Modern Optics IV-coherence
Special topics course in IAMS
Lecture speaker: Wang-Yau Cheng
2006/4
Outline
Spatial and Temporal Coherence
Beams can be coherent or only partially coherent (indeed, even incoherent) in both space and time.
Spatial and
Temporal
Coherence:
Temporal
Coherence;
Spatial
Incoherence
Spatial
Coherence;
Temporal
Incoherence
Spatial and
Temporal
Incoherence
The Temporal Coherence Time and �the Spatial Coherence Length
The temporal coherence time is the time over which the beam wave-fronts remain equally spaced. Or, equivalently, over which the field remains sinusoidal with a given wavelength:
The spatial coherence length is the distance over which the beam wave-fronts remain flat:
Since there are two transverse dimensions, we can define a coherence area.
The spatial coherence depends on the emitter size and its distance away.
The van Cittert-Zernike Theorem states that the spatial
coherence area Ac is given by:
where d is the diameter of the light source and D is the distance away.
Basically, wave-fronts smooth
out as they propagate away
from the source.
Starlight is spatially very coherent because stars are very far away.
What is the coherence of light?
Classical idea
Coherent length
Coherent time
Δτ = Δl/c
Phase stability is the key role! ！
Linewidth should be narrow！
A nice tool for detecting the subtle structures of our material words
Frequency-stabilized lasers is the highest coherent light source that human being ever use
The coherence time is the reciprocal of the bandwidth.
The coherence time is given by:
where Δν is the light bandwidth (the width of the spectrum).
Sunlight is temporally very incoherent because its bandwidth is
very large (the entire visible spectrum).
Lasers can have coherence times as long as about a second,
which is amazing; that's >1014 cycles!
Orthogonal polarizations don’t interfere.
The most general plane-wave electric field is:
where the amplitude is both complex and a vector:
The irradiance is:
Orthogonal polarizations don’t interfere (cont’d)
Because the irradiance is given by:
combining two waves of different polarizations is different from combining
waves of the same polarization.
Different polarizations (say x and y):
Same polarizations (say x and x, so we'll omit the x-subscripts):
Therefore: Cross term!
Mach-Zehnder Interferometer
The Mach-Zehnder interferometer is usually operated “misaligned” and with something of interest in one arm.
Mach-Zehnder Interferogram
Nothing in either path
Plasma in one path
The Sagnac Interferometer
The two beams automatically take the same path around the
interferometer. The paths can differ, however, if the device
is rotating.
The Sagnac interferometer senses rotation.
Sagnac Interferometer Math
Suppose that the beam splitter moves by a distance, d, in the
time, T, it takes light to circumnavigate the Sagnac interferometer.
As a result, one beam will travel more, and the other less distance.
If R = the interferometer radius, and Ω = its angular velocity:
Thus, the Sagnac Interferometer's sensitivity to rotation depends on its area. And it need not be round!
Newton's Rings
Newton's Rings
Get constructive interference when an integral number of half wavelengths occur between the two surfaces (that is, when an integral number of full wavelengths occur between the path of the transmitted beam and the twice reflected beam).
This effect also causes the colors in bubbles and oil films on puddles.
Multiple-beam interference: The Fabry-Perot Interferometer or Etalon
A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and forth. An etalon is a type of Fabry-Perot etalon, and is a piece of glass with parallel sides.
The transmitted wave is an infinite series of multiply reflected beams.
Transmitted wave:
Incident wave: E0
Reflected �wave: E0r
δ = round-trip phase delay
inside medium
Transmitted �wave: E0t
r, t = reflection, transmission coefficients from glass to air
n
n = 1
n = 1
The Etalon (cont'd)
The transmitted wave field is:
The transmittance is:
where:
Dividing numerator and denominator by
Etalon Transmittance vs. Thickness, Wavelength, or Angle
The transmittance varies significantly with thickness or wavelength.�We can also vary the incidence angle, which also affects δ.
As the reflectance of each surface (r2) approaches 1, the widths of the high-transmission regions become very narrow.
Transmission maxima occur when:
2πL/λ = 2mπ
or:
The Etalon Free Spectral Range
λFSR
λFSR =
Free Spectral
Range
The Free Spectral Range is the wavelength range between transmission maxima.
Etalon Linewidth and Finesse
The Linewidth δLW is a transmittance peak's full-width-half-max (FWHM).
Setting δ equal to δLW/2 should yield T = 1/2:
For δ << 1, we can make the small argument approx:
The Finesse, F, is the ratio of the �Free Spectral Range and the Linewidth:
Substituting we have:
The Finesse is the number of wavelengths the interferometer can resolve.
δ = 2π corresponds to one FSR
taking
Applications of Fabry-Perot interferometers and etalons
To frequency filter a beam (this is often done inside a laser).
�To measure the wavelength or spectrum of a beam (but you must know it in advance to within a Free Spectral Range, and you must scan the thickness of the interferometer and watch for the transmission vs. thickness).
�Money is now coated �with interferometric inks �to help foil counterfeiters.�Notice the shade of the�“20,” which is shown �from two different angles.
Anti-reflection Coating
Notice that the center of the round glass plate looks like it’s missing.
It’s not! There’s an “anti-reflection coating” there (on both the front and back of the glass).
Anti-reflection Coating Math
Consider a beam incident on a piece of glass (n = ns) with a layer of material (n = nl) if thickness, h, on its surface.
It can be shown that the Reflectance is:
Notice that R = 0 if:
Multilayer coatings
Typical laser mirrors and camera
lenses use many layers.
The reflectance and transmittance
can be tailored to taste!