Laser physics M. A. Bouchene Laboratoire « Collisions, Agrégats, Réactivité », Université Paul Sabatier, Toulouse, France

Outlook 1- Basic introduction 2- Optical cavity 3- Energetic model of the interaction 4- Laser oscillation (CW laser) 5- Laser frequency 6- Pulsed regime (summary)

1- Basic introduction

Basic Ideas AbsorptionSpontaneous emission Stimulated emission

Basic Ideas AbsorptionSpontaneous emission Stimulated emission

Basic Ideas AbsorptionSpontaneous emission Stimulated emission

Mirror Population inversion

Mirror Spontaneous emission

Mirror Stimulated emission

Mirror Feed-back by the cavity

Mirror Stimulated emission

Mirror Feed-back by the cavity

Mirror Laser beam After several round trips… Photons with: - same energy : Temporal coherence - same direction of propagation : Spatial coherence …..

Mirror Laser beam After several round trips… Photons with: - same energy : Temporal coherence - same direction of propagation : Spatial coherence

External medium Active medium L ight A mplification by S timulated E mission of R adiation Optical cavity (Resonator)

Pumping process External medium Active medium L ight A mplification by S timulated E mission of R adiation Optical cavity (Resonator)

Pumping process External medium Active medium - Population inversion - Amplification by stimulated emission L ight A mplification by S timulated E mission of R adiation Optical cavity (Resonator)

Pumping process External medium Active medium - Population inversion - Amplification by stimulated emission L ight A mplification by S timulated E mission of R adiation Optical cavity (Resonator) - Feed-back - Spectral and spatial shape

Some properties… Laser Spectral lamp CW: 10MHz (standard) 10GHz CW kW/cm 2

2- Optical cavity

Transfer matrix Optical system Optical axis x

Transfer matrix Optical system Optical axis x

Transfer matrix Optical system Optical axis x

Transfer matrix In the paraxial approximation Optical system Optical axis x

Some examples 0 d z z Free propagation mirror ……………….. …. M1M1 M2M2 Mn M = M n......M 2 M 1 z

Geometric stability Condition: StableUnstable

Geometric stability

g1g1 g2g2 Unstable g 1 g 2 = 1 1 1

Geometric stability g1g1 g2g2 Unstable g 1 g 2 = R 1 =R 2 = R 1 =R 2 = L/2 concentric Plan R 1 =R 2 = L confocal

Cavity Modes Elementary solution of Maxwell equation in vacuum

Cavity Modes Arbitrary (plane-wave) solution of Maxwell equation in vacuum

Cavity Modes Arbitrary (plane-wave) solution of Maxwell equation in cavity : Define longitudinal modes

Cavity Modes Arbitrary solution of Maxwell equation in cavity : beyond plane-wave approximation : Define longitudinal modes z x y

Cavity Modes Arbitrary solution of Maxwell equation in cavity : beyond plane-wave approximation : Define transverse modes : Define longitudinal modes z x y

Cavity Modes Arbitrary solution of Maxwell equation in cavity : beyond plane-wave approximation : Define transverse modes : Define longitudinal modes Mode: elementary solution of Maxwell equation inside cavity Complete basis set

Transverse modes Monochromatic wave:

Transverse modes Monochromatic wave: Solution of propagation equation inside the cavity within the paraxial approximation

Transverse modes Monochromatic wave: One possible solution: Gaussian beam Solution of propagation equation inside the cavity within the paraxial approximation

Transverse modes Monochromatic wave: One possible solution: Gaussian beam Condition: cavity is stable Solution of propagation equation inside the cavity within the paraxial approximation

FUNDAMENTAL MODE: GAUSSIAN BEAM Z AXIS 1/e 2 of I

FUNDAMENTAL MODE: GAUSSIAN BEAM Z=0 plane wave Z AXIS 1/e 2 of I 0

FUNDAMENTAL MODE: GAUSSIAN BEAM Z AXIS 1/e 2 of I

FUNDAMENTAL MODE: GAUSSIAN BEAM Z AXIS 1/e 2 of I

FUNDAMENTAL MODE: GAUSSIAN BEAM Z AXIS 1/e 2 of I

Z AXIS Everything depends on the waist How to get w 0 and z=0 in a cavity ??

