EE485 Introduction to Photonics

Transcription

1 Pattern formed by fluorescence of quantum dots EE485 Introduction to Photonics Photon and Laser Basics 1. Photon properties 2. Laser basics 3. Characteristics of laser beams Reading: Pedrotti 3, Sec. 1.2, , 8.8,

2 Historical Landmark: Photo-electric Effect (Einstein, 195) Lih Y. Lin 2 Clean metal V Different metals, same slope Light I ν Slope = h/e ν = c/λ Current flows for λ < λ, for any intensity of light Apply stopping potential V Higher voltage required for shorter λ The slopes are independent of light intensity, but current is proportional to light intensity. hν = ev + w Energy of light = (Kinetic energy of the electron) + (Work function) E ph = hν Energy of quantum of light (Smallest energy unit): Photon

3 Lih Y. Lin 3 Photon Energy E = hν = ω h = = h / 2π 34 Joule Sec : Plank s constant λ ( µ m) = 1.24 E (ev)

4 Lih Y. Lin 4 Photon Momentum Plane wave Photon momentum Er (, t) = Aexp( ik r ω i t)ˆ e p = k p = k Radiation Pressure (assume the photons are absorbed) p = h λ Force: Pressure: = h λ p h λ h = N t t λ N: # of photons per second Force Area Exercise: An atom with mass 1-25 kg emits a photon with energy 2.48 ev. (1) What is the recoil velocity? (2) Compare this with the rootmean-square thermal velocity of the atom at T = 3K.

5 Lih Y. Lin 5 Optical Tweezers Forces arising from momentum change of the light F P t = a A 2 µm In (a-b) q p Out (c-d) p q b p P B Q 2 µm q pp Q c C Resultant gradient force D d E.g., λ = 164 nm, P = 1 mw, diameter of polystyrene sphere = 5 µm F = 3.18 x 1-12 N. Ashkin, et al., Observation of radiation pressure trapping of particles by alternating laser beams, Phys. Rev. Lett., V. 54, p , 1985

6 Lih Y. Lin 6 Photon Position Heisenberg s Uncertainty Principle ( z) ( p) 4 Photon position cannot be determined exactly. It needs to be determined with probability. Probability p( r) da I( r) da Example: (1) Photon position probability in a Gaussian beam. (2) Transmission of a single photon through a beam-splitter. (1) (2) or or

7 Lih Y. Lin 7 Photon Polarization Linearly Polarized Photons Photon polarized along x-direction Er (, t) = ( A xˆ + A yˆ)exp[ i( kz ωt)] x Er (, t) = ( A xˆ' + A yˆ')exp[ i( kz ωt)] x' y y' ( ), ' ( ) 1 1 A = A A A = A + A 2 2 x' x y y x y Example: Probability of observing a linearly polarized photon after a polarizer. 2 p( θ ) = cos θ

8 Lih Y. Lin 8 Quantum Communication Secured information transmission with single photons

9 Photon Polarization Circularly Polarized Photons Lih Y. Lin 9 ( ˆ ˆ ) Er (, t) = A e + A e exp[ i( kz ωt)] R R L L 1 1 eˆl = ( xˆ + iyˆ) eˆr = ( xˆ iyˆ) 2 2 Example: (1) A linearly-polarized photon transmitting through a circular polarizer. (2) A right-circularly-polarized photon transmitting through a linear polarizer.

10 Lih Y. Lin 1 Single Photon Interference Assume the mirrors and beam-splitters are perfectly flat and lossless. The beam-splitters are 5/5 beam-splitters. The path length difference is d. Probability of finding the photon at the detector? If we don t find the photon, where is it?

11 Lih Y. Lin 11 Photon Flux Density and Photon Flux Monochromatic light of frequency ν and intensity I(r) Photon flux density I ( r) photons ϕ( r) = hν sec area Photon flux P Φ= ϕ( r) da =, P = I( ) da : Optical power (watts) A r A hν Quasi-monochromatic light of central-frequency Mean photon flux density Mean photon flux I ( r) ϕ( r) = hν P Φ= hν ν

12 Lih Y. Lin 12 Laser Early History Laser: Light Amplification by the Stimulated Emission of Radiation Theoretical background: Stimulated emission, Einstein, First Maser ( M Microwave): Charles H. Townes and coworkers, First laser (Ruby laser): Theodore H. Maiman, 196. Theodore H. Maiman and the first ruby laser.

