Optics Fundamentals

Optics is the branch of physics that studies the behavior and properties of light, including its interactions with matter through phenomena like reflection, refraction, and diffraction. It encompasses the design and analysis of optical systems such as lenses, mirrors, and prisms, which are critical in everyday technologies and scientific research. This guide provides a comprehensive exploration of optics fundamentals, covering all basic formulas (lens formula, magnification, mirror formula, Snell’s law, critical angle), detailed examples, and practical applications to illustrate how light shapes our understanding of the world.

Lens Formula (with Magnification, Mirror Formula, Snell’s Law, Critical Angle)

The lens formula relates the object distance (\( u \)), image distance (\( v \)), and focal length (\( f \)) for thin lenses, assuming a sign convention (negative for virtual, positive for real):

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

Related Formulas:

Magnification (Lens): \( m = \frac{v}{u} \)
Mirror Formula: \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \) (same form, with sign convention)
Snell’s Law: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) (where \( n \) is refractive index, \( \theta \) is angle)
Critical Angle: \( \sin\theta_c = \frac{n_2}{n_1} \) (for \( n_1 > n_2 \))

Where:

\( f \): Focal length (m)
\( u \): Object distance (m)
\( v \): Image distance (m)
\( m \): Magnification (unitless)
\( n \): Refractive index (unitless)
\( \theta \): Angle (degrees)
\( \theta_c \): Critical angle (degrees)

Examples

Example 1: Lens Formula (Convex Lens)

A convex lens with \( f = 20 \, \text{cm} \), object at \( u = -30 \, \text{cm} \) (real object):

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] \[ \frac{1}{20} = \frac{1}{-30} + \frac{1}{v} \] \[ \frac{1}{v} = \frac{1}{20} - \frac{1}{30} \] \[ \frac{1}{v} = \frac{3 - 2}{60} \] \[ \frac{1}{v} = \frac{1}{60} \] \[ v = 60 \, \text{cm} \]

Example 2: Magnification (Lens)

With \( v = 60 \, \text{cm} \), \( u = -30 \, \text{cm} \):

\[ m = \frac{v}{u} \] \[ = \frac{60}{-30} \] \[ = -2 \]

Example 3: Mirror Formula (Concave Mirror)

A concave mirror with \( f = -10 \, \text{cm} \), object at \( u = -15 \, \text{cm} \):

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] \[ \frac{1}{-10} = \frac{1}{-15} + \frac{1}{v} \] \[ \frac{1}{v} = \frac{1}{-10} - \frac{1}{-15} \] \[ \frac{1}{v} = \frac{-3 + 2}{30} \] \[ \frac{1}{v} = -\frac{1}{30} \] \[ v = -30 \, \text{cm} \]

Example 4: Snell’s Law

Light from air (\( n_1 = 1.0 \)) to glass (\( n_2 = 1.5 \)), \( \theta_1 = 30^\circ \):

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \] \[ 1.0 \cdot \sin(30^\circ) = 1.5 \cdot \sin\theta_2 \] \[ 0.5 = 1.5 \cdot \sin\theta_2 \] \[ \sin\theta_2 = \frac{0.5}{1.5} \] \[ \sin\theta_2 = 0.333 \] \[ \theta_2 \approx 19.47^\circ \]

Example 5: Critical Angle

Critical angle for glass (\( n_1 = 1.5 \)) to air (\( n_2 = 1.0 \)):

\[ \sin\theta_c = \frac{n_2}{n_1} \] \[ \sin\theta_c = \frac{1.0}{1.5} \] \[ \sin\theta_c = 0.6667 \] \[ \theta_c \approx 41.81^\circ \]

Applications

Optics is foundational to numerous technologies and scientific fields. Below are examples with calculations:

Example 1: Camera Lens

A camera lens with \( f = 50 \, \text{mm} \), object at \( u = -100 \, \text{mm} \). Find image distance:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] \[ \frac{1}{50} = \frac{1}{-100} + \frac{1}{v} \] \[ \frac{1}{v} = \frac{1}{50} - \frac{1}{100} \] \[ \frac{1}{v} = \frac{2 - 1}{100} \] \[ \frac{1}{v} = \frac{1}{100} \] \[ v = 100 \, \text{mm} \]

Example 2: Telescope Mirror

A concave mirror with \( f = -200 \, \text{cm} \), object at \( u = -300 \, \text{cm} \). Find image distance:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] \[ \frac{1}{-200} = \frac{1}{-300} + \frac{1}{v} \] \[ \frac{1}{v} = \frac{1}{-200} - \frac{1}{-300} \] \[ \frac{1}{v} = \frac{-3 + 2}{600} \] \[ \frac{1}{v} = -\frac{1}{600} \] \[ v = -600 \, \text{cm} \]

Example 3: Microscope Magnification

Objective lens with \( v = 16 \, \text{cm} \), \( u = -4 \, \text{cm} \). Find magnification:

\[ m = \frac{v}{u} \] \[ = \frac{16}{-4} \] \[ = -4 \]

Example 4: Refraction in Water

Light from air (\( n_1 = 1.0 \)) to water (\( n_2 = 1.33 \)), \( \theta_1 = 45^\circ \):

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \] \[ 1.0 \cdot \sin(45^\circ) = 1.33 \cdot \sin\theta_2 \] \[ 0.707 = 1.33 \cdot \sin\theta_2 \] \[ \sin\theta_2 = \frac{0.707}{1.33} \] \[ \sin\theta_2 \approx 0.5316 \] \[ \theta_2 \approx 32.06^\circ \]

Example 5: Fiber Optics (Critical Angle)

Critical angle for glass (\( n_1 = 1.5 \)) to cladding (\( n_2 = 1.4 \)):

\[ \sin\theta_c = \frac{n_2}{n_1} \] \[ \sin\theta_c = \frac{1.4}{1.5} \] \[ \sin\theta_c = 0.9333 \] \[ \theta_c \approx 69.31^\circ \]

Example 6: Eyeglass Lens

A lens with \( f = 50 \, \text{cm} \), object at \( u = -100 \, \text{cm} \). Find image distance: