are crossed (transmission axes at 90°), no light passes through: Physics: Foundations of Physical Science

produce a completely dark field. Inserting a third polarizer between them at 45° to each will restore some light, which appears counterintuitive but follows directly from Malus’s Law applied sequentially to each polarizer.

Example — Malus's Law

at θ = 35°. I = 20 × cos²(35°) = 20 × (0.8192)² = 20 × 0.671 = 13.4 W/m².

Liquid crystal displays (LCDs) work by sandwiching a liquid crystal layer between two crossed polarizers. The liquid crystal can rotate the polarization of light by 90° when no voltage is applied (letting light through) or leave it unchanged when voltage is applied (blocking light). Switching individual pixels on and off with voltage produces the images on every LCD screen.

28.11 Evidence That Light Is Transverse

The existence of polarization is definitive proof that light is a transverse wave, not a longitudinal one.

A longitudinal wave — like sound — oscillates parallel to its propagation direction; there is only one possible oscillation direction, and polarization is impossible. A transverse wave oscillates perpendicular to propagation, allowing oscillations in many directions — and thus allowing polarization.

Sound cannot be polarized; light can. This experimental fact establishes that light must be transverse. The direction of the electric field’s oscillation is the polarization direction of light. Electromagnetic wave theory confirms this: the electric and magnetic fields in a light wave are always perpendicular to the direction of propagation and to each other.

Brewster’s angle (also called the polarizing angle) provides another demonstration.

reflected light is completely polarized parallel to the surface (s-polarized). The refracted light is partially polarized.

Brewster discovered this in 1815; it is explained by electromagnetic wave theory and is only possible because light is transverse.

(Brewster’s angle)

28.12 Real-World Applications of Wave Optics

Wave optics underpins an enormous range of practical technologies, many of which would be impossible to engineer without understanding interference, diffraction, and Physics: Foundations of Physical Science polarization.

Anti-reflection coatings Camera lenses, microscope objectives, and eyeglasses are coated with thin dielectric films (typically MgF₂ or multiple layers of other materials) designed so that reflected

reflection from 4% to less than 0.5% for a specific wavelength. Multi-layer coatings (10–50 layers) can reduce reflection to below 0.1% across the entire visible spectrum.

Holography A hologram records the interference pattern between a reference beam and a beam scattered from an object. When the hologram is illuminated by a laser, it reconstructs the wavefronts from the object, producing a three-dimensional image. Credit card security holograms, medical imaging, and data storage systems all use holographic principles.

Interferometers Interferometers split a beam into two paths, send them along different routes, and recombine them to measure path-length differences with sub-wavelength precision.

The Michelson interferometer (1887) was used to show that the speed of light is the same in all directions, contributing to the development of special relativity. The LIGO (Laser Interferometer Gravitational-Wave Observatory) uses two 4 km arms; when a gravitational wave passes, one arm stretches by 10 ¹⁸ m — less than one-ten- thousandth the diameter of a proton — which is detected as a change in the interference pattern.

Optical data storage (CD, DVD, Blu-ray) CDs use pits and lands (0.83 μm wide) etched in a reflective aluminum layer. A laser reads the disc: when the beam crosses a pit edge, reflected light from the pit and the surrounding land interfere destructively, signaling a bit transition. DVD uses smaller features (0.74 μm) and a shorter-wavelength laser for higher density. Blu-ray uses a 405 nm blue laser for even finer features (0.32 μm) and higher capacity.

X-ray crystallography X-rays have wavelengths (0.01–1 nm) comparable to interatomic spacings in crystals.

When X-rays diffract from a crystal’s periodic lattice, they produce a diffraction pattern from which the crystal structure can be determined with atomic resolution. DNA,

crystallography.

Physics: Foundations of Physical Science — Condensed from the history of crystallography

28.13 Common Mistakes in Wave Optics Analysis

occur at simple integer wavelengths. These are different formulas for different geometries.

while the double slit has two. The grating produces much sharper, brighter maxima.

