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Laser Theory: Come On, How Hard Can It Be ?. Cockcroft Institute Laser Lectures April 2008. Graeme Hirst STFC Central Laser Facility. Lecture 3 Plan. Stimulated emission and laser gain Rate equations and gain saturation Linewidth Laser cavities Spectrum (axial modes)

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Laser theory come on how hard can it be l.jpg
Laser Theory:Come On, How HardCan It Be ?

  • Cockcroft Institute Laser Lectures

  • April 2008

Graeme HirstSTFC Central Laser Facility

Lecture 3 plan l.jpg
Lecture 3 Plan

  • Stimulated emission and laser gain

  • Rate equations and gain saturation

  • Linewidth

  • Laser cavities

  • Spectrum (axial modes)

  • Gaussian beams (transverse modes)

  • Modelocking

  • Conclusions

Elements of a laser l.jpg
Elements of a Laser



Gain medium

All lasers contain a medium in which optical gain can beinduced and a source of energy which pumps this medium

Many also contain optical elements whichmodify the laser beam

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A21 = an3B21

B21 = B12

Einstein A and B Coefficients

“The formal similarity of the curve of the chromatic distributionof black-body radiation and the Maxwell velocity-distribution istoo striking to be hidden for long …” *

Consider a gas of two-level atomsin equilibrium with photons.

Einstein showed that the black bodyspectrum and the Maxwell distributioncould be reconciled by adding to thefamiliar processes of spontaneousemission and absorption a third“induced radiation” process.

* Opening lines of A Einstein,On the Quantum Theory of Radiation, Physikalische Zeitschrift 18 121 (1917)

Laser gain l.jpg
Laser Gain

Neglecting other loss processes, changes to the number ofcoherent photons, n, will be controlled by the balancebetween absorption and stimulated emission:

Where N1 and N2 are the populations of the respective levels

Since B21 = B12this simplifies to:

Laser gain therefore requires N2 > N1

i.e. population inversion, which is a non-equilibrium condition

Population dynamics l.jpg


Population Dynamics


Consider a two-level gain mediumin the presence of light

Neglecting thermal excitation thefull rate equations for the levelpopulations are:





A21, B21





g (inverse lifetime) characterises relaxations to all other levels

NB these equations do not fullydescribe the evolution of n

Achieving inversion l.jpg
Achieving Inversion

Considering the conditions for steady-state inversionin the absence of lasing (these are easiest to achieve)

, n  0


Inversion requires N2/N1 > 1 and depends on a combinationof

selective pumping


a favourable lifetime ratio

Even with the best selective pumping, i.e. R1=0, the equationfor N1 shows that cw inversion requires g1/A21 > 1 i.e. thethe lower level must empty faster than it’s being filled.

Without this the only possibilitywill be transient lasing.

Gain saturation l.jpg

The difference between them is the population inversion:

Gain Saturation

The steady-state solutions of the rate equations with lasing(i.e. with n0) and with negligible R1 can be shown to be:

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Gain Saturation

Saturation occurs when stimulated emission becomescomparable with spontaneous emission

Efficient, low-noise laser operation, requiring stimulated emission to dominate spontaneous emission, dependson strong saturation and, therefore, operation at low gain

Any loss processes can then become very wasteful

CW lasers, or those where the interpulse period is less than1/(g2+A21), are characterised by Isat hn(g2+A21)/s W/m2

In pulsed lasers Esat  hn/s J/m2 (for emission cross-section s)

Isat and Esat depend only on the lasing species and not, forexample, on the pumping rate, inversion density etc.

Saturation suppresses ASE butdistorts pulse shapes

Linewidth l.jpg



Broadening is Homogeneouswhen it affects all “atoms” equallyand Inhomogeneous when it splitsthem into sub-groups


In fact emission and absorption depend on photon frequency andare characterised by a linewidth

  • Line broadening types include:

  • Natural, from finite lifetime (H)

  • Phonon, from lattice vibrations (H)

  • Collisional, in gases (H)

  • Strain, from static lattice inhomogeneities (I)

  • Doppler, in gases (I)

  • and the laser levels mayalso be closely spacedmanifolds with more orless inter-level coupling