13 Lih Y. Lin 13 Einstein s Theory of Light-Matter Interaction RSt. Abs. = B12N1 ρν ( ) ρν ( ) : Spectral energy density R = Sp. Em (), 2 2 assume only spontaneous emission τ 1/A : Spontaneous radiative lifetime 21 A N N t = N e R A t Output photon: Same energy, direction, polarization and phase as the input photon. A B B.. = B21N2 ρν ( ) 3 8πhv = 3 c = B St Em

14 Lih Y. Lin 14 Essential Elements of a Laser Structure of a ruby laser Pump: Creates population inversion Laser medium: Provides amplifying medium Resonator: Provide optical feedback and determine lasing wavelength

15 Laser Operation Lasing process in a ruby laser Process: 1. Population inversion Through pumping and the existence of a metastable state. 2. Seed photons From spontaneous emission and initiate the stimulated emission process. 3. Optical cavity Resonant enhancement and define the output wavelength. 4. Gain saturation Population inversion decreases as stimulated emission increases. Steady state. 5. Output coupling Let some of the photons out at each round trip. Lih Y. Lin 15

16 Lih Y. Lin 16 Fabry-Perot Cavity and Laser Resonators Laser resonator Fabry-Perot cavity = Laser amplifier +

17 Lih Y. Lin 17 Mode Suppression with a Fabry-Perot Etalon Exercise: A He-Ne laser has a cavity length of 6 cm. The gain medium is capable of supporting the laser light of wavelengths in the range from λ 1 = nm to λ 2 = nm. (a) Determine the gain bandwidth in frequency. (b) How many longitudinal modes can exist in the laser cavity? (c) Find the maximum length d of an etalon that could be used to limit this laser to single-mode operation.

18 Lih Y. Lin 18 Monochromaticity Fluorescence (spontaneous emission) Lasing Finite bandwidth at each energy level. Lorentzian lineshape: v gv () = 2 π [( v v ) + ( v / 2) ] 2 2 Linewidth narrowed by stimulated emission and laser cavity

19 Directionality Divergence of a laser beam Fraunhofer diffraction of a plane wave through a circular aperture Quiz: What makes the difference? (More on Gaussian beams later.) Lih Y. Lin 19

20 Lih Y. Lin 2 Intensity of Light Sources A 5-mW green laser of 532 nm wavelength emits from a.5 mm diameter aperture. Determine its intensity at 1 meter distance. How does this translate to number of photons per unit area per second? (a) Find out the divergent angle. (b) Determine the beam spot size at 1 meter distance. (c) Calculate the intensity. (d) Translate this into number of photons. A 1-W light bulb has an average wavelength of 5 nm. Consider this as a point source, determine its intensity at 1 meter distance. How does this translate to number of photons per unit area per second? (a) Find out the surface area at 1 meter distance. (b) Calculate the intensity. (c) Translate this into number of photons.

21 Lih Y. Lin 21 Focusability of Light Ideal situation Focusing an incoherent light Focusing a laser light λ φ ~ 1.27 d Reversibility λ d 1.27 λ, approximately φ Diffraction limit

22 Lih Y. Lin 22 Prelude to Gaussian Beam Optics Review: 3-D wave equation n E E = c 2 t 2 Er (,) t = E Plane wave solution i( kr ωt) Spherical wave solution E Er (,,, ) ( θφ θφ t =, ) e r e i( kr ωt) Paraxial approximation at far field: At any point (x, y) on the transverse plane z = R, 1/ x + y x + y r = R 1+ R+ 2 R 2R ikr Exy (, ) = (constant) e z= R (constant) e 2 2 ikr i( k / 2)[( x + y ) / R] e

23 Lih Y. Lin 23 Laser-Beam Mode Structures Measurement of a fundamental-mode laser beam: Intensity 3-D beam profile Contour plot of intensity distribution Gaussian distribution: I = / w Ie ρ

