1. Fundamentals of Photonics
Chapter 4 Laser
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2. Fundamentals of Photonics
LASERS
In 1958 Arthur Schawlow, together with Charles Townes, showed how to extend the principle of the maser to the optical region. He shared the 1981 Nobel Prize with Nicolaas Bloembergen. Maiman demonstrated the first successful operation of the ruby laser in 1960.
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3. Fundamentals of Photonics
LASERS
an oscillator is an amplifier with positive feedback
Two conditions for an oscillation: 1. Gain greater than loss: net gain 2. Phase shift in a round trip is a multiple of 2π
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4. Fundamentals of Photonics
Stable condition: gain = loss
If the initical amplifier gain is greater than the loss, oscillation may initiate. The amplifier then satuates whereupon its gain decreases. A steady-state condition is reached when the gain just equals the loss.
Steady-state
power
Gain
Loss
Power
An oscillator comprises:
◆ An amplifier with a gain-saturation mechanism
◆ A feedback system
◆ A frequency-selection mechanism
◆ An output coupling scheme
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5. Light amplifier with positive feedback
Gain medium (e.g. 3-level system w population inversion)
When the gain exceeds the roundtrip losses, the system goes into oscillation + g +
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6. Gain medium (e.g. 3-level system w population inversion)
LASERS
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Amplified once
Gain medium (e.g. 3-level system w population inversion)
Initial photon
Reflected
Amplified twice
Output
Reflected
Amplified Again
Partially reflecting Mirror
Light
Amplification through
Stimualted
Emission
Radiation
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7. Partially transmitting mirror
Active medium d
Partially transmitting mirror
Laser output
A laser consists of an optical amplifier (employing an active medium) placed within an optical resonator. The output is extracted through a partially transmitting mirror.
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8. 5.1-43 Small signal Gain Coefficient 5.1-42 Saturated Gain Coefficient
Optical amplification and feedback
★ Gain medium
The laser amplifier is a distributed-gain device
characterized by its gain coefficient
Small signal Gain Coefficient
Saturated Gain Coefficient
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9. Fundamentals of Photonics
Phase-shift Coefficient (Lorentzian Lineshape)
Figure Spectral dependence of the gain and phase-shift coefficients for an optical amplifier with Lorentzian lineshape function
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10. Optical Feedback-Optical Resonator
Feedback and Loss: The optical resonator
A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains the medium (refractive index n). Travel through the medium introduces a phase shift per unit length equal to the wavenumber
In traveling a round trip through a resonator of length d, the photon-flux density is reduced by the factor R1R2exp(-2asd). The overall loss in one round trip can therefore be described by a total effective distributed loss coefficient ar, where
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11. Fundamentals of Photonics
Loss coefficient
Photon lifetime
ar represents the total loss of energy (or number of photons) per unit length, arc represents the loss of photons per second
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12. Fundamentals of Photonics
Resonator response
Resonator modes are separated by the frequency
and have linewidths
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13. Conditions for laser oscillation
Gain condition: Laser threshold
Threshold Gain Condition
where or
Threshold Population Difference
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14. For a Lorentzian lineshape function, as
If the transition is limited by lifetime broadening with a decay time tsp
As a numerical example, if l0=1 mm, tp=1 ns, and the refractive index n=1, we obtain Nt=2.1×107 cm-3
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15. Fundamentals of Photonics
Conditions for laser oscillation(2)
Phase condition: Laser Frequencies
Frequency Pulling or
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16. Fundamentals of Photonics
Laser Frequencies
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17. Fundamentals of Photonics
The laser oscillation frequencies fall near the cold-resonator modes; they are pulled slightly toward the atomic resonance central frequency n0.
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18. Characteristics of the laser output
Internal Photon-Flux Density
Gain Clamping
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19. Fundamentals of Photonics
Laser turn-on
Time
Steady state
ar Loss coefficient
g(u) Gain coefficient
Photon-flux density
Determination of the steady-state laser photon-flux density f.
The smaller the loss, the greater the value of f.
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20. Steady-State Laser Internal Photon-Flux Density
Steady Photon Density
Since
and
Steady-State Laser Internal Photon-Flux Density
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21. Fundamentals of Photonics
Steady-state values of the population difference N, and the laser internal photon-flux density f, as functions of N0 (the population difference in the absence of radiation; N0, increases with the pumping rate R).