Linewidth effects l.jpg
Linewidth Effects

A laser beam of intensity I (W/m2), propagating in thez direction through a medium with gain coefficient g (m-1)grows in intensity as I = I0 exp(gz)

Since g depends on wavelength, this process will increase theintensity for wavelengths near line-centre faster than for thosein the wings, leading to gain-narrowing of the spectrum

In inhomogeneously broadened media a laser beam at onewavelength will not saturate the whole population inversion

The part that remains can support lasing at other wavelengths,making single-wavelength operation hard to achieve

Inhomogeneously broadened media tend to saturate as1/(1+I/Isat)½ rather than 1/(1+I/Isat)

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The analysis so far reproduces the basic features of reallasers. However in reality:

  • Many more energy levels and pathways are involved

  • Levels have “degeneracy” which must be accounted for

  • Optical losses (in addition to lower-to-upper levelabsorption) need to be included

  • Pulsed lasers and transient effects in cw ones are verycommon and merit separate treatment (although the cwresults are surprisingly relevant)

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Optical Cavities

In the absence of external optics the light from a laser mediumwill be more or less bright ASE with the following properties:

  • SPECTRUM: set by the medium’s spectral gain profile,perhaps subject to gain-narrowing and effects ofinhomogeneous broadening

  • TEMPORAL PROFILE: spiky (incoherent) with spike widthscorresponding to the Fourier transform of the spectrum

  • TRANSVERSE PROFILE: divergence (coherence)limited by the physical extent of the gain medium - long,thin media may yield quite low-divergence beams

  • POLARISATION: random unless crystal effects causethe gain to be polarisation sensitive

Optics can “improve” all ofthese parameters

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Generic cavity:

Real cavity:

Cavity Elements

Optical cavities take control of a laser by feeding back verymuch more light than is present from optical noise

If the feedback is high enough(i.e. cavity losses are lowenough) there will be netround-trip gain andoscillation. The intracavitylight will saturatethe gain until itjust balancesthe loss. Intensities insidecavities can be very high indeed.




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Generic cavity:




Intracavity Fields

Consider the electric field in an x-y plane inside the cavity.After one round trip the field will be transformed fromE(x0, y0) to E(1)(x, y) by a propagation integral of the form:

The kernel K(x,y,x0,y0) describes the whole transformation,including free-space “Huygens” propagation, the effects ofapertures, optics etc except for the axial phase change e-j2kL

The equation can be solved numericallyfor particular cases

Intracavity fields16 l.jpg

With gain, the steady-state modes can satisfy

Intracavity Fields

Consider the eigenmodes of the cavity i.e. those fielddistributions Enm(x, y) which satisfy:

where the eigenvalue e-j2kLgnm is a complex numberdescribing the overall phase and amplitude changeexperienced by the n,mth mode for each round trip.

Without laser gain the amplitude of gnm will be less than onebecause of “diffraction losses” around the mirrors’ edges(diffraction redistributes energy from the sharp-edged fielddistribution immediately after reflection into a larger areawhen the beam returns).

Spectral control l.jpg

(typically several million )

i.e. the optical cavitysupports longitudinal(axial) modes with a verylarge number of closelyspaced frequencies



Laser gain can select just a few(one ?) of these

Spectral Control

In practice the phase, f, of gnm is almost independentof the photon wavenumber, k, for different modes. Thisallows the overall phase of e-j2kLgnm to be separated fromthe amplitude. The solutions of (1) then need to satisfy:

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Generic cavity:

This is the equation for aspherical wavefront, inthe paraxial approximation,where R(z) = R0+z-z0 is the slightly contrived radius of curvature


Transverse Profile

It is, perhaps, intuitive that themodes’ wavefronts will be spherical,since this will allow them to matchexactly the surfaces of the sphericalmirrors and thus be reflected back along their incoming paths

This wave has the undesirable property that the fieldamplitude is independent of x and y, so the wavefrontsextend to infinity away from the optical axis, whichis unphysical given the finite sizesof the mirrors

Gaussian beams l.jpg

R(z) is now a real radius of curvatureand w(z) is a transverse spot size, asis clear when (3) is substituted into (2)


w(z), R(z) and (z) can all be expressed in terms of a Rayleighrange, zR, which with the wavelength defines the beam uniquely