24 Lih Y. Lin 24 Towards Gaussian Beam Solution Substitute into the wave equation, ( ω ) Ert (,) = U(, xyze, ) i kz t 2 2 U U U + + 2ik = 2 2 x y z Recall a far-field spherical wave under paraxial approximation, 2 2 Exy (, ) (constant) e ikr e i k x y R z= R and take an educated guess: U ( xyz,, ) = Ee ( / 2)[( + ) / ] 2 2 ipz { ( ) + [ kx ( + y) / 2 qz ( )]} p i q =, = 1 z q z Consider the Gaussian field distribution as well, Guess a complex radius of curvature q(z), 1 1 λ = + i qz ( ) Rz ( ) π wz ( ) 2 E= Ee ( x y )/ w

25 Lih Y. Lin 25 TEM, Gaussian Beam Solution 2 w E ρ = E exp wz 2 ( ) w ( z ) Amplitude factor (Gaussian) 2 ρ exp ik 2 Rz ( ) Radial phase exp i kz 1 ω tan z z t Longitudinal phase 1/e points of the field

26 Radius of Curvature, Spot Size, and Intensity Distribution Define z = as where the Gaussian beam wavefront is planar. qz ( ) = q() + z R( z) = z 1 + z z z wz ( ) = w 1+ z 2 πw q( z) = z i z iz λ 2 πw z : Rayleigh range λ z = is where the beam waist is. wz ( ) = 2w 2 2 w 2ρ 2 I( ρ, z) = I exp wz ( ) w ( z) 2 2 z = z = z z = 2z θ Lih Y. Lin 26

27 Lih Y. Lin 27 Far Field and Divergence Angle Far field region, z z Rz ( ) wz ( ) = z λ = π w z Divergence angle θ FF λ = π w

28 Lih Y. Lin 28 Exercise: Gaussian Beam in a Laser Cavity He-Ne laser, λ = nm Determine: (a) Beam radius w at the beam waist. (b) Rayleigh range z (c) Beam radius w at the rear laser mirror. (d) Divergence angle.

29 Lih Y. Lin 29 Correspondence between R(z) and q(z) Extend the correspondence to simple lens law: Ray-transfer matrix for laser propagation through arbitrary optical systems

30 Lih Y. Lin 3 ABCD Law for Gaussian Beam Propagation y R = α Example 27-2: Determine the waist radius and its location for the He-Ne laser system depicted below. (a) Determine q 1 at the flat mirror. (b) Determine the ABCD matrix of the optical system. (c) Relate q 2 at the beam waist to q 1. y2 A B y1 = 2 C D α α1 AR1 + B R2 = CR + D q 2 = Aq Cq B + D

31 Ray-transfer Matrix of Basic Optical Elements (I) Lih Y. Lin 31

32 Ray-transfer Matrix of Basic Optical Elements (II) Lih Y. Lin 32

33 Lih Y. Lin 33 Matrices of Cascaded Optical Components n n f y θ M 1 y y 1 = M θ 1 1 θ M 2 y2 y1 = M2 θ2 θ1 M 3 M N y θ N N y y A B N = M M= MN MM 2 1 = N C D θ θ Works particularly well with computer programming n A useful tip: o Det M = AD BC = n f

34 Lih Y. Lin 34 Gaussian Beam Through a Thin Lens z ' zz ' 1 1 z+ z' A B 1 z' 1 z f f 1 C D = 1 = z f 1 f f z ' zz ' 1 q+ z+ z' f f q ' = 1 z q + 1 f f Conditions: 2 ' 2 π W ( W ) ( ) q = i π, q' = i λ λ Equate real parts and imaginary parts.

35 Focusing a Gaussian Beam z' = f + M ( z f) 2 For a plane wave, z' = f For a Gaussian beam, z' f Beam waist at the lens z = W ' = W 1+ z f f z' = f 1+ z 2 2 If z Assume W is approximately equal to the radius of the lens, W ' f λ W f f ' = = θ πw z' = f 2 fλ 2 λ( f #) = = π(2 W ) π f f # : f-number of the lens D Lih Y. Lin 35