1. Rays and phase fronts
2. Interference
3. Form of waves
Electric field
Need four quantities to describe
| Amplitude | A |
| Wavelength (or frequency) | l = c/f (in free space) |
| Phase | f |
| Polarization |  1 (unit vector) |
4. Auxiliary relations
5. Electric and Magnetic Fields and Intensity
1. Refractive index
Index of Refraction:
|
2. Dispersion
For most materials, n decreases with decreasing frequency or increasing wavelength
This change in velocity can be used to manipulate wavefronts and waves and is the principal used for the operation of the lens and the prism
1. Some Simple Rules
Simple Rule #1: The angle of incidence equals the angle of reflection. |
Simple Rule #2: by
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2. Refraction
Refraction is the bending of light rays by an interface between two materials of differing refractive indices due to the difference in the phase velocity in each case.
3. More Simple Rules and Refraction
Simple Rule #3: qc for waves at oblique incidence is given by
for n 2 < n1. |
Simple Rule #4: qb for waves at oblique incidence is given by for both n 2 < n1 and n2 > n1. |
Simple Rule #5:
|
4. Reflection at Any Angle
In general, the power reflection coefficients R|| and R^ for parallel and perpendicular polarization can be found and written in a simple format. The result is
Rule #6:
where the angles
q1 and q2 are the angle and incidence and the angle of refraction, respectively, related by Simple Rule #5.[Note that R
|| (above) is the same as Rp (of the text), likewise for R^ and Rs.]
5. Applications:
a. Total internal reflection–prisms and fibers
b. Lens-beam expander
c. Mirrors
d. Polarizers (Brewster angle mirrors)
1.
Diffraction is the spatial spreading of light due to propagation or scattering by an object. Since the distribution of light far from a source is proportional to the Fourier transform of the source distribution, we see that small (measured in wavelengths) sources have a large amount of spreading while large (measured in wavelengths) sources have a small amount of spreading. Diffraction is an attribute of all sources of waves and is responsible for the spreading in free space of microwave beams, optical beams and acoustics. Mathematically, this is given by the relation
Beamwidth ~ l/D (l = wavelength of wave, D = diameter of beam) as stated previously.
2.
A comparison of diffraction and refraction for typical optical waves is useful. Since for most optical materials the index of refraction decreases with increasing wavelength, by Snell's law it is clear that for refraction, higher frequency (e.g., blue) light is usually bent more than for lower frequency (e.g., red) light. However, for diffraction, larger wavelength (e.g., red) light is bent more by scattering from a grating or edge than smaller wavelength (e.g., blue) light. That is, diffraction and refraction (from normally dispersive materials) have opposite behavior with variations in wavelength or frequency.
3. Ray Picture of Diffraction
If two rays are considered which emanate from the edge of an aperture of width D illuminated by a plane wave of wavelength l, they destructively interfere at a half spot size on each side of the optic axis (see next page). Simple trigonometry demonstrates that
where F is the beamwidth (in radians). For small angels, this leads directly to our previously stated relation
F ~ l/D.
Note:
Nulls occur when destructive interference appears from to rays which are 180° (= p radians) out of phase.
4. Physical or Wave Optic Viewpoint
An alternative viewpoint, which we shall investigate later in dealing with optical signal processing, shows that the far-field pattern is the Fourier transform of the source distribution. Below is given a summary of this result. Details are given in the last section of these notes.
The diffracted optical field y(x,y,z) is given in terms of integration or summation over the aperture (in the z = 0+ plane) of the aperture field y0(x',y',0) (see next page). This is known as Huygen's principle. Here k (=2p/l) is the wavenumber of the incident wave and q is the angle of the observer with respect to the z axis.
In the far-field this expression becomes
Note:
| Example: Find the far-field pattern for
a two dimensional square aperture of side L as shown above. Solution:
|
5. Resolution and Diffraction
Diffraction is the limiting effect to the ultimate resolution of optical instruments and systems. Resolution is a measure of the ability of a system to distinguish between signals which are closely spaced in wavelength.
The resolving power Rp of an optical instrument or system is defined by
Derivations for a number of devices are carried out in the detailed notes, here we summarize these results.
DEVICE
PARAMETERS
RESOLVING POWER
TYPICAL RESOLUTION
Prism
Beamwidth B
l
Dispersion dn/dB dn/d
l10,000
Grating
Beamwidth B
Spacing dB/d
100,000
Interferometer
Finesse
F
Cavity length L
Wavelength l2
FL/l10,000,000
1. Coherence
Coherence is the ability of light to interfere with a delayed or displaced version of itself. Interference with a delayed version produces a measure of longitudinal coherence and yields a measure of coherence time tc and its associated longitudinal coherence length lc = ctc. Interference with a displaced version produces a measure of transverse coherence and yields a measure of the coherence area and the transverse coherence length lt. Together, one can envision a coherence volume composed of the coherence length lc and a cross-sectional area given by lt x lt.
2. Longitudinal Coherence
Optically, one uses the Michelson interferometer the measure the degree of longitudinal coherence. This is described in the handout "Notes on Lasers and Light." Mathematically, it is the autocorrelation function which measures the degree to which a wave is like a delayed or time-shifted version of itself. The self coherence function G(t) as a function of time delay t is defined as the autocorrelation of the light wave field u(t).
A normalized version, the coherence function g(t) has an envelope which is called the fringe visibility V. Here,
The width of the visibility V is called the coherence length. Rigorously, one defines the longitudinal coherence time as
From the fundamental properties of autocorrelations and Fourier transforms, the definition above and from numerous examples, one finds
where Dn is the spectral width of the light wave field. This yield the longitudinal coherence length as
3.
Transverse Coherencet is approximated byTransverse coherence is measured by examining the maximum separation distance two pinholes can be placed in a beam and still demonstrate interference. It can be shown that the transverse coherence length l
for a source of angular dimension Dq from the observer and of wavelength l. This is due to the interference from different portions of a source as seen from the observer. Clearly, a point source (e.g., a distant light) has a large coherence length. The coherence area is given by
where DW is the solid angle subtended by the source at the observer.
longitudinal coherence length lc = ctc. Interference with a displaced version produces a measure of transverse coherence and yields a measure of the coherence area and the transverse coherence length lt. Together, one can envision a coherence volume composed of the coherence length lc and a cross-sectional area given by lt x lt.