Gaussian Beams

A better choice of spherical wave turns out to be of the sameform, but with a complex radius of curvature, q(z), where:

Gaussian beams20 l.jpg
Gaussian Beams

  • This beam is the lowest order (TEM00) eigenmode of the propagation integral for empty spherical-mirror cavities (in fact for the “stable” sub-set of such cavities)

  • There is an infinite series of higher-order “Hermite-Gaussian” modes, making up a complete basis set which can be used to analyse any field distribution in the cavity

  • In real cavities different transverse modes experience different gains and losses. “Mode discrimination” can quickly lead to few-mode or single-mode operation.

  • The sometimes surprising properties of Gaussian laser beams (forms of R(z), positions of beam waists etc) persist outside the optical cavity: remember this when transporting them !

Temporal control l.jpg


A switchable “shutter” in thecavity will stop lasing and allowa large inversion to build up

When the shutter opens the high gain establishes a very stronglaser beam which depletes the inversion in a “giant pulse”,(typically lasting for nanoseconds). In somemedia self-Q-switching can occur.



Gain medium

Temporal Control

For applications needing high peak power it is necessary tooperate the laser in a pulsed mode.

Simply pulsing the pump may be sufficient if, for example,the cw power limit is a thermal one. But repetition rates arelikely to be low and each pulse will need to start from noise.

Temporal control22 l.jpg


The axial modes of an optical cavity are ideal building blocksfor a pulse train at f=c/2L (typically tens of MHz to GHz)provided a way can be found to lock their phases


Temporal Control


If the laser output is tobe a regular train of veryshort pulses (d-functions)then the spectrum mustalso tend towards acomb of phase-lockedd-functions (short pulsesneed bandwidth)

Modelocking l.jpg





Active modelocking

A loss-modulator put intothe cavity will introduce“sidebands” on each of theaxial modes with an offsetequal to the drive frequencyand a fixed relative phase.

If wmod = c/2L then the sideband radiation can compete withnoise and “take control” of the phase of adjacent modes.

In the time domain the modulator can be seen as a shutterwhich opens once per cavity round trip. Light which arrivesjust when the shutter opens will be preferentially amplified.Modulation depth need not be large butfrequency matching is critical.

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Gain medium


Saturable absorbers needto have sufficiently fast recovery


Passive modelocking

Nonlinear opticalprocesses can lowercavity losses for highintensity pulses,forming “shutters”which automaticallyopen at the correct time.

Modelocking25 l.jpg

Aperture selectsfor tight focus



f set by intensityi.e. by pulse length

Kerr lens modelocking is thepreferred technique for Ti:S lasers


Passive modelocking

Nonlinear opticalprocesses can lowercavity losses for highintensity pulses,forming “shutters”which automaticallyopen at the correct time.

Intracavity dispersion can stretch a pulse making it lesseffective at “opening the shutter”. Dispersion compensationis critical if the shortest pulses are to be generated.

In the frequency domain dispersion changes the axialmode spacing, preventing lockingacross the full spectrum.

Polarisation l.jpg

This can be controlled easily using intracavity elementsand lasers will often be linearly polarised

  • because the laser gain is polarisation-sensitive (crystals)

  • because optics in the cavity have particularly low lossfor one polarisation (Brewster windows, dielectric mirrors …)

  • because the cavity contains elements which use polarisation(Pockels cell, AO modulator, intracavity doubler, diffractiongrating, Lyot filter …)

  • to avoid problems in beam propagation outside the cavity

  • to facilitate nonlinear optics outside the cavity

Polarisation ratios of >100:1 are typical

Conclusions l.jpg

  • Gain requires population inversion which is achieved by acombination of selective pumping and favourable lifetimes.

  • Lasers consist of an optical gain medium and a pumpsource. Extra optics can modify the laser beam’s properties.

  • Laser gain is saturable. Saturation is needed for efficiencybut can distort beams’ spatial and spectral profiles.

  • Lasers have linewidth.

  • Laser cavity modes influence the beam’s spectral and temporal properties (axial) and spatial profile (transverse).

  • Gaussian beams have distinctive properties and cannotbe reliably treated with simple geometric optics.