Basic optical processes

1. Basic optical processes Interference Coherence Light pressure Optical absorption Amplification Atom-light interaction classical Spectral lines Mode locking

2. Interference interference - the effect from a number of individual sources (electromagnetic waves), is larger or smaller than the sum of individual effects (constructive or destructive) suppose N atoms each emitting a light wave ak is the (time-independent) amplitude of the k-th wave Let's add up all the contributions to obtain the total amplitude and then take the square to obtain the total effect (intensity)

3. when atomic emission is not coordinated - phases vary randomly and in the end average to zero this is the same as if there was no interference and the total effect (intensity) is the same as the sum of individual effects (intensities) if all the phases are the same (laser) this is N times more intense than when the phases are random the sources are coherent

4. Coherence light is a 3D spatial wave evolving in time - two different coherences: • temporal coherence concerns the same "beam of light" at the same spatial point, but two different times • spatial coherence concerns two or more different points of the wave, but at the same time a monochromatic wave with infinite coherence time a wave whose phase drifts – short coherence time coherence - very important for obtaining interference

5. temporal coherence Michelson interferometer monochromatic source – the spread in frequency of light involved, dn, is smaller than n if the time delay between the beams is dt than fringes are formed only if dtdn < 1 the same beam interferes with itself -at two different times for a lamp dn ~ 108 s-1 so that dt ~108 s-1 the spatial coherence length isdl = c x 10-8 s = 3 m for a laser, dn ~ 104 s-1 so thatdl =30 x 103 m

6. Spatial coherence – Young’s double slit experiment the same light illuminates two slits “close” to each other the light is of size ds and the angle between the source and the slits dq interference fringes will form only if dqds < l the source is composed of many point sources – lead to different fringe patterns since all the sources are out of phase with each other we have to add all the point intensities at the end to obtain the total fringe pattern if the distance between the slits increases – the patterns become more and more out of step – fringe disappearance

7. Light pressure a charge q interacts with an electric field E through the force and with the magnetic field B through the Lorentz force v is the charge velocity the combined effect of the two forces leads to radiation pressure (P) - the force on an area Aper that area force is also the rate of change of momentum (Dp)

8. the momentum density is given by the Poynting vector, S, divided by c2. The volume is AcDt considered just normal incidence, for isotropic radiation in a cavity

9. radiation pressure using Einstein's rate equations the photons that make up an EM wave of wavevector k each carry a momentum in a medium of refractive index the photon momentum is given Suppose now that a photon is absorbed by an atom of mass M. It then gains the velocity of

10. the atom then decays via stimulated emission - the emitted photon carries away the momentum in the same direction as the original photon spontaneous emission - direction of the momentum of the emitted photon anywhere within a 4p solid angle. The atom therefore recoils in some random direction. On average – there is no cancellation of the momentum previously gained as the net spontaneous emission transfer averages to zero

11. so in a cycle of absorption and emission we have the net transfer of from the photons to the atoms, in the direction of the incident beam • every absorption = one momentum kick in laser direction • • net effect: laser pushes atom • • the momentum transfer to atoms gives rise to radiation pressure

12. the number of atoms Nis large enough to produce small time dependencies in the atomic populations and that Pis the total atom momentum • The rate of change of Pin the presence of radiative energy density <W> at a frequency w(resonant with the ground and the excited state of the atoms) is proportional to the difference between the absorbed and the stimulated emission rates the rate of change of momentum is negative for N2 > N1 the number of atoms in steady state

13. for strong fields we have that WS << <W> this is the saturation value for transfer rate for very strong beams once saturation has been reached, any increase of beam strength produces very little change in the momentum transfer rate the steady state momentum transfer rate is equal to the force acting on each atom how can this force be measured?