Forgetting phase shifts on reflection in thin-film problems. Light reflecting from a medium of higher index undergoes a 180° phase shift (like a string reflecting from a fixed end). If one reflected beam has a phase shift and the other does not, the constructive and destructive conditions are reversed. Always check each interface.

with the m = 1 minimum.

polarizer orientation.

a criterion (a convention), not a physical law. Super-resolution techniques can

molecule localization.

Physics: Foundations of Physical Science in the material must be compared to the wavelength in that material, not the vacuum wavelength.

Chapter Summary

Wave optics is required when wavelength is comparable to relevant dimensions. Ray optics cannot explain interference, diffraction, or polarization.

and destructive interference (δ = (m + ½)λ).

Δy = λL/d. It established light as a wave.

(m ≠0). Narrower slit →wider pattern.

wavelength gives better resolution.

wavelengths for spectroscopy.

Polarization proves light is transverse. Unpolarized light through a polarizer gives I/2. Malus’s Law: I = I₀ cos²θ for incident polarized light.

angle.

Applications: anti-reflection coatings, holography, interferometers, optical storage, X-ray crystallography, LCD screens, polarizing sunglasses.

Key Terms

wave optics (physical optics): The treatment of light as a wave, necessary when wavelength is comparable to relevant dimensions; explains interference, diffraction, Physics: Foundations of Physical Science and polarization. coherence: The property of two wave sources that maintain a fixed phase relationship over time; required for stable interference patterns. path-length difference (δ): The difference in distances traveled by two interfering waves from their sources to the observation point; determines whether interference is constructive or destructive. constructive interference: Interference

amplitudes add; bright fringe in optical interference.

amplitudes cancel; dark fringe in optical interference. thin-film interference: Interference of light reflected from the front and back surfaces of a thin transparent film; produces colored patterns in soap bubbles, oil slicks, and coatings. reflection phase shift: A 180° phase change that occurs when light reflects from a medium of higher index of refraction; equivalent to a half-wavelength path difference.

Young's double-slit experiment: The landmark 1801 experiment by Thomas Young demonstrating that light produces interference fringes, establishing its wave nature. interference fringe: A bright or dark band in an interference pattern; bright fringes are constructive maxima, dark fringes are destructive minima. fringe spacing (Δy): The distance between adjacent bright (or dark) fringes in a double -slit pattern: Δy = λL/d. diffraction: The bending and spreading of a wave as it passes through an aperture or

aperture size. diffraction limit (Rayleigh criterion): The minimum angular separation of two resolvable point objects: θ_min = 1.22λ/D for a circular aperture of diameter D.

producing a central maximum and side minima at a sinθ = mλ. diffraction grating: An optical element with many equally spaced slits or grooves; produces sharp spectral maxima obeying d sinθ = mλ. polarization: The restriction of a transverse wave’s oscillation to a preferred direction; only possible for transverse waves (not longitudinal). polarizing filter (polarizer): An optical element that transmits only the component of the electric field along its transmission axis.

light passes through it at angle θ to the polarization axis.

Physics: Foundations of Physical Science light is completely linearly polarized parallel to the surface. interferometer: An optical instrument that splits and recombines a beam of light to measure path-length differences with sub-wavelength precision. hologram: A recording of the interference pattern between a reference beam and a scattered beam; reconstructs a three-dimensional image when illuminated.

Concept Review Questions

Answer in complete sentences. No calculations required unless specified.

1. Explain why geometric (ray) optics fails to account for interference and

diffraction. Give a specific example where the ray model makes a prediction that contradicts experiment.

2. Describe Young's double-slit experiment. What pattern did Young observe, and

why is it impossible to explain with the particle model of light?

3. Why does a thin soap film appear colored in white light? Would the film appear

the same color from all viewing angles? Explain why or why not.

4. Explain why the central fringe in a double-slit pattern is bright. A student says:

“The two paths are equal, so they cancel.” What is wrong with this reasoning?

5. Why do astronomers build telescopes with large mirrors or lenses? Use the

diffraction limit formula to explain why doubling the aperture improves resolution.

6. Why is polarization evidence that light is transverse? Explain why sound cannot

be polarized.

7. Two polarizing filters are crossed (90° apart) and no light passes through. A third

filter is inserted between them at 45° to each. Explain why some light now passes through, using Malus’s Law.

