diffraction patterns produced by a circular aperture sinq(1) = 1.22(l/d) q(1) - angular position of the first order diffraction minima, l - wavelength of incident light, d - diameter of the aperture q(1) @ 1.22(l/d) Thus, if two objects are at a distance D apart from each other and at a distance L from an observer, the angle between them is: q = D / L D(0) = 1.22(lL/d) where D(0) is the minimum separation distance between the objects that will allow them to be resolved

Stimulated Emission Depletion (STED( Microscopy A phenomenon that stops fluorescence (=spontaneous emission) is that of stimulated emission

The STED-microscope S. W. Hell and J. Wichmann Opt. Lett.19, 780 (1994).

Improvement beyond the diffraction limit

Classical treatment of atom-light interaction One of the best analogues to understand the interaction between radiation and matter is a pendulum representing an atom, and a string representing the field The two can oscillate separately without affecting each other Suppose that the pendulum is stationary and the string is vibrating; if the string is coupled to the pendulum, then the initial oscillations of the string will be transmitted and the pendulum will start to oscillate as well at the same frequency If the two have similar frequencies, then the oscillations will be large, and all the energy of the string will be transferred to the pendulum — this corresponds to atom absorbing a photon After some time the energy will go back to the string which is the reverse of the previous process, corresponding as it does to the emission of a photon by the atom

Dipole radiation the atom - a mass (electron) on a spring (attached to the nucleus) the spring is contracted and extended as it interacts with light - an EM wave as the spring extends the energy from the EM field gets stored - absorption 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

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 lets look at the Fourier spectrum by taking a Fourier Transform

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

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 Where do these come from? atomic collisions Doppler broadening lifetime (natural) broadening

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 the uncertainty principle DEDt > /2p Dw = DE/ > 1/t this broadening is intrinsic to the transition – natural broadening and the spectrum corresponds to Lorentzian lineshape

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 lifetime than we need to replace t by tcol in Dwlifetime = 1/t – resulting in additional broadening Based on the kinetic theory of gases tcol is given by ss - collision cross section P - pressure 1/tcol and Dw are proportional to P collisional broadening  pressure broadening

at standard temperature and pressure (STP) typical tcollision ~ 10−10 s - much shorter than typical radiative lifetimes corresponds to linewidths of ~ 1010 rad s−1

Doppler broadening the Maxwell-Boltzmann velocity dis. due to random motion of the atoms in the gas  Doppler shifts in 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 Doppler linewidth of the 589. 0 nm line of Na at 300 K is 1 the Doppler linewidth of the 589.0 nm line of Na at 300 K is 1.3 GHz (~0.04 cm-1), about two orders of magnitude larger than the 10 MHz (3.3 x 10-4 cm-1) natural broadening due to the radiative lifetime of 16 ns The dominant broadening in low pressure gases at room temperature is usually Doppler broadening and the lineshape is closer to Gaussian

Line broadening in solids the spectra are 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

This inhomogeneous broadening is related to environmental broadening The 1064 nm transition of a Nd:YAG crystal at room temperature is homogenously broadened to around 120 GHz (4 cm-1) by phonon emission The equivalent transition in Nd:glass is 40–60 times broader owing inhomogeneity of the glass medium on the emission frequency of the Nd3+ ions This inhomogeneous broadening is related to environmental broadening

Nonlinear Optics What are nonlinear-optical effects and why do they occur? Maxwell's equations in a medium Nonlinear-optical media Second-harmonic generation Sum- and difference frequency generation Conservation laws for photons ("Phase-matching")

Nonlinear Optics optical frequencies of lasers can be doubled in nonlinear optical crystals for an optical wave passing through a crystal, the atomic electrons are induced into forced oscillations for small electric field amplitudes E of the wave the elongations of the oscillating electrons are small and the restoring forces are proportional to the elongation (linear range) The induced dipole moments p = α . E are proportional to the field amplitude and the component Pi of the dielectric polarization of the medium induced by the light wave depends linearly on E ij are the components of the tensor  of the electric susceptibility c of a material is a measure of how easily it polarizes in response to an electric field

The field amplitude of the sunlight reaching the earth at l ~ 500 nm within a bandwidth of 1 nm is about E ≈ 3V/m A typical laser generates a peak electric field strength of 107 -1010 V/m the Coulomb field binding the electron to the nucleus is 1011 V/m then the response of the medium to the laser field is (very nearly) linear and the (total) polarization is

Maxwell's Equations in a Medium The induced polarization, P, contains the effect of the medium: The polarization is proportional to the field: This has the effect of simply changing the dielectric constant:

