1. Today’s summary
• Multiple beam interferometers: Fabry-Perot resonators
– Stokes relationships
– Transmission and reflection coefficients for a dielectric slab
– Optical resonance
• Principles of lasers
• Coherence: spatial / temporal
MIT 2.71/2.710 Optics
10/20/04 wk7-b-1

2. Fabry-Perot interferometers
MIT 2.71/2.710 Optics
10/20/04 wk7-b-2

3. 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
MIT 2.71/2.710 Optics
10/20/04 wk7-b-3

4. Proof using time reversal
air
glass
air
glass
MIT 2.71/2.710 Optics
10/20/04 wk7-b-4

5. Fabry-Perot interferometers
reflected
transmitted
incident
Resonance condition: reflected wave = 0
⇔ all reflected waves interfere destructively
wavelength in free space
refractive index
MIT 2.71/2.710 Optics
10/20/04 wk7-b-5

6. Calculation of the reflected wave
incoming
transmitted
transmitted
reflected
reflected
transmitted
reflected
transmitted
reflected
transmitted
reflected
air
glass
air
MIT 2.71/2.710 Optics
10/20/04 wk7-b-6

7. Calculation of the reflected wave
Use Stokes relationships
MIT 2.71/2.710 Optics
10/20/04 wk7-b-7

8. Transmission & reflection coefficients
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10/20/04 wk7-b-8

9. Transmission & reflection vs path
Path delay
Path delay
Reflection
Transmission
Path delay
Path delay
MIT 2.71/2.710 Optics
10/20/04 wk7-b-9

10. Fabry-Perot terminology
free
Spectral
range
band
width
Transmission coefficient
resonance
frequencies
Frequency v
MIT 2.71/2.710 Optics
10/20/04 wk7-b-10

11. Fabry-Perot terminology
FWHM Bandwidth is inversely proportional to the finesse F (or quality factor) of the cavity
Transmission coefficient
free spectral range
bandwidth
finesse
MIT 2.71/2.710 Optics
10/20/04 wk7-b-11

12. Spectroscopy using Fabry-Perot cavity
Goal: to measure the specimen’s absorption as function of
frequency ω
Experimental measurement principle:
Scanning stage
(controls cavity length L )
Spectrum of light beam is modified by substance
Light beam of known spectrum
Power meter
Transparent windows
Container with specimen to be measured
Partially-reflecting mirrors (FP cavity)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-12

13. Spectroscopy using Fabry-Perot cavity
Goal: to measure the specimen’s absorption as function of
frequency ω
Experimental measurement principle:
Spectrum of light beam is modified by substance
Electro-optic (EO) modulator
(controls refr. Index n)
Light beam of known spectrum
Power meter
Transparent windows
Container with specimen to be measured
Partially-reflecting mirrors (FP cavity)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-13

14. Spectroscopy using Fabry-Perot cavity
transmissivity
Unknown
spectrum
Sample measured:
MIT 2.71/2.710 Optics
10/20/04 wk7-b-14

15. Spectroscopy using Fabry-Perot cavity
aaaFabry–Perot
transmissivity
Unknown
spectrum
Sample measured:
MIT 2.71/2.710 Optics
10/20/04 wk7-b-15

16. Spectroscopy using Fabry-Perot cavity
aaaFabry–Perot
transmissivity
Unknown
spectrum
Sample measured:
MIT 2.71/2.710 Optics
10/20/04 wk7-b-16

17. Spectroscopy using Fabry-Perot cavity
unknown spectrum
width should not exceed the FSR
aaaFabry–Perot
transmissivity
Unknown
spectrum
Sample measured:
MIT 2.71/2.710 Optics
10/20/04 wk7-b-17

18. Spectroscopy using Fabry-Perot cavity
aaaFabry–Perot
transmissivity
spectral resolution is determined by the cavity bandwidth
Unknown
spectrum
Sample measured:
MIT 2.71/2.710 Optics
10/20/04 wk7-b-18

19. Lasers
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10/20/04 wk7-b-19

20. Absorption spectra Atmospheric transmission human vision
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10/20/04 wk7-b-20

21. Semi-classical view of atom excitations
Energy
Atom in ground state
Energy
Atom in excited state
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10/20/04 wk7-b-21

22. Light generation Energy excited state equilibrium: most atoms
in ground state
ground state
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10/20/04 wk7-b-22

23. Light generation Energy excited state
A pump mechanism (e.g. thermal excitation or gas discharge) ejects some atoms to the excited state
ground state
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10/20/04 wk7-b-23

24. Light generation Energy excited state The excited atoms radiatively
decay, emitting one photon each
ground state
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10/20/04 wk7-b-24