Output photon-flux density
Optical Intensity of Laser Output
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22. Fundamentals of Photonics
Optimization of the output photon-flux density
From
We obtain
When,
use the approximation
Then
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23. Fundamentals of Photonics
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24. Spectral Distribution
Determined both by the atomic lineshape and by the resonant modes
Number of Possible Laser Modes
Linewidth ?
Schawlow-Townes limit
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25. Fundamentals of Photonics
ar Loss
Gain B
Allowed modes
n1n2……nM
Resonator modes nF
Figure (a) Laser oscillation can occur only at frequencies for which the gain coefficient is greater than the loss coefficient (stippled region). (b) Oscillation can occur only within dn of the resonator modal frequencies (which are represented as lines for simplicity of illustration ).
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26. Homogeneously Broadened Medium
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27. Fundamentals of Photonics
Inhomogeneously Broadened Medium
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Frequency n ar
(a)
(b)
Figure (a) Laser oscillation occurs in an inhomogeneously broadened medium by each mode independently burning a hole in the overrall spectral gain profile. (b) Spectrum of a typical inhomogeneously broadened multimode gas laser.
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28. Hole burning in a Doppler-broadened medium
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29. Hole burning in a Doppler-broadened medium
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30. Spatial distribution and polarization
Spherical mirror
Laser intensity
x,y
The laser output for the (0,0) tansverse mode of a spherical-mirror resonator takes the form of a Gaussian beam.
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31. Fundamentals of Photonics
TEM0,0
a11
a00
B11
B00 n
(1,1) modes
(0,0) modes
Laser output
TEM1,1 n
Figure The gains and losses for two transverse modes, say (0,0) and (1,1), usually differ because of their different spatial distributions.
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32. Fundamentals of Photonics
Two Issues: Polarization, Unstable Resonators
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33. Fundamentals of Photonics
Mode Selection
Selection of
1. Laser Line
2. Transverse Mode
3. Polarization
4. Longitudinal Mode
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34. High reflectance mirror
Laser output
Aperture
Active medium
Output mirror
High reflectance mirror
Prism
Unwanted line
Figure A paticular atomic line may be selected by the use of a prism placed inside the resonator. A transverse mode may be selected by means of a spatial aperture of carefully chosen shaped and size.
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35. High reflectance mirror Polarized laser output
Output mirror
Polarized laser output qB
Brewster window
Active midum
Figure The use of Brewster windows in a gas laser provides a linearly polarized laser beam. Light polarized in the plane of incidence (the TM wave) is transmitted without reflection loss through a window placed at the Brewster angle. The orthogonally polarized (TE) mode suffers reflection loss and therefore does not oscillate.
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36. High reflectance mirror
Selection of Longitudinal Mode
Etalon
Active midum
High reflectance mirror
Output mirror d1 d
Resonator loss
c/2d
Resonator mdoes
Etalon mdoes
c/2d1
Laser output
Figure Longitudianl mode selection by the use of an intracavity etalon. Oscillation occurs at frequencies where a mode of the resonator coincides with an etalon mode; both must, of course, lie within the spectral window where the gain of the medium exceeds the loss.
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37. Fundamentals of Photonics
Multiple Mirror Resonators
(a)
(b)
(c)
Figure Longitudinal mode selection by use of (a) two coupled resonators (one passive and one active); (b) two coupled active resonators; (c) a coupled resonator-interferometer.
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38. Characteristics of Common Lasers
Solid State Lasers: Ruby, Nd3+:YAG, Nd3+:Silica, Er3+:Fiber, Yb3+:Fiber
Gas Lasers: He-Ne, Ar+; CO2, CO, KF;
Liquid Lasers: Dye
Plasma X-Ray Lasers
Free Electron Lasers
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39. Fundamentals of Photonics
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40. Fundamentals of Photonics
Pulsed Lasers
Method of pulsing lasers
External Modulator or Internal Modulator? t
Average power
CW power
Modulator
(a)
(b)
Figure Comparison of pulsed laser outputs achievable with (a) an external modulator, and (b) an internal modulator
Gain switching Q-Switching
3. Cavity Dumping 4.Mode Locking
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41. Fundamentals of Photonics
Gain Switching
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42. Fundamentals of Photonics
Q- Switching t
Modulator
Loss
Gain
Laser output
Figure Q-switching.
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43. Fundamentals of Photonics
Cavity Dumping
Figure Cavity dumping.
