1. What optics studies

Optics is the branch of physics that studies the behavior of light and its interactions with matter. In practice, the subject is usually split into:

Geometric optics: treats light as rays and is accurate when optical components are much larger than the wavelength.
Wave optics: treats light as a wave and is needed for interference, diffraction, and polarization.

Light is an electromagnetic wave. In vacuum it travels at speed

c \approx 3.00 \times 10^8 \ \text{m/s}

In a medium, the speed is lower:

v = \frac{c}{n}

where $n$ is the refractive index.

Core ideas

Rays show the direction of energy flow in the short-wavelength limit.
Reflection sends light back into the original medium.
Refraction changes the ray direction when light crosses an interface.
Lenses and mirrors form images by redirecting rays.
Wave effects appear when the aperture or obstacle size is comparable to the wavelength.

2. Light as a wave and a ray

Ray model

The ray model is a geometric approximation. It works well when:

Optical parts are much larger than the wavelength.
Surfaces are smooth on the scale of the wavelength.
You only need image location, size, and orientation.

In ray diagrams:

A ray is drawn along the direction of propagation.
A point source emits rays in all directions.
Image formation is found by intersecting rays or extending them backward.

Wave model

A monochromatic plane wave can be described by

y(x,t) = A \cos(kx - \omega t + \phi)

with

k = \frac{2\pi}{\lambda}, \qquad \omega = 2\pi f

and

v = \lambda f

For light in vacuum,

c = \lambda f

Refractive index

The refractive index is

n = \frac{c}{v}

Typical values:

Air: approximately $1.0003$
Water: about $1.33$
Glass: often between $1.5$ and $1.9$

The larger the refractive index, the slower light travels in the material.

3. Geometric optics

Geometric optics is built on three main principles:

Light travels in straight lines in a uniform medium.
Rays obey the law of reflection.
Rays obey Snell's law at interfaces.

Paraxial approximation

Many optical formulas assume small angles relative to the optical axis. Under this approximation:

\sin \theta \approx \theta, \qquad \tan \theta \approx \theta

This makes ray tracing and lens formulas much simpler.

Image types

Image type	Rays actually meet?	Can be projected on a screen?	Typical sign behavior
Real image	Yes	Yes	Usually inverted
Virtual image	No, rays only appear to meet	No	Usually upright

Object and image conventions

Different courses use slightly different sign conventions. A common one for lenses and mirrors is:

Real object distances are positive.
Real image distances are positive for mirrors, but lens conventions can vary.
Focal length sign depends on whether the element is converging or diverging.

The safest approach is to state the convention being used before solving.

4. Reflection and mirrors

Law of reflection

The angle of incidence equals the angle of reflection:

\theta_i = \theta_r

Both angles are measured from the normal to the surface.

Plane mirrors

A plane mirror forms a virtual image behind the mirror at the same distance as the object in front:

$$ d_i = -d_o $$

The image is:

Virtual
Upright
Same size as the object
Laterally inverted

Spherical mirrors

Two important types:

Concave mirror: converging
Convex mirror: diverging

For a spherical mirror, the focal length is related to the radius of curvature by

f = \frac{R}{2}

Mirror equation

For paraxial rays:

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Magnification is

m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}

Interpretation:

$m < 0$: inverted image
$m > 0$: upright image
$|m| > 1$: enlarged
$|m| < 1$: reduced

Ray tracing for mirrors

Use these principal rays:

A ray parallel to the axis reflects through the focal point.
A ray through the focal point reflects parallel to the axis.
A ray through the center of curvature reflects back on itself.

Common mirror cases

Object beyond the center of curvature: real, inverted, reduced image between $f$ and $R$.
Object at the center of curvature: real, inverted, same size at $R$.
Object between $R$ and $f$: real, inverted, enlarged image beyond $R$.
Object inside $f$: virtual, upright, enlarged image behind the mirror.

5. Refraction and Snell's law

Refraction occurs when light crosses into a medium with a different refractive index. The speed changes, and the direction usually changes too.

Snell's law

n_1 \sin \theta_1 = n_2 \sin \theta_2

where angles are measured from the normal.

Physical meaning

If $n_2 > n_1$, the ray bends toward the normal.
If $n_2 < n_1$, the ray bends away from the normal.

Critical angle and total internal reflection

When light moves from a higher index medium to a lower index medium, there is a critical angle:

\sin \theta_c = \frac{n_2}{n_1} \qquad (n_1 > n_2)

For $\theta_1 > \theta_c$, the light undergoes total internal reflection.

Applications:

Fiber optics
Prisms
Endoscopy

Dispersion

The refractive index depends on wavelength, so different colors travel and refract differently. In normal dispersion:

Shorter wavelengths usually have larger $n$.
Blue light bends more than red light.

This is why prisms spread white light into a spectrum.

6. Thin lenses and image formation

Thin lenses are idealized as lenses with negligible thickness compared to object and image distances.

Lens types

Converging lens: thicker in the middle, positive focal length
Diverging lens: thinner in the middle, negative focal length

Thin lens equation

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Magnification:

m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}

Ray tracing for lenses

Use three principal rays:

A ray parallel to the axis passes through the far focal point in a converging lens.
A ray through the near focal point emerges parallel to the axis.
A ray through the center of the lens continues approximately straight.