8. Explain how a diffraction grating works as a spectrometer. Why can a grating

separate colors more precisely than a prism?

Interference Problems

Physics: Foundations of Physical Science

9. Two slits are separated by 0.30 mm. Red light (λ = 660 nm) illuminates the slits

and a screen is placed 1.5 m away. Find (a) the fringe spacing, (b) the position of the 3rd bright fringe, (c) the position of the 2nd dark fringe.

10. A double-slit experiment uses green light (λ = 532 nm). The fringe spacing is 3.2

mm and the screen is 1.8 m from the slits. Find the slit separation.

11. A thin film of oil (n = 1.45) 200 nm thick lies on water (n = 1.33). White light

reflects from the film. Which visible wavelengths (400–700 nm) undergo constructive interference in the reflected light? (Note: one phase shift on

12. An anti-reflection coating of MgF₂ (n = 1.38) is applied to glass (n = 1.52). For

what minimum coating thickness is required? (Both reflections have phase shifts: air–MgF₂ and MgF₂–glass.)

Diffraction Exercises

13. A single slit of width 0.40 mm is illuminated by λ = 600 nm light at a screen

1.50 m away. Find (a) the angular half-width of the central maximum, (b) the

width of the central maximum on the screen.

14. A diffraction grating has 500 lines/mm. Find the angle of the first-order

maximum for (a) red light (λ = 700 nm) and (b) violet light (λ = 400 nm).

15. A telescope objective has diameter D = 150 mm. Find the angular resolution

telescope resolve two stars separated by 0.8 arcseconds?

16. X-rays with λ = 0.154 nm (characteristic copper X-ray) diffract from a crystal with

reflection from crystal planes.) Polarization Analysis

17. Unpolarized light of intensity 80 W/m² passes through a polarizer. What intensity

emerges?

Physics: Foundations of Physical Science

18. Light polarized vertically at intensity 60 W/m² passes through an analyzer with

its axis at (a) 0°, (b) 30°, (c) 60°, (d) 90° from vertical. Find the transmitted intensity in each case.

19. Horizontally polarized light passes through three consecutive polarizers: the first

W/m², find the intensity after each polarizer.

20. Find Brewster's angle for light going from air (n = 1.00) into glass (n = 1.55). At

this angle, what is the angle of refraction in the glass? (Use Snell's Law to verify

Applied Wave Optics Problems

21. A CD has a track spacing of 1.6 μm. It is read with a λ = 780 nm laser. What

angle does the first-order diffracted beam make with the normal to the disc surface?

22. The LIGO gravitational wave detector uses λ = 1064 nm infrared laser light in

10 ¹⁸ m. Express this as a fraction of the wavelength. What is the corresponding change in path-length difference between the two arms?

23. A butterfly’s wing produces iridescent blue color (peak λ

450 nm) through

thickness that produces this color for near-normal reflection. (Assume one phase

24. An LCD pixel is formed by two crossed polarizers with a liquid crystal layer

between them. In the ‘on’ state, the crystal rotates polarization by 90°, allowing

why the pixel appears bright in the on state and dark in the off state. What percentage of the original unpolarized light reaches the viewer in the bright state (accounting for both polarizers)?

Before moving on, confirm you can do each of the following.

Physics: Foundations of Physical Science I can… See Section Explain why wave optics is needed and give an example where ray optics fails 28.1 Define coherence and explain why it is required for stable interference 28.2 State the conditions for constructive and destructive

28.3 Explain thin-film interference and the role of phase shifts on reflection 28.4 Describe Young's double-slit experiment and calculate fringe positions and spacing 28.5 Explain diffraction and the Rayleigh criterion for resolution 28.6

28.7 Apply the grating equation d

and calculate diffraction angles 28.8 Define polarization and explain why it is evidence that light is transverse 28.9 Apply Malus's Law to calculate transmitted intensity through an analyzer 28.10 Calculate Brewster's angle and state what happens at that angle 28.11 Describe three technological applications of wave optics 28.12 Identify and avoid common errors in wave optics calculations 28.13