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Optics Formulas

Complete collection of optics formulas with detailed explanations.

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Geometric Optics

Lens Formula

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

Reciprocal of focal length equals sum of reciprocals of object and image distances.

Notation:

f:Focal length
u:Object distance
v:Image distance

Units:

f, u, v:m

Applications:

  • •Camera lenses
  • •Microscopes
  • •Telescopes

Limitations:

Thin lens approximation

Magnification

M=−vu=h′hM = -\frac{v}{u} = \frac{h'}{h}

Magnification equals negative image distance divided by object distance or image height divided by object height.

Notation:

M:Magnification
v:Image distance
u:Object distance
h':Image height
h:Object height

Units:

M:dimensionless
v, u:m
h', h:m

Applications:

  • •Optical instruments
  • •Image analysis
  • •Microscopy

Limitations:

Paraxial rays

Power of Lens

P=1fP = \frac{1}{f}

Power of lens equals reciprocal of focal length.

Notation:

P:Power
f:Focal length

Units:

P:diopters (D)
f:m

Applications:

  • •Eyeglass prescriptions
  • •Contact lenses
  • •Optical systems

Limitations:

Thin lens approximation

Combined Power

Ptotal=P1+P2P_{total} = P_1 + P_2

Total power of two thin lenses in contact equals sum of individual powers.

Notation:

P_total:Total power
P₁, P₂:Individual powers

Units:

P_total, P₁, P₂:diopters (D)

Applications:

  • •Compound lenses
  • •Optical design
  • •Lens combinations

Limitations:

Thin lenses in contact

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Reflection and Refraction

Snell's Law

n1sin⁡(θ1)=n2sin⁡(θ2)n_1\sin(\theta_1) = n_2\sin(\theta_2)

Product of refractive index and sine of angle in first medium equals product in second medium.

Notation:

n₁:Refractive index of medium 1
θ₁:Angle of incidence
n₂:Refractive index of medium 2
θ₂:Angle of refraction

Units:

n₁, n₂:dimensionless
θ₁, θ₂:degrees or radians

Applications:

  • •Light bending
  • •Prism optics
  • •Fiber optics

Limitations:

Planar interface

Law of Reflection

θi=θr\theta_i = \theta_r

Angle of incidence equals angle of reflection.

Notation:

θ_i:Angle of incidence
θ_r:Angle of reflection

Units:

θ_i, θ_r:degrees or radians

Applications:

  • •Mirror optics
  • •Reflection measurements
  • •Optical instruments

Limitations:

Smooth surface

Critical Angle

θc=sin⁡−1(n2n1)\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)

Critical angle equals inverse sine of ratio of refractive indices.

Notation:

θ_c:Critical angle
n₁:Higher refractive index
n₂:Lower refractive index

Units:

θ_c:degrees or radians
n₁, n₂:dimensionless

Applications:

  • •Total internal reflection
  • •Fiber optic cables
  • •Optical fibers

Limitations:

n₁ > n₂

Refractive Index

n=cvn = \frac{c}{v}

Refractive index equals speed of light in vacuum divided by speed in medium.

Notation:

n:Refractive index
c:Speed of light in vacuum
v:Speed of light in medium

Units:

n:dimensionless
c:3 × 10⁸ m/s
v:m/s

Applications:

  • •Material characterization
  • •Optical properties
  • •Light propagation

Limitations:

Transparent media

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Interference and Diffraction

Young's Double Slit

dsin⁡(θ)=mλd\sin(\theta) = m\lambda

Path difference equals slit separation times sine of angle equals integer times wavelength.

Notation:

d:Slit separation
θ:Angle
m:Order number
λ:Wavelength

Units:

d:m
θ:radians
m:integer
λ:m

Applications:

  • •Interference patterns
  • •Wave nature of light
  • •Optical measurements

Limitations:

Monochromatic light

Single Slit Diffraction

asin⁡(θ)=mλa\sin(\theta) = m\lambda

Slit width times sine of angle equals integer times wavelength for minima.

Notation:

a:Slit width
θ:Angle
m:Order number
λ:Wavelength

Units:

a:m
θ:radians
m:integer
λ:m

Applications:

  • •Diffraction patterns
  • •Optical resolution
  • •Wave phenomena

Limitations:

Fraunhofer diffraction

Diffraction Grating

dsin⁡(θ)=mλd\sin(\theta) = m\lambda

Grating spacing times sine of angle equals integer times wavelength.

Notation:

d:Grating spacing
θ:Angle
m:Order number
λ:Wavelength

Units:

d:m
θ:radians
m:integer
λ:m

Applications:

  • •Spectroscopy
  • •Wavelength measurement
  • •Spectral analysis

Limitations:

Normal incidence

Optical Path Length

OPL=nLOPL = nL

Optical path length equals refractive index times geometric length.

Notation:

OPL:Optical path length
n:Refractive index
L:Geometric length

Units:

OPL:m
n:dimensionless
L:m

Applications:

  • •Interference calculations
  • •Optical design
  • •Phase differences

Limitations:

Uniform medium

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Polarization

Malus's Law

I=I0cos⁡2(θ)I = I_0\cos^2(\theta)

Transmitted intensity equals incident intensity times cosine squared of angle between polarizer axes.

Notation:

I:Transmitted intensity
I₀:Incident intensity
θ:Angle between axes

Units:

I, I₀:W/m²
θ:degrees or radians

Applications:

  • •Polarized light
  • •Optical filters
  • •Light intensity control

Limitations:

Perfect polarizers

Brewster's Angle

θB=tan⁡−1(n2n1)\theta_B = \tan^{-1}\left(\frac{n_2}{n_1}\right)

Brewster's angle equals inverse tangent of ratio of refractive indices.

Notation:

θ_B:Brewster's angle
n₁:Refractive index of medium 1
n₂:Refractive index of medium 2

Units:

θ_B:degrees or radians
n₁, n₂:dimensionless

Applications:

  • •Polarization by reflection
  • •Optical coatings
  • •Anti-reflection surfaces

Limitations:

Non-absorbing media

Birefringence

Δn=ne−no\Delta n = n_e - n_o

Birefringence equals difference between extraordinary and ordinary refractive indices.

Notation:

Δn:Birefringence
n_e:Extraordinary index
n_o:Ordinary index

Units:

Δn, n_e, n_o:dimensionless

Applications:

  • •Crystal optics
  • •Wave plates
  • •Polarization control

Limitations:

Uniaxial crystals

Try Interactive Calculators

Snell's LawThin Lens EquationMirror EquationCritical AngleBrewster's AngleYoung's Double SlitLens PowerTelescope Magnification