14. suppose that an atom beam passes perpendicularly to a strong laser beam • the atoms in the beam interact with light and absorb and emit radiation • deflection occurs when atoms absorb light followed by spontaneous emission, in which case they gain momentum perpendicular to their direction of motion • the resulting deflection is about 10-5rad • the two isotopes have different transition frequencies • if the laser is tuned to a resonant transition of one isotope • then only the on-resonant atoms will be deflected, and therefore the two isotopes would separate into two different beams

15. Optical absorption light propagating through a medium is absorbed and reemitted by atoms - its intensity will decrease the intensity of radiation is I= uv u- energy density of the radiation field v- field velocity for small distances traveled by light, the change in intensity has to be proportional to the traveled medium length and to the intensity itself Kn - absorption coefficient (depends on frequency and medium)

16. Beer’s law the intensity changes exponentially as it propagates Using the fact that I= vu we can write the equation for dI from Einstein’s relationship ( instead of n)

17. Fuchbauer- Ladendurg formula for N1 N2we obtain dI = 0 no absorption - saturation for N2 >> N1amplification– thebasis of laser operation this condition, cannot be achieved by exciting a two level atom - when we have saturation then the No. of ground state atoms > No. of excited state atoms can be obtained in a three level atom

18. Amplification: three level system the atom population can be stored in the third level - decay to the second level and lase to the first level

19. S – transition probability due to both radiative and non-radiative processes total No. of atoms conserved use levels 1 and 2 for lasing, so the equations are solved for the steady state case (i.e. , dNi /dt = 0)

20. where N= N1+ N2+ N3is the total number of atoms in all three levels The number of atoms arriving at the level 2 per unit time is: Let’s express the numerator of N2 - N1 (the denominator is positive at all times). It is N( 􀀀 - A21) The condition for lasing Population inversion 3-level systems

21. Ruby-laser Maiman (1960): cavity L =n l Ruby: Al2O3 + Cr Xe t=0.003 s coherent monochromatic collimated

22. four level laser • atoms are pumped from the ground state to level 4 from where they decay rapidly to level 3, creating population inversion with respect to level 2 • the pumping to level 4 can be optical (from a flashlamp or another laser) or electrical • the decay rate from level 2 to ground state (level 1) must be fast to prevent atoms accumulating in that level and destroying the population inversion

23. Nd:YAG (Nd:Y3AI5O12) laser

24. HeNe laser • pump He to metastable state (20.61 eV) • transfer excitation to Ne metastable state (20.66 eV) • laser transition • spontaneous emission (2 times) to deplete lower level ( low pumping) • not very efficient! (20.6 eV vs 2 eV)

25. Classical treatment of atom-light interaction the atom is represented as a mass (electron) on a spring (attached to the nucleus) this spring is then contracted and extended as it interacts with light - an EM wave as the spring extends the energy from the EM field gets stored -absorption of radiation and is then released when the spring contracts - radiation emission F is the force on the electron due to the field F = qE0 cos(wt) the atom oscillates at the frequency of the driving field the highest amplitude of oscillation is when the field is on resonance - the driving frequency is the same as the natural oscillator

26. Radiation damping the solution decays exponentially to zero - all oscillations must eventually die away as energy dissipated into the environment the solution is not a monochromatic wave - more than one frequency component is present in its expansion We look at the Fourier spectrum by taking a Fourier Transform

27. there are many frequencies in the spectrum and not just that of the driving field the intensities of various frequencies is given by Lorentzian broadening

28. Spectral lines • Atomic states have in principle well defined energies - these energy levels, when analyzed spectroscopically, appear to be broadened • The shape of the emission line is is described by the spectral lineshape function gw(w), which peaks at the line center defined by hw0 = E2 - E1 • Where do these come from? atomic collisions Doppler broadening lifetime (natural) broadening

29. lifetime broadening light is emitted when an electron in an excited state drops to a lower level by spontaneous emission The rate of decay is determined by the Einstein A coefficient  determines the lifetime t the finite lifetime of the excited state leads to broadening of the spectral line according to uncertainty principle DEDt > /2p Dw =AE/ > 1/t this broadening is intrinsic to the transition – natural broadening and the spectrum corresponds to Lorentzian lineshape

30. collisional (pressure) broadening The atoms in a gas frequently collide with each other and with walls of the containing vessel, interrupting the light emission and shortening the effective lifetime of the excited state If the mean time between collisions, tcol , is shorter than the radiative lifetimethan we need to replacet by tcol in Dwlifetime = 1/t – resulting in additional broadening Based on the kinetic theory of gases tcol is given by ss is the collision cross section and P the pressure 1/tcol and Dw are proportional to P collisional broadening  pressure broadening

