Some Topics in Optics

Transcription

1 Some Topics in Optics The HeNe LASER The index of refraction and dispersion Interference The Michelson Interferometer Diffraction Wavemeter Fabry-Pérot Etalon and Interferometer

2 The Helium Neon LASER A LASER consists of mainly three components An energy pump HeNe LASER uses DC HV for glow discharge exciting He and Ne atoms from their ground states into higher energy states A medium that provides optical gain He/Ne (about 5:1) gas mixture with higher energy states overpopulated compared to ground state (population inversion) A resonating cavity Inducing emission through photons of the same wavelength in optical resonator

3 HeNe LASER Energy Levels LASER process in multiple steps Excitation of He ground state atoms into metastable 2 1 S and 2 3 S states Excitation of Ne ground state atoms through collision transfer of energy from He atoms (which de-excite into ground state) Overpopulated Ne 3s and 2s states with respect to their 3p and 2p states will slowly de-excite by emission of photons The 3p and 2p states depopulate fast into the 1s state by spontaneous emission, maintaining the population inversion Slow 3s and 2s transitions can be accelerated by stimulated emission via other photons

4 Stimulated Emission Optical resonating cavity stores photons by reflection, which will interact with the optical active medium several times Photons from stimulated emission have same wavelength, polarization, and direction of emission Mirrors are 100% reflecting on one side and 99% reflecting on the other to decouple some photons from resonator for LASER beam Population inversion important to maintain amplification

5 LASER Beam Profile The intensity distribution of a LASER beam can be described in the fundamental mode (TEM 00 ) by a Gaussian profile I (r, z) = I 0 w ( 0 ) 2 2 r exp( 2 ) w (z) w 2 (z) r = radial distance from the beam s center axis z = axial distance from the beam s narrowest point (waist) w (z) = intensity dropped to 1/e 2 w 0 = w (0) = intensity dropped to 1/e 2 (waist size of beam)

6 Dispersion The index of refraction also depends of the wavelength This leads to different refraction for different colors of light having different wavelength

7 Tabulated Indexes of Refraction The speed of light in a medium is given by v = c n = λ f This leads to dispersion c = n λ f = λ 0 f with λ 0 the wavelength of light in the vacuum

8 Constructive and Destructive Interference Path difference r 2 r 1 = m λ 1 2 constructive ( m + ) λ destructive with m = 0, ±1, ±2, ±3,

9 Two-Source Interference of Light (Young s Experiment) For constructive interference we find r 2 r 1 = d sinθ = m λ

10 Two-Source Interference of Light For the distance between the central maximum and the m-th maximum (constructive interference) we find m λ d for small angles θ y m = R tanθ R sinθ = R

11 The Michelson Interferometer Assume AC is parallel to the motion of the Earth inducing an ether wind. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions. After reflection the beams recombine at A slightly out of phase and are viewed by telescope E. For constructive interference n λ = 2 (l 1 - l 2 )

12 The Analysis Time t 1 from A to C and back: t 1 = l 1 c + v + l 1 c v = 2l 1 c Time t 2 from A to D and back: 1 1 v 2 /c 2 t 2 = 2l 2 c 2 v = 2l 2 2 c 1 1 v 2 /c 2 So the difference in times is: Δt = t 2 t 1 = 2 c l 2 1 v 2 /c l v 2 /c 2 Assuming Galilean Transformation

13 Analysis (continued) Upon rotating the apparatus, the optical path lengths l 1 and l 2 are interchanged producing a different time difference between the two paths (note the change in denominators): Δ t = t 2 t1 2 = c l 2 1 v 2 /c 2 l 1 1 v 2 /c 2 Thus the time difference between rotations is given by: Δ t Δt = 2 c l 1 + l 2 1 v 2 /c l 1 + l v 2 /c 2 and using a binomial expansion, assuming v/c << 1, this reduces to Δ t Δt v 2 ( l c l 2 )

14 Results Estimating the Earth s orbital speed as: together with v = m/s l 1 l 2 = 1.2 m the time difference becomes Δt - Δt s Compare to visible light λ / c = s Although a very small number, it was within the experimental range of measurement for light waves.

15 Diffraction from a Slit

16 Fraunhofer Diffraction sinθ = m λ a y m = x m λ a m = ±1, ±2, ±3,

17 Intensity in a Single-Slit Pattern

18 Intensity Maxima Pattern E P = E 0 sin (β / 2) β / 2 Amplitude in single diffraction sin (β / 2) I = I 0 β / 2 Intensity in single diffraction 2 2 π β = λ I = I 0 a sin θ sin (π a (sin θ ) / λ ) π a (sin θ ) / λ 2

19 Multiple Slit Interference

20 Diffraction Grating Intensity maxima for multiple slits d sin θ = m λ m = 0, ±1, ±2, ±3,

21 Grating Monochromator Light entering the monochromator is reflected of the collimator C onto the diffraction grating D Multiple image of different wavelength are reflected by mirror E onto the exit slit The exit slit selects the image of the desired wavelength

22 The Grating Spectrograph In a grating spectrograph a grating is used like a prism to disperse light depending on the wavelength Spectroscopy analyzes the presence or absence of certain wavelength Energy transitions inside atoms and molecules have characteristic wavelengths Astronomy and chemistry make use of the characteristic light to identify chemical elements

23 The Grating Spectrograph

24 m λ = 4 n d Wavemeter Input LASER with unknown wavelength λ creates m fringes for the distance d moved by the retro-reflector (n index of refraction for λ in air) Reference LASER with known λ 0 Eliminating d gives λ = λ 0 m 0 n m n 0 m 0 λ 0 = 4 n 0 d Burleigh WA-1000 Wavemeter

25 Fabry-Pérot Interferometer An incident wave of wavelength λ partially reflects and transmits between two plane mirrors with the reflectance R separated by a distance l (fixed in the case of an etalon and adjustable for an interferometer) The medium between the two mirrors has a refractive index n m λ m = 2 n l cos θ Phase difference between each reflection δ = 2 π λ δ = 2 m π 2 n l cos θ

26 Fabry-Pérot Interferometer The transmittance function of the etalon is given by ( 1 R ) 2 1 T e = = 1 + R 2 2 R cos δ 1 + F sin 2 (δ/2) With the coefficient of Finesse given by F = 4 R ( 1 R ) 2

27 Fabry-Pérot Interferometer Maximum transmission T e = 1 is obtained for m λ 0 = 2 n l cos θ The Free Spectral Range Δλ is the wavelength separation between transmission peaks Δλ λ n l cos θ For the case of n = 1 and θ = 0 Δλ = 2 l or for the frequency Δ f = c 2 l

28 Fabry-Pérot Interferometer The Finesse is defined as the FSR Δλ over the FWHM δλ Δλ F = δλ π F For R > 0.5 the Finesse becomes F = 2 π R 1 R