The effect of an induced polarization on a wave requires solving Maxwell’s Equations The induced polarization in Maxwell’s Equations yields another term in the wave equation this is the “Inhomogeneous Wave Equation” The polarization is the driving term for a new solution to this equation

For much larger light intensities the nonlinear range of electron elongations can be readily reached The dielectric polarization is: c(n) is the nth order susceptibility, represented by a tensor of rank (n+1) c(n) decrease rapidly with increasing n for high field amplitudes, E, the higher order terms can be no longer neglected They form the basis of nonlinear optical phenomena

When a monochromatic light wave passes through the medium, the frequency spectrum of the induced polarization P contains in addition to ω, higher harmonics mω (m = 2, 3,. . . ) the amplitudes A(mω) of these emitted waves depends on the magnitude of the c(n) coefficients and on the amplitude E0 of the incident light wave

Optical Frequency Doubling If the light wave passes through an isotropic medium we obtain from for the location z =0, the x-component of the dielectric polarization, when high order terms are neglected using the relation The dielectric polarization contains a constant term, a linear term with frequency ω and the nonlinear term with 2ω each of the atoms hit by the incident wave radiates a scattered wave containing the frequencies ω and 2ω

Conservation laws for photons in nonlinear optics Energy must be conserved: the ħ’s canceled Momentum must also be conserved Unfortunately, may not correspond to a light wave at frequency wsig! Satisfying the two relations simultaneously - phase-matching

Conservation laws for photons in Second-Harmonic Generation Energy must be conserved: w1 wsig Energy w1 Momentum must also be conserved: To simultaneously conserve energy and momentum: The phase-matching condition for SHG!

Phase-matching Second-Harmonic Generation The phase-matching condition for SHG: Unfortunately, dispersion prevents this from ever happening! Frequency Refractive index

Phase-matching Second-Harmonic Generation using birefringence Birefringent materials have different refractive indices for different polarizations. “Ordinary” and “Extraordinary” refractive indices can be different by up to 0.1 for SHG crystals the phase-matching condition can be satisfied Use the extraordinary polarization for w and the ordinary for 2w: Frequency Refractive index ne depends on propagation angle, so we can tune for a given w Some crystals have ne < no, so the opposite polarizations work

Light created in real crystals Far from phase-matching: SHG crystal output beam Input beam Closer to phase-matching: SHG crystal output beam Input beam the SH beam is brighter as phase-matching is achieved

Optical interferometry with second harmonic generation for measuring widths of ultrashort pulses the time resolution even for fast optical detectors is limited to about 100 ps measurement of short pulses can no longer be performed with conventional devices optical interferometry The laser beam is split into two parts, recombined after having traveled along two different paths with slightly different path lengths The superposition of the two parts with variable time delay t and intensities

The total intensity depends on the relative phase between the two optical waves, i.e., on t the detector cannot follow the fast optical waves, but can measure the time dependent interference pattern I(t), if the change of t is slow if the spectral width of the short pulse is large, it contains a superposition of many monochromatic carrier waves with a nearly continuous frequency spectrum in this case there will be no clear interference pattern and the detector would measure the sum of the two intensities I1+ I2, independent on their separation the frequency-doubling of the fundamental wavelength in a nonlinear crystal helps the intensity of the second harmonic: the detector measures the time average of the pulses, giving the true pulse profile I(t)

Sum- and difference-frequency generation Suppose there are two different-color beams present So: 2nd-harmonic gen 2nd-harmonic gen Sum-freq gen Diff-freq gen dc rectification Note also that, when wi is negative inside the exp, the E in front has a *.

Sum frequency mixing Frequency doubling Down conversion

Multiphoton Processes The nonlinear phenomena are collective effects of nonlinear media which exhibit higher order nonlinearities But can each individual atom behave in a non-linear way? each atom is a harmonic oscillator and, when the motion becomes anharmonic, the atom starts to behave in a nonlinear fashion

One photon photoelectric process the energy conservation law for photons knocking out electrons from a metal the photoelectric effect will therefore be observed only if the lowest frequency able to achieve the effect is Could we observe the effect at lower frequencies? Before the advent of lasers the answer would definitely be "no"

multiphoton photoelectric process and the threshold frequency is N times smaller than according to Einstein's formula This process is possible in reality, but intensities need to be high so that the probability to absorb N photons be high

Confocality detector Pinhole

detector Pinhole

The two beams within the birefringent cystal are referred to as the ordinary and extraordinary ray  The polarization of the ex ray lies in the plane containing the direction of propagation and the optic axis, and the polarization of the or ray is perpend.