25. Light amplification: 3-level system
Energy
Super-excited state
excited state
ground state
equilibrium: most atoms
in ground state; note the existence
of a third, “super-excited” state
MIT 2.71/2.710 Optics
10/20/04 wk7-b-25

26. Light amplification: 3-level system
Energy
Super-excited state
excited state
Utilizing the super-excited state
as a short-lived “pivot point,” the
pump creates a population inversion
ground state
MIT 2.71/2.710 Optics
10/20/04 wk7-b-26

27. Light amplification: 3-level system
Energy
Super-excited state
excited state
ground state
When a photon enters, ...
MIT 2.71/2.710 Optics
10/20/04 wk7-b-27

28. Light amplification: 3-level system
Energy
Super-excited state
excited state
When a photon enters, it “knocks”
an electron from the inverted population
down to the ground state, thus creating
a new photon. This amplification process
is called stimulated emission
ground state
MIT 2.71/2.710 Optics
10/20/04 wk7-b-28

29. w population inversion)
Light amplifier
Gain medium
(e.g. 3-level system
w population inversion)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-29

30. Light amplifier w positive feedback
Gain medium
(e.g. 3-level system
w population inversion)
When the gain exceeds the roundtrip losses, the system goes into oscillation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-30

31. w population inversion)
Laser
initial photon
Gain medium
(e.g. 3-level system
w population inversion)
Partially
reflecting
mirror
Light
Amplification through
Stimulated
Emission of
Radiation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-31

32. w population inversion)
Laser
amplified once
initial photon
Gain medium
(e.g. 3-level system
w population inversion)
Partially
reflecting
mirror
Light
Amplification through
Stimulated
Emission of
Radiation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-32

33. w population inversion)
Laser
amplified once
initial photon
Gain medium
(e.g. 3-level system
w population inversion)
reflected
Partially
reflecting
mirror
Light
Amplification through
Stimulated
Emission of
Radiation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-33

34. w population inversion)
Laser
amplified once
initial photon
Gain medium
(e.g. 3-level system
w population inversion)
reflected
amplified twice
Partially
reflecting
mirror
Light
Amplification through
Stimulated
Emission of
Radiation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-34

35. w population inversion)
Laser
amplified once
initial photon
Gain medium
(e.g. 3-level system
w population inversion)
reflected
output
amplified twice
reflected
Partially
reflecting
mirror
Light
Amplification through
Stimulated
Emission of
Radiation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-35

36. w population inversion)
Laser
amplified once
initial photon
Gain medium
(e.g. 3-level system
w population inversion)
reflected
output
amplified twice
reflected
amplified again etc.
Partially
reflecting
mirror
Light
Amplification through
Stimulated
Emission of
Radiation
MIT 2.71/2.710 Optics
10/20/04 wk7-b-36

37. Confocal laser cavities
diffraction angle
waist w0
Beam profile:
2D Gaussian function
“TE00mode”
MIT 2.71/2.710 Optics
10/20/04 wk7-b-37

38. Other “transverse modes”
(usually undesirable)
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10/20/04 wk7-b-38

39. Types of lasers • Continuous wave (cw) • Pulsed – Q-switched
– mode-locked
• Gas (Ar-ion, HeNe, CO2)
• Solid state (Ruby, Nd:YAG, Ti:Sa)
• Diode (semiconductor)
• Vertical cavity surface-emitting lasers –VCSEL–(also sc)
• Excimer(usually ultra-violet)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-39

40. CW (continuous wave lasers)
Laser oscillation well approximated by a sinusoid
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

41. 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

42. Two types of incoherence
temporal
incoherence
spatial
incoherence
matched paths
point source
waves with equal paths
but from different points
on the wave front do not interfere
waves from unequal paths
do not interfere
MIT 2.71/2.710 Optics
10/20/04 wk7-b-42

43. 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)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-43

44. Coherence time and coherence length
‧ much shorter than “coherence length” ctc
Sharp interference fringes
Intensity
incoming
laser
beam
‧ much longer than “coherence length” ctc
no interference
Michelson interferometer
Intensity
MIT 2.71/2.710 Optics
10/20/04 wk7-b-44

45. 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)
MIT 2.71/2.710 Optics
10/20/04 wk7-b-45

46. 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”): MHz
Typical average power: 1-2W; peak power ~MW-GW
MIT 2.71/2.710 Optics
10/20/04 wk7-b-46

47. 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)
Pulsed: quasi-monochromatic,
spatially coherent
(e.g. Q-switched, mode-locked)
Light emitting diodes (LEDs):
monochromatic, spatially incoherent
~nsec
~psec to few fsec
pulse duration
mono/poly-chromatic = single/multi color
MIT 2.71/2.710 Optics
10/20/04 wk7-b-47