Optics Development History and Light Interference

Optics Development History

17th-18th Centuries: Geometric optics, characterized by the rectilinear propagation of light and the laws of reflection and refraction, with Newton's corpuscular theory as its representative.
19th Century: Wave optics, dealing with the phenomena of light interference, diffraction, polarization, and the electromagnetic nature of light. Key figures include T. Young, Fresnel, and Maxwell.
Early 20th Century: Quantum optics, addressing the wave-particle duality of light.
1960s: Modern optics, including:
- Nonlinear optics, dealing with phenomena such as frequency doubling, mixing, stimulated emission, and self-focusing, where the superposition principle does not hold.
- Information optics (Fourier optics).
- Applications in optical communication, storage, information processing, and laser-induced fusion ignition.
- Laser technology, characterized by high energy density, monochromaticity, and directionality.
- Research into optical computers, aiming to enhance storage capacity and processing speed through optical parallel processing, with optoelectronic conversion as a critical technology.

Young's Double-Slit Experiment

Thomas Young (1773-1829):
- British physicist, physician, archaeoligist. One of the founders of the wave theory of light.
- Contributions:
  - Wave Optics: Young's double-slit interference experiment.
  - Physiological Optics: Three-primary-color principle.
  - Material Mechanics: Young's modulus of elasticity.
  - Archaeology: Deciphered inscriptions on ancient Egyptian steles.

Experimental Setup

A single wave front is divided into two to create two coherent light sources.
Observed interference pattern includes maximum intensity fringes.
In 1801, Young devised a method to split a single wavefront into two, ensuring a fixed phase difference between the two light sources to study light interference.
He explained the interference phenomenon using the superposition principle and made the first measurement of the wavelength of light, which laid the foundation for the wave theory of light.

Quantitative Derivation

The optical path difference $\delta$ between light emitted from $S1$ and $S2$ to a point $P$ on the screen is given by: $\delta = r2 - r1$ , where $r1$ and $r2$ are the distances from $S1$ and $S2$ to $P$ , respectively.
The phase difference $\Delta \varphi$ at point P is: $\Delta \varphi = \frac{2 \pi}{\lambda} \delta = \frac{2 \pi}{\lambda} (r2 - r1)$ .

Interference Conditions

Constructive Interference (Bright Fringes):
- Phase difference: $\Delta \varphi = \pm 2 \pi k$ , where $k = 0, 1, 2, 3, …$
- Optical path difference condition: $\delta = \pm k \lambda$ .
Destructive Interference (Dark Fringes):
- Phase difference: $\Delta \varphi = \pm (2k + 1) \pi$ , where $k = 0, 1, 2, 3, …$
- Optical path difference condition: $\delta = \pm (k + \frac{1}{2}) \lambda$ .
The central point $O$ on the screen is a bright spot because $\delta = 0$ when $k = 0$ .

Geometric Relationship

Approximations: Since d << D and $\theta$ is small, $sin \theta \approx tan \theta \approx \theta$ .
From the geometry: $r1 = \sqrt{D^2 + (x - \frac{d}{2})^2} \approx D + \frac{(x - \frac{d}{2})^2}{2D}$ and $r2 = \sqrt{D^2 + (x + \frac{d}{2})^2} \approx D + \frac{(x + \frac{d}{2})^2}{2D}$ .
Optical path difference: $\delta = r2 - r1 \approx \frac{xd}{D}$ .
Bright fringe positions: $x = \pm \frac{k \lambda D}{d}$ , where $k = 0, 1, 2, 3, …$ .
Dark fringe positions: $x = \pm \frac{(2k + 1) \lambda D}{2d}$ , where $k = 0, 1, 2, 3, …$ .

Discussion

Fringe Movement:
- The position of interference fringes depends only on the vertical coordinate $x$ and is independent of the horizontal coordinate $y$ .
- Moving the double slits along the y-direction shifts the fringe pattern in the same direction without altering the vertical distribution.
Replacing the point source with a line source (parallel to the y-axis) and the double holes with narrow slits (parallel to the y-axis) increases the brightness of the fringes without changing the distribution.
Fringe Order:
- The central fringe at the center of the screen is the zero-order bright fringe (k=0).
- Bright and dark fringes are symmetrically and alternately distributed on either side of the central bright fringe.
- Bright fringe positions: $x = \pm \frac{k \lambda D}{d}$ , where $k = 0, 1, 2, 3, …$ .
- Dark fringe positions: $x = \pm \frac{(2k + 1) \lambda D}{2d}$ , where $k = 0, 1, 2, 3, …$ .
Fringe Spacing:
- The distance between adjacent bright fringes is given by: $\Delta x = x{k+1} - xk = \frac{\lambda D}{d}$ .
To discern fringes $\lambda$ should be small ( $0.4 \mu m$ ~ $0.76 \mu m$ ), d small and D big
- The fringe width is proportional to the wavelength $\lambda$ . Different colors produce interference patterns that do not overlap, but the central bright fringe remains at the center.

Light Intensity Distribution

The total light intensity at any point P is $I = E^2$ , where $E$ is the amplitude of the electric field vector.
If comparing light intensities at different positions within the same medium, the intensity is directly proportional to the square of the amplitude.
The average light intensity over time $\tau$ is: $ = \frac{1}{\tau} \int_0^{\tau} I dt$ .
The combined light intensity at any point P is $I = I1 + I2 + 2\sqrt{I1 I2} \cos(\Delta \varphi)$ , where $\Delta \varphi$ is the phase difference.
For incoherent light, the integral of $\cos(\Delta \varphi)$ over time is zero: $ = I1 + I2$ .
For coherent light, $\Delta \varphi$ is constant.
- If $I1 = I2 = I$ , then $I = 4I_0 \cos^2(\frac{\Delta \varphi}{2})$ .