In a cavity: The wavefront adapts to the shape of the mirors ( stability)

High order modes Solutions : Hermite-gauss beams (a complete basis set)

High order modes Solutions : Hermite-gauss beams (a complete basis set) TEM m,n Nodes in x,y x y

Longitudinal modes Condition of resonance :

Longitudinal modes Condition of resonance : ………

Longitudinal modes Condition of resonance : ………

Cavity losses

transmission

Cavity losses Diffusion, Spont.emission

Cavity losses absorption

Cavity losses diffraction

Cavity losses transmission Diffusion, Spont.emission diffraction absorption

Cavity losses Consequence: broadenning of the longitudinal modes transmission Diffusion, Spont.emission diffraction absorption

3- Energetic model of the Interaction

1- Evolution of populations Absorption 1 2 Two-level open system

1- Evolution of populations Absorption Stimulated emission 1 2

1- Evolution of populations Absorption Relaxation Stimulated emission 1 2

1- Evolution of populations Absorption Relaxation Stimulated emission 1 2 Rate equations Pumping

1- Evolution of populations Absorption Relaxation Stimulated emission 1 2 Rate equations Pumping

Stimulated emissionAbsorption 2- Evolution of Intensity Stimulated emissionAbsorption 1 2

2- Evolution of Intensity Stimulated emissionAbsorption 1 2

Inversion of population ? In a CLOSED two-level system, an incoherent excitation can never accomplish population inversion Arb. parameter 1 2 N2N2 1/2

Solution : More levels FOUR LEVEL SYSTEM

Solution : More levels pumping

Solution : More levels fast relaxation

Solution : More levels Inversion of population Laser transition

Solution : More levels A 21 Spontanous emission

Solution : More levels relaxation CONDITION: A 21

Solution : More levels relaxation CONDITION: A 21 Three level system Laser transition

Solution : More levels fast relaxation CONDITION: A 21 Three level system Laser transition

4- Laser Oscillation (CW laser)

Amplifier in an optical cavity Active medium Gain

Amplifier in an optical cavity Active medium Losses<0 (spread) Laser equations

: Not sufficient! Gain must overcome the losses

: Not sufficient! Gain must overcome the losses Laser works (Stationnary regime): Gain = losses Photons created on a round trip = photons lost on a round trip

Laser equations

Two regimes :

Pumping Stimulated emission Losses

Pumping Stimulated emission Losses Increases the gain (Pop. Inv)

Pumping Stimulated emission Losses Decreases the gain Increases the gain

Pumping Stimulated emission Losses equilibrium Fixes the laser intensity Increases the gain Decreases the gain

5- Laser frequency

1 2

1 2 Frequency Spectral line

1 2 Frequency Gain cavity modes Spectral line

1 2 Frequency Gain Losses cavity modes

1 2 Frequency Gain Losses cavity modes Possible laser frequencies

Ring laserLinear laser Homogenous linewidth MonomodeMultimode Inhomogenous linewidth Multimode travelling waveSpatial stationnary wave

6- Pulsed regime (summary)

Relaxation regime: µs pulses (10 -6 s) Q-switch regime : Nanosecond pulses (10 -9 s) Mode-locking Regime: Picosecond ( s) and femtosecond pulses ( s)

Bibliography -YARIV: "Quantum Electronics" (Willey interscience 1975) - RABIN&TANG: "Quantum Electronics" (Academic Press 1975) - A.E. SIEGMAN: "Lasers" (University Science Books 1986) - P.W. MILONNI & J.H. EBERLY "Lasers" (John Wiley & Sons 1988) - - M. SARGENT III, M.O. SCULLY, W.E. LAMB,Jr. :"Laser Physics" (Addison-Wesley Publishing Company 1974)

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