3.1 The Plane Mirror Resonator 3.2 The Spherical Mirror Resonator 3.3 Gaussian modes and resonance frequencies 3.4 The Unstable Resonator

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1 Quantum Electronics Laser Physics Chapter 3 The Optical Resonator 3.1 The Plane Mirror Resonator 3. The Spherical Mirror Resonator 3.3 Gaussian modes and resonance frequencies 3.4 The Unstable Resonator 1

2 Introduction The optical resonator (OR) is the optical counterpart of an electronic resonant circuit: it confines and stores light at certain frequencies. Most important application: OR as a container within which laser light is generated. LASER=OR containing a light amplifying medium OR determines frequency and spatial distribution of the laser beam

3 Introduction From Fundamentals of Photonics, Saleh and Teich, Wiley, chap 10, p.366 3

4 Mirror resonators: or 3 mirrors, D or 3D cavities Dielectric Resonators: use TIR instead of mirrors: Fiber rings and integrated optic rings Microdisks, microspheres, etc (Whispering Gallery modes) Micropillars Introduction Photonic Crystals Currently: Nanolasers with quantum confinement of carriers (e.g electrons) or photons 4

5 Introduction Two key parameters: Modal volume V: volume occupied by confined optical mode Quality factor Q: proportional to storage time in units of optical period V and Q represent the degrees of spatial and temporal confinements, respectively Large Q means low-loss resonator 5

6 3.1- Plane Mirror Resonator Fabry-Perot interferometer: pair of plane mirrors separated by distance d. z z = 0 Monochromatic plane: u(! r ) = Re U(! r )e πiνt Satisfies Helmholtz equation: [ ] z = d U + k U = 0 with k = π ν c 6

7 3.1- Plane Mirror Resonator Standing wave solution is obtained for the boundary conditions: U r! r ( ) = 0 at z = 0 and U (! ) = 0 at z = d U! r ( ) = A sin kz with kd = qπ k = q π d, q = 1,,3,... q is the mode number, mode frequencies are: ν q = q c d Arbitrary wave = superposition of modes U( r) = A q sin k q z q Constant frequency difference between adjacent modes (free spectral range): ν F = c d 7

8 3.1- Plane Mirror Resonator The resonance wavelengths in the optical medium are: λ q = c d = qλ q ν Examples: q d = 30 cm, n =1 (air), free spectral range = 500 MHz d = 3 microns, n = 1(air), 50 THz (7 modes in visible range: q = 8,...,14, λ q = 750,..., 49 nm) Free spectral range can be adjusted by placing resonators in series 8

9 3.1- Plane Mirror Resonator Calculation of light intensity in the resonator: Summation of multiply reflected amplitudes Phase shift after one round trip of propagation (d) is d = kd ϕ = π λ Wave reproduces itself after a round trip, thus: π λ d = kd = qπ, q = 1,,3... 9

10 3.1- Plane Mirror Resonator U U 1 U 0 Calculation of light intensity in the resonator: z = 0 Summation of multiply reflected amplitudes for perfect non lossy resonator: r r r z = d z U 1 =U 0 e i π λ d,u =U 1 e i π λ d =U 0 e i π λ 4d,etc... Total amplitude: U =U 0 +U 1 +U

11 3.1-Plane Mirror Resonator If resonator has losses, amplitude reduction upon reflection is taken into account (r reflection coefficient): Total amplitude :U =U 0 + ru 1 + r U +... r = complex reflection coefficient (overall amplitude attenuation) Intensity : I = U I 0 I = # 1+ F & % ( $ π ' I 0 = U 0 # ϕ & sin % ( $ ' = incident intensity = transmitted intensity R = r, reflectivity of lossy mirror (or overall losses over round trip) F = π R 1 R Intermode spacing, Finesse of resonator 3.Optical = Resonator Width of a mode = ν F δν 11