Light Source and Spectrum

Light Waves:
- Light propagates as waves; light waves are electromagnetic waves.
- Visible light range: 4000 Å to 7600 Å.
Propagation Characteristics:
- Independence: After superposition, waves continue in their original direction and parameters.
- Superposition: Light vibrations combine in the overlapping region.
- Superposition requires independence.
Coherent Light:
- Same frequency & vibration direction, and constant phase difference.
- Coherent phenomena: The combined vibration intensity varies periodically in space.
- It is difficult to maintain constant phase differences between independent light sources.
Ordinary Light Sources:
- Light waves are the sum of emissions from numerous atoms and molecules.
- Each atom emits light independently for a short duration (10^-9 s): wave trains.
- There is no fixed phase relationship between emissions, thus no interference is observed, only average intensity: $I = I1 + I2$ .
Temporal Coherence:
- Thin film interference is limited by the emission mechanism of ordinary light sources.
- If the time difference between reflected waves is too great, interference cannot occur.
- The light path difference must be smaller than the length of the wave train.
- The coherence length (lc) is the maximum optical path difference for interference.
  - $l_c = ct$ , where t is the time over which the atom emits light
- The longer the coherence length, the better the light's coherence.
- This limitation due to the intermittent nature of light emission is called temporal coherence.
Spatial Coherence:
- The width of the light source influences spatial coherence.
The condition for observing interference fringes is \Delta l < \Delta x , which implies \Delta S < \frac{\lambda b}{d} .
This relationship between the light source's width and coherence is termed spatial coherence.
Visibility:
- When $I{max} = 4I$ and $I{min} = 0$ , the fringes have the most distinct contrast, and the visibility is $V = 1$ .
- If $I1 \neq I2$ , then $I_{min} \neq 0$ , reducing the fringe contrast and visibility.
- Visibility is defined as: $V = \frac{I{max} - I{min}}{I{max} + I{min}}$ .

Applications of Double-Slit Interference

Measuring Wavelength:
Measuring Thickness and Refractive Index of Thin Films:
Measuring Small Changes in Length:

Sample Problems

In Young's double-slit experiment, the distance between the double slits is 0.60 mm, the distance between the slits and the screen is 1.50 m, and the fringe width is measured to be 1.50 mm. Find the wavelength of the incident light.
- Solution: Using the formula for the fringe spacing, $\Delta x = \frac{\lambda D}{d}$ , we can solve for the wavelength: $\lambda = \frac{\Delta x \cdot d}{D} = \frac{1.50 \times 10^{-3} \times 0.60 \times 10^{-3}}{1.50} = 6.00 \times 10^{-7} m = 600 nm$ .

Fresnel Biprism Experiment

Fresnel proposed a method of obtaining coherent light.

Fresnel Double Mirror Experiment

Fresnel proposed a method of obtaining coherent light.

Lloyd's Mirror Experiment

"Half-wave Loss": When light travels from an optically rarer medium to an optically denser medium, the reflected light undergoes a phase change of $\pi$ , or an additional optical path difference of $\lambda/2$ .
- Observed at the point there should theoretically be constuctive interference, there is destructive interference, verifying the half-wave loss.

Optical Path

The distance light travels in a medium converted to its equivalent length in a vacuum.
Optical Path = $nr$ , where n is the refractive index and r is the geometric path length.
If a light beam propagates through a medium, the ultimate phase change depends not only on the geometric path but also on the refractive index.
Optical Path Difference: $\Delta = n2 r2 - n1 r1$ .

Interference with transparent film

Zero-order bright fringe appears on P, screen inteference fringes shift upward
Number of fringes shifted: $\Delta k = \frac{(n-1)t}{\lambda}$
Displacement distance: $OP = \Delta k \cdot \Delta x$

Conditions for Interference

Phase Difference: \Delta \φ = \frac{2\pi}{\lambda} (n2r2 - n1r1) = k \pi .
Optical Path Difference: Integer multiples of the wavelength.

Application-Determining mica thickness based on fringe movement

With no mica, zero-order bright fringes are at point $O$ ; with mica, light path 1 increases by $(n-1)b$ .
Due to zero-order bright fringes corresponding optical path difference is 0.After S1 seam covering mica, zero-order bright edges will move from point $O$ to point P), which is the position of the original ninth-level bright fringes.
(n-1)b = k\lambda
b = \frac{k\lambda}{(n-1)} = \frac{9 \times 550 \times 10^{-9}}{(1.58-1)} = 8.53 \times 10^{-6}m

Lens not causing additional optical path differences

Parallel light wavefronts at A $,$ B $,$ C),… Same phase, same optical path.

Phase sudden change and additional optical path difference of reflected light

When light is reflected on the interface of two media, in some cases, the phase of the reflected light is opposite to the phase of the incident light, that is, the phase of the light changes suddenly during reflection. Because the phase change of $n$ corresponds to the optical path difference of $\lambda/2$ , the sudden change of the phase of light during reflection is often referred to as half-wave loss.
The requirements to have half-wave loss is. for light from the less dense medium to the denser medium.
- Half-wave loss happens only in reflected light

Thin film interference formula.

Optical path difference $\delta = 2n_2dcosr + \frac{\lambda}{2}$

Equal Inclination Interferece

Equal Inclination Bright and Dark Stripes under Interferece

$2dn2\sqrt{1-\frac{n1}{n_2}Sin^2i} = k\lambda$

equal light intensity, interefence

Under the condition of incident angly, the intensity of the interference is the same.