31. Emission of all the atoms atoms moving toward the observer Doppler broadening originates from the random motion of the atoms in the gas Doppler shifts in the observed frequencies the Maxwell-Boltzmann velocity dis. where N(v) is the number of atoms moving with v The line shape is a Gaussian the Doppler broadening gives a Gaussian profile rather than a Lorentzian its half width at half maximum is The dominant broadening in low pressure gases at room temperature is usually Doppler broadening and the lineshape is closer to Gaussian

32. Line broadening in solids • the spectra will be subject to lifetime broadening as in gases – a fundamental property of radiative emission • the atoms are locked in their positions – neither pressure nor Doppler broadening are relevant • The emission and absorption lines can be broadened by other mechanisms non-radiative transitions (phonons) The non-radiative transitions shorten the lifetime of the excited state according to the phonon emission times in solids are often very fast – substantial broadening inhomogeneity of the host medium

33. More detailed principles of laser laser elements: • a cavity with two or more highly reflecting mirrors • a gain medium - support inverted atomic population • an energy source which can excite the atoms in the gain medium to achieve the population inversion • a loss mechanism by which the stored energy is dissipated

34. absorption and amplification in a certain medium, but now • an electro-magnetic field propagates through a medium, the medium responds to it – polarization • the medium is composed of a bunch of classical oscillating springs • the microscopic dipole is then p= 􀀀ex P= Np= 􀀀Nex the driven harmonic oscillator equation for polarization is the proportionality constant between P and E - a complex number the measured quantity is the intensity which is the mod square of E the real part represents the intensity decay due to the absorption in the medium, while the imaginary part the oscillatory behavior of E solutions

35. the dielectric susceptibilty E a plane wave the response of the medium is linear, therefore the wavevector is also is a complex number, so is the refractive index the real part n’ is the "normal" refractive index, the imaginary component absorption or gain in vacuum in medium

36. the electric field is modified as it propagates. It has the form the last term is equal to which proves Beer’s law in terms of intensity the absorption (gain) coefficient

37. the laser cavity has two laser mirrors if intensity Iimpinges on a mirror of reflectivity r will return with the intensity rI the gain due to the medium for combination of one round trip reflections and the gain coefficient where For oscillations to build up The maximum value of the gain coefficient is threshold population inversion

38. The size of population difference N2 -􀀀N1 amplification is achieved for rate of emission into a cavity mode exceeding the rate of loss of photons from the cavity in that mode The No. of photons into a modeis where The rate of loss of photons is where is the lifetime of a photon within the laser cavity and r is the reflectivity of the mirrors (the same for both mirrors) The amplification condition is very different from the “zero" coming out from Einstein's analysis (but still small compared to the number of atoms which is of the order of 1023

39. Mode locking • radiation in a cavity will be a mixture of many different modes oscillating at different frequencies (rates) • they will all oscillate with different phases as the atoms in the cavity mirrors and the lasing medium radiate randomly at different times  contributing to different initial phases • why do not all these interfere destructively, canceling each other out and leaving no signal? • the state of the field • the intensity of radiation to obtain is proportional to the number of modes

40. mode locking condition the frequency separation between different modes is the mode-locked intensity is arrange so that all of the fields in different modes have the same phase the peak intensity is N times stronger than without mode locking the maximum happens the time between the maxima if modes can be locked, then high intensities and small pulse widths can be obtained the width of the pulse it is allowed to achieve shorter and shorter pulses (ultra-short pulses) by "locking in" more and more modes (N)

41. problem The organic dye Rhodamine 6G can lase at wavelengths from 550 to 630 nm. Estimate the shortest possible pulse duration that can be achieved by mode-locking this material in a laser. How many optical cycles does this represent? If the cavity of the laser has a length d = 1.5 m, and the average power output is 100 mW, what is the peak power of a mode-locked pulse having the shortest duration allowed by the gain medium.