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44. Mode locking Laser modes coupling together
Lock their phases to each other

45. Rate equation for the photon-number density
photon lifetime
Probability density for induced absorption/emission
From
We have
Photon-Number Rate Equation
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46. Population-difference rate equation (Three-level system)
Rate equation for the Population Difference
For a three level system
Note
Then
Where the small signal population difference
Substituting
Population-difference rate equation (Three-level system)
We have
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47. Situation for the gain switching
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48. Situation for the Q-switching
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49. Fundamentals of Photonics
Dynamics of a Q-Switching process
Dynamics of the Q-switching process
Note the time relationship between the photon density and the
population inversion variations!
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50. Fundamentals of Photonics
Determination of the peak power, energy, width and shape of the optical pulse
Dividing
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51. Fundamentals of Photonics
Power
Peak pulse power
When Ni>>Nt
It is clear So
The larger the initial population inversion,
the higher the Q-switched pulse peak power.
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52. Fundamentals of Photonics
c. Pulse energy:
The final population difference Nf
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53. Fundamentals of Photonics
d. Pulse width:
A rough estimation of the pulse width is the ratio
of the pulse energy to the peak pulse power.
When Ni>>Nth and Ni>>Nf
The shorter the photon life time,
the shorter the Q-switched pulses.
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54. Fundamentals of Photonics
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55. Techniques for Q-switching
1. Mechanical rotating mirror method:
Q-switching principle: rotating the cavity mirror results in
the cavity losses high and low, so the Q-switching is obtained.
Advantages: simple, inexpensive.
Disadvantages: very slow, mechanical vibrations.
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56. 2. Electro-optic Q-switching
Disadvantages:
complicate and expensive
Advantages:
very fast and stable.
Pockels effect: applying electrical field in a uniaxial crystal results in additional birefringence, which changes the polarization of light when passing through it.
Electro-optic effects: optical properties change with the applied electrical field strength. In the case of Pockels cell, the polarization of the light changes with the applied electrical field strength.
Combined with a polarization beam splitter, one can use them to realize the Q-switching.
Q-switching principle: placing an electro-optic crystal between crossed
polarizers comprises a Pockels switch. Turning on and off the electrical
field results in high and low cavity losses.
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57. Fundamentals of Photonics
Electro-optic Q-switch operated at (a) quarter-wave and (b) half-wave retardation voltage
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58. 3. Acousto-optic Q-switching
1. Part of the light intensity is deflected away by the acoustic wave. The stronger the acoustic wave, the stronger the deflection.
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59. Fundamentals of Photonics
Diffracted light
Incident light q q L
Sound
Transmitted light
Piezoelectric transducer RF
1. Part of the light intensity is deflected away by the acoustic wave. The stronger the acoustic wave, the stronger the deflection.
Bragg scattering: due to existence of the acoustic wave, light
changes its propagation direction.
Q-switching principle: through switching on and off of the acoustic
wave the cavity losses is modulated.
Advantages: works even for long wavelength lasers.
Disadvantages: low modulation depth and slow.
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60. 4. Saturable absorber Q-switching
What’s a saturable absorber?
Absorption coefficient of the material is
reversely proportional to the light intensity.
Is: saturation intensity.
Saturable absorber Q-switching:
Insertion a saturable absorber in the
laser cavity, the Q-switching will be
automatically obtained.
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61. General characteristics of laser Q-switching
Pulsed laser output:
Pulse duration – related to the photon lifetime.
Pulse energy - related to the upper level lifetime.
Laser operation mode:
Single or multi-longitudinal modes.
Active verses passive Q-switching methods:
Passive: simple, economic, pulse jitter and intensity fluctuations.
Active: stable pulse energy and repetition, expensive.
Comparison with chopped laser beams:
Energy concentration in time axis.
Function of gain medium
Energy storage
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62. Fundamentals of Photonics
Laser mode-locking
Aims:
Familiarize with the principle of laser mode-locking.
Familiarize with different techniques of achieving laser
Mode-locking.
Outlines:
Principle of laser mode-locking.
Methods of laser mode-locking.
Active mode-locking.
Passive mode-locking.
Transform-limited pulses.
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63. Principle of laser mode-locking
1. Lasing in inhomogeneously broadened lasers:
i) Laser gain and spectral hole-burning.
ii) Cavity longitudinal mode frequencies.
iii) Multi-longitudinal mode operation.
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64. Fundamentals of Photonics
2. Laser multimode operation:
Single mode lasers:
Multimode lasers:
Mode-frequency separations:
Phase relation between modes:
Random and independent!
Total laser intensity fluctuates with time !
The mean intensity of a
multimode laser remains
constant, however, its
instant intensity varies with
time.