12 y = 3.1- Plane Mirror Resonator sin ( 100πx) Spectral Response of the lossy resonator: I I 0 = " 1 + F # $ π 1 % & ' sin " π ν % # $ ν F & ' ν F = c d, I 0 Incident intensity 1 = round-trip time ν F y = sin ( 100πx) Spectral width of a mode ν F F Spectral width depends strongly on finesse (F). Cf. F=100 vs 10 1

13 3.1- Plane Mirror Resonator The two principal sources of loss in the optical resonator are Absorption and scattering in the medium between the mirror (see laser amplifier): Round trip power attenuation: exp α S d Losses arising from imperfect reflection at the mirrors (necessary transmission + finite size effects): Overall distributed-loss written as: exp( α r d) = R 1 R exp( α S d) ( ) α S : linear absorption coefficient of the medium Mirrors of reflectance : R 1 and R Overall round trip loss of intensity : R 1 R exp( α S d) r α r = α S + 1 d Log 1 R 1 R Ultimately (after maths): F π α r d if α r d << 1 (small loses) 13

14 3.1 Plane Mirror Resonator The resonance linewidth is inversely proportional to the loss factor (α r d) Finesse is by definition F = ν F δν δν c d π α r d = cα r π α r is the loss per unit length, cα r is the loss per unit time The resonator lifetime or photon lifetime in cavity is: τ p = 1 cα r, thus δν = 1 πτ p 14

15 3.1 Plane Mirror Resonator The Quality factor Q can be used to characterize the losses: Stored energy Q = π Energy loss per cycle In the case of an optical resonator (laser), one can show that : Q = πν 0 τ p Q = ν 0 F, ν 0 = frequency of one of the modes ν F Since the resonator frequencies are much larger than the mode spacing, then Q >> F 15

16 3.1-Plane Mirror Resonator What are the requirements for a laser: Assume 3-D resonator (3 pairs of parallel mirrors, closed resonator), equivalent to black-body cavity. Number and frequency of modes is given by the particle in the box model (photons): dn V = 8πν dν c 3 V = 1cm 3,ν = Hz, dν = Hz dn = 10 9 modes 16

17 3.1-Plane Mirror Resonator All the modes would have comparable Q in the 3D resonator To be avoided in a laser as it would cause all the atoms to emit power into a large number of modes (would differ in their frequency and spatial characteristics) Large, open resonators consisting of opposite flat/curved reflectors must be used: Energy of the vast majority of modes lost after a single pass Surviving modes are near the axis 17

18 3. Spherical Mirror Resonator Ray confinement: Concave R<0, Convex R>0, Only meridional (lie in a plane passing through the optical axis) and paraxial rays are considered Geometric optics is sufficient to find the condition for the existence of the confined modes R 1 d R g 1 = 1+ d R 1, g = 1+ d R P Chapter z

19 3. Spherical Mirror Resonator Condition for the existence of confined modes: Outside this domain the resonator is said to be unstable 0 g 1 g 1 For same radii, stability condition becomes: R 1 = R g 1 = g = g 1 g +1 0 Three resonators of practical interest: confocal concentric, confocal/planar d ( R) 19

20 3.-Spherical Mirror Resonator Stability condition: R 1 = R g 1 = g = g 1 g +1 0 d ( R) 0

21 3.3 Gaussian Modes and resonance frequencies Gaussian beams are stable modes of the spherical mirror resonator: wavefronts and phase match exactly the boundary conditions imposed by spherical mirror resonator (Helmholtz paraxial equation). Gaussian beam retraces incident beam if the radius of the wavefronts is exactly the same as the mirror radius. Phase of Gaussian beam: ( ) = kz ζ ( z) + kρ R( z) with ρ = x + y On-axis: ϕ ( 0, z) = kz ζ ( z) ϕ R, z ( ) = arctan z & ' z 0 ζ z % ( ) * : phase retardation with respect to plane wave 1

22 3.3 Gaussian Modes and resonance frequencies z 0 = d # % R $ d 1 & ( ' 1 W 0 = λd # % π R $ d 1 & ( ' λd W 1 