Lecture 3
Interferometers. Coherence
Today’s summary
• Different kinds of interferometers
• Multiple beam interferometers: Fabry-Perot resonators
– Stokes relationships
– Transmission and reflection coefficients for a dielectric slab
– Optical resonance
• Coherence: spatial / temporal
• Kinds of lasers
Interferometers
Michelson Interferometer
Mach-Zender Interferometer
Twyman-Green Interferometer
Fabry-Perot interferometers
Relation between r, r’and t, t’
air
glass
air
glass
Proof: algebraic from the Fresnel coefficients
or using the property of preservation of the
field properties upon time reversal
Stokes relationships
Proof using time reversal
air
glass
air
glass
Fabry-Perot interferometers
reflected
transmitted
incident
Resonance condition: reflected wave = 0
⇔ all reflected waves interfere destructively
wavelength in free space
refractive index
Calculation of the reflected wave
incoming
reflected
transmitted
transmitted
reflected
transmitted
reflected
transmitted
reflected
transmitted
reflected
air
glass
air
Calculation of the reflected wave
Calculation of the reflected wave
Use Stokes relationships
Transmission & reflection coefficients
reflection
coefficient
transmission
coefficient
Reflection
Transmission
Transmission & reflection vs path
Path delay
Reflection
Transmission
Path delay
Path delay
Path delay
 Różnica
częstotliwości pomiędzy
sąsiednimi modami:
 Liczba
modów
gdzie Δλ jest szerokością połówkową linii a
λ0 – długością centralną linii
Fabry-Perot terminology
Transmission coefficient
free
Spectral
range
band
width
resonance
frequencies
Frequency v
Transmission coefficient
Fabry-Perot terminology
FWHM Bandwidth is inversely proportional
to the finesse F (or quality factor) of the cavity
Fabry-Perot terminology
free spectral range
bandwidth
finesse
Fabry-Perot using options
K 
Spectrum line
Ne
Potential
mods

Generation
level
K+2 K K-2
K+3 K+1 K-1 K-3

2d
K
Fabry-Perot using options


Every mod is a superposition of plane waves, which is due to
the diffraction losses depend on the x and y coordinates can
not give stationary field, and after many reflections the fixed
configuration A (x, y) can be achived. The field distribution
in the resonator of the two transverse axes of symmetry can
be analyzed separately for each axis. Distribution of field for
each axis can be described by a function of HermiteGaussian
In the above equations show that higher-order transverse
modes in addition to the curvature of the wavefront
described by kr2/2R there are phase jumps for π (change
the sign of the amplitude) and then for different modes that
occures on different places in the wave front. Number of
strokes along the axis of symmetry of the phase corresponds
to the values of the mode index.
 The
intensity of higher order modes
reach significant values in a larger area
than the primary mode, which means that
the laser beam of a higher order takes
larger surface on the resonator mirrors,
and further has a greater divergence.
Confocal laser cavities
diffraction
angle
waist w0
Beam profile:
2D Gaussian function
“TE00mode”
Transverse modes
(usually undesirable)
Lasers
Atmospheric transmission
Absorption spectra
human vision
Absorption spectra
CW (continuous wave lasers)
Typical sources:
• Argon-ion: 488nm (blue) or 514nm (green); power ~1-20W
• Helium-Neon (HeNe): 633nm (red), also in green and yellow; ~1-100mW
• doubled Nd:YaG: 532nm (green); ~1-10W
Quality of sinusoid maintained over a time duration known as
“coherence time” tc
Typical coherence times ~20nsec (HeNe), ~10μsec (doubled Nd:YAG)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-40
Two types of incoherence
temporal
incoherence
spatial
incoherence
matched
paths
point
source
Michelson interferometer
Young interferometer
poly-chromatic light
(=multi-color, broadband)
mono-chromatic light
(= single color, narrowband)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-41
Coherent vs incoherent beams
Mutually coherent: superposition field amplitude
is described by sum of complex amplitude
Mutually incoherent: superposition field intensity
is described by sum of intensities
(the phases of the individual beams vary
randomly with respect to each other; hence,
we would need statistical formulation to
describe them properly — statistical optics)
Coherence time and coherence length
‧
much shorter than
“coherence length” ctc
Sharp interference fringes
Intensity
incoming
laser
beam
Michelson interferometer
‧
much longer than
“coherence length” ctc
no interference
Intensity
Coherent vs incoherent beams
Coherent: superposition field amplitude
is described by sum of complex amplitudes
Incoherent: superposition field intensity
is described by sum of intensities
(the phases of the individual beams vary
randomly with respect to each other; hence,
we would need statistical formulation to
describe them properly — statistical optics)
Mode-locked lasers
Typical sources: Ti: Sa lasers (major vendors: Coherent, Spectra Phys.)
Typical mean wavelengths: 700nm –1.4μm (near IR)
can be doubled to visible wavelengths
or split to visible + mid IR wavelengths using OPOs or OPAs
(OPO=optical parametric oscillator;
OPA=optical parametric amplifier)
Typical pulse durations: ~psec to few fsec
(just a few optical cycles)
Typical pulse repetition rates (“rep rates”): 80-100MHz
Typical average power: 1-2W; peak power ~MW-GW
Overview of light sources
non-Laser
Laser
Thermal: polychromatic,
spatially incoherent
(e.g. light bulb)
Continuous wave (or cw):
strictly monochromatic,
spatially coherent
(e.g. HeNe, Ar+, laser diodes)
Gas discharge: monochromatic,
spatially incoherent
(e.g. Na lamp)
Light emitting diodes (LEDs):
monochromatic, spatially
incoherent
Pulsed: quasi-monochromatic,
spatially coherent
(e.g. Q-switched, mode-locked)
~nsec
~psec to few fsec
pulse duration
mono/poly-chromatic = single/multi color
Types of lasers
Mode of operation:
• Continuous wave (cw)
• Pulsed
– Q-switched
– mode-locked
Types of lasers
Lasing medium:
• Gas (Ar-ion, HeNe, CO2)
• Metal-vapour lasers (HeCd, HeHg, HeAg, HeSe …)
• Solid state (Ruby, Nd:YAG, Ti:Sa)
• Dye (liquid)
• Excimer (193 nm (ArF), 248 nm (KrF), 308 nm (XeCl), 353 nm
(XeF))
•
•
•
•
Gas dynamic laser
FEL
Raman laser
Semiconductor lasers
• Diode (semiconductor)
• Vertical cavity surface-emitting lasers –VCSEL
Types of lasers