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This is summarization of Chapter 1: Ray Optics of ‘Fundamentals of photonics’
2.1. Postulates of Wave Optics
Refractive indext
Speed of light in a homogeneous transparent medium such as glass will be reduced by
Wave equation
It is the partial differential equation that any valid optical wavefunction must satisfy in a source-free medium.
- : The real scalar wavefunction describing the light at position and time .
- : The Laplacian operator () representing spatial curvature.
- : Time.
Source-free medium: a region of space (or material) where there are no internal "generators" creating or driving the optical wave (such as electric charges, currents, or light sources like lasers/bulbs).
Optical Intensity
Connects the mathematical wavefunction to the physically measurable quantity, optical intensity (watts/cm²). The factor of 2 is a convention to simplify later complex notation.
- : Optical intensity (or irradiance).
- : Time average over an interval much longer than an optical cycle.
Optical Power
Total power flowing through a specific area
Optical Energy
Total energy collected over a specific period of time. (units of joules)
2.2. Monochromatic waves
2.2.A. Complex Representation and the Helmholtz Equation
Harmonic Representation
A monochromatic wave has a single fixed frequency and oscillates harmonically in time.
- : Amplitude (position dependent).
- : Phase (position dependent).
- : Frequency (Hz).
- : Angular frequency (radians/s)
- : Period (s)
Complex Wavefunction
To simplify calculations, we use a complex function instead of the real function .
- : Complex Amplitude (static in time).
is just the real part of
Helmholtz Equation
Wavenumber:
representing spatial frequency.
Intensity of Monochromatic Wave
The reason why 2 is multiplied at optical intensity is explained with below.
Wavefronts:
The surfaces of equal phase, = constant.
2.2.B. Elementary Waves
Plane Wave
A wave where wavefronts are parallel planes perpendicular to the direction of propagation vector .
- : complex envelope (strength of the wave)
- : Wavevector
- Magnitude of wavevector = = wavenumber
*Wavelength
*Inside a medium of index
Frequency remains the same.
Spherical Wave
A wave originating from a point source, expanding as concentric spheres.
- : Distance from the origin.
- : Indicates amplitude decays as
- : inversely proportional to the square of the distance
Fresnel Approximation (Paraboloidal Wave)
For points near the z-axis (paraxial), a spherical wave can be approximated as:
Denoting , we use an approximation based on the Taylor-series expansion:
Fresnel number:
- : raidus of circlle centered about z axis. →
- : maximum angle
2.4 Simple Optical Components
Reflection
Refraction
Snell's Law:
Transmission Through Optical Components
- : Complex amplitude transmittance (ratio of output to input field).
- : Phase shift introduced by the material relative to air.
Diffraction Gratings
- : Period of the grating.
- : Diffraction order integer. ()
2.5. Interference
Interference Equation of Two Waves
- : Intensities of the individual waves.
- : Phase difference between the two waves.
Interferometers
2.6. Polychromatic and Pulsed Light
Complex Analytic Signal
: The Fourier transform of the original wavefunction .
Intensity of Polychromatic Light
Pulsed Plane Wave
- : Complex envelope function defining the pulse shape and duration.
- : Carrier (central) frequency.