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65. Fundamentals of Photonics
3. Effect of mode-locking:
(i) Supposing that the phases of all modes are locked together:
(ii) Supposing that all modes have the same amplitude:
purely for the convenience of the
mathematical analysis
(iii) Under the above two conditions, the total electric field
of the multimode laser is:
For the sack of simplicity, we assume that all laser modes have the same amplitude.
As there exist a lot of modes, we select the mode at the center of the mode frequency distribution as the reference frequency, and denote all the other mode in relation to this reference mode. The reason for doing this is for the easy of mathematical treatment.
where
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66. Fundamentals of Photonics
0 is the frequency of the central mode, N is the number
of modes in the laser, c is the mode frequency separation.
i is the frequency of the i-th mode.
Calculating the summation yields:
Note this is the optical
field of the total laser
Emission !
The optical filed can be thought to consist of a carrier wave of
frequency 0 that amplitude modulated by the function
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67. Fundamentals of Photonics
4. Characteristics of the mode-locked lasers:
The intensity of the laser field is:
The output of a mode-locked laser consists of a series of pulses. The time separation between two pulses is determined by RT and the pulse width of each pulse is tp.
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68. Fundamentals of Photonics
5. Properties of mode-locked pulses:
i) The pulse separation RT:
The round-trip time of the cavity!
ii) The peak power:
Considering
when  is small,
N times of the average power. N: number of modes.
The more the modes the higher the peak power of the
Mode-locked pulses.
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69. Fundamentals of Photonics
iii)The individual pulse width:
a: bandwidth of
the gain profile.
Narrower as N increases.
The mode locked pulse width is reversely proportional to
the gain band width, so the broader the gain profile, the shorter
are the mode locked pulses.
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70. Techniques of laser mode-locking
Active mode-locking:
Actively modulating the gain or loss of a laser cavity in a periodic way,
usually at the cavity repetition frequency c/2nL to achieve mode-locking.
Amplitude modulation:
A modulator with a transmission function of
is inserted in the laser cavity to modulate the light. Where  is the
modulation strength and  < 0.5. Under the influence of the modulation
phases of the lasing modes become synchronized and as a consequence
become mode-locked.
Operation mechanism of the technique:
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71. Fundamentals of Photonics
Time domain analysis:
Consider the extreme case where a shutter is inside the cavity, and the opens only for a short time every second. Is the cavity round trip time. In this case only a pulse with pulse width narrower than the opening time can survive in the cavity, all the CW type of operation will be blocked by the shutter. To have a pulse moving in the cavity the phase of all lasing modes must be synchronized. The laser modes will arrange themselves to realize such a state.
It is a natural competition and the fittest will survive.
Shutter losses
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72. Fundamentals of Photonics
Frequency domain analysis:
Amplitude modulation
Sidebands generation
Electrical filed of each mode
Amplitude of each mode is modulated
Sidebands are generated by the modulation m
m+
m-
Without modulation
After amplitude modulation
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73. Fundamentals of Photonics
In the case of a multimode laser
As all modes are modulated by the same frequency, the
sidebands of one mode will drive its adjacent modes, and
as a consequence, all modes will oscillate with locked phase.
From both the time domain and the frequency domain analysis it is easy to understand why the modulation frequency must be exactly the cavity longitudinal mode separation frequency.
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74. Fundamentals of Photonics
Passive mode-locking:
Inserting an appropriately selected saturable absorber inside the laser
cavity. Through the mutual interaction between light, saturable absorber
and gain medium to automatically achieve mode locking.
A typical passive mode locking laser configuration:
laser medium
satur
able
absorber
Mechanism of the mode-locking:
i) Interaction between saturable absorber and laser gain:
Survival takes all!
ii)Balance between the pulse shortening and pulse broadening:
Final pulse width.
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75. Transform limited pulses
Gaussian pulses:
In our analysis we have assumed
that En=E0
The real gain line has a Gaussian
profile,which results in that the lasing
mode amplitudes have a Gaussian
distribution.
A Gaussian gain line shape function a Gaussian
mode-locked pulse intensity variation, namely a Gaussian pulse.
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76. Fundamentals of Photonics
Intensity of the pulse:
Gaussian intensity profile!
Transform limited pulses:
If the product of pulse width and spectral bandwidth of a Gaussian
pulse equals 0.441, then the Gaussian pulse is called a transform
limited pulse as in this case the pulse width is purely determined
by the Fourier transformation of the pulse spectral distribution.
For transform limited
Gaussian pulses!
tp: pulse width, L: spectral bandwidth
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