Image behavior

Converging lens

Object beyond $f$: real, inverted image on the opposite side.
Object at $f$: image at infinity.
Object inside $f$: virtual, upright, enlarged image on the same side as the object.

Diverging lens

Always forms a virtual, upright, reduced image on the object side.

Lensmaker's equation

For a thin lens in air,

\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)

where $R_1$ and $R_2$ are the radii of curvature using the chosen sign convention.

Multiple optical elements

For lenses in series:

Find the image formed by the first element.
Use that image as the object for the next element.
Keep track of signs and distances carefully.

For thin elements close together, optical powers add:

P = \frac{1}{f}

P_{total} = P_1 + P_2 + \cdots

with $f$ in meters and power in diopters $(\text{D})$.

7. Optical instruments

Human eye

The eye acts like a variable-focus imaging system. Important terms:

Cornea and lens: provide most of the focusing power
Retina: image plane
Accommodation: change in lens shape to focus at different distances

Common vision defects:

Myopia: nearsightedness, image focuses in front of the retina
Hyperopia: farsightedness, image focuses behind the retina
Astigmatism: different focal lengths in different meridians

Corrective lenses move the focal point onto the retina.

Magnifiers

A simple magnifier is a converging lens used to form a larger angular image. The key idea is angular magnification, not linear magnification.

Microscopes

A compound microscope uses:

An objective lens to create a real, enlarged intermediate image
An eyepiece to act as a magnifier for that intermediate image

Telescopes

Telescopes are used to view far objects under increased angular size.

Refracting telescope: uses lenses
Reflecting telescope: uses mirrors

The essential idea is to collect light with a large aperture and form a manageable image for the eye or detector.

Camera and sensor systems

For imaging systems:

Aperture controls light gathering and depth of field.
Focal length controls field of view and magnification.
Sensor size influences framing and effective perspective in a given setup.

8. Wave optics

Wave optics becomes essential when the wavelength cannot be neglected relative to the size of the slit, aperture, or obstacle.

Superposition principle

When waves overlap, the resulting displacement is the sum of the individual displacements:

y_{total} = y_1 + y_2 + \cdots

Interference and diffraction are both consequences of superposition.

Coherence

Stable interference requires coherent sources, meaning:

Same frequency
Constant phase difference

Laser light is highly coherent compared with ordinary lamps.

Path difference

Interference depends on path difference $\Delta r$ and phase difference $\Delta \phi$:

\Delta \phi = \frac{2\pi}{\lambda}\Delta r

9. Interference

Interference is the constructive or destructive addition of waves.

Double-slit interference

For slit separation $d$ and screen distance $L$ with small angles:

d \sin \theta = m \lambda

for bright fringes, where $m = 0, 1, 2, \dots$

For dark fringes:

d \sin \theta = \left(m + \frac{1}{2}\right)\lambda

On a distant screen,

y_m \approx \frac{m \lambda L}{d}

Thin-film interference

Thin films produce interference because light reflects from the top and bottom surfaces. Whether the reflected waves reinforce or cancel depends on:

Path length through the film
Phase changes upon reflection
Refractive indices of the layers

Useful rule:

Reflection from a boundary to a higher refractive index medium adds a phase shift of $\pi$.

Many interference patterns arise from varying film thickness or geometry. The exact formulas depend on the setup, but the workflow is always:

Find the optical path difference.
Include any phase reversal on reflection.
Apply bright/dark conditions.

10. Diffraction

Diffraction is the spreading of waves around obstacles and through apertures.

Single-slit diffraction

For a slit of width $a$, dark fringes occur at

a \sin \theta = m \lambda \qquad m = 1, 2, 3, \dots

The central maximum is the widest and brightest part of the pattern.

Diffraction grating

For a grating with line spacing $d$,

d \sin \theta = m \lambda

This is the same mathematical condition as double-slit bright fringes, but with many slits the maxima are much sharper.

Resolution

Diffraction limits the resolving power of optical instruments. A common criterion is the Rayleigh criterion:

\theta_{min} \approx 1.22 \frac{\lambda}{D}

where $D$ is the aperture diameter.

Implications:

Larger aperture improves resolution.
Shorter wavelength improves resolution.

Common misconceptions

Diffraction is not the same as interference, but both are wave effects and often occur together.
A smaller aperture does not just "dim" the image; it also increases blur from diffraction.

11. Polarization

Polarization describes the orientation of the electric field in a transverse wave.

Types of polarization

Linear polarization: electric field oscillates in one fixed plane
Circular polarization: field rotates with constant magnitude
Elliptical polarization: most general case

Unpolarized light has random polarization directions over time.

Polarizers

An ideal polarizer transmits only the component of the electric field along its transmission axis.

For incident intensity $I_0$ and angle $\theta$ between the light's polarization direction and the transmission axis:

I = I_0 \cos^2 \theta

This is Malus's law.

Brewster's angle

At Brewster's angle, reflected light is perfectly polarized:

\tan \theta_B = \frac{n_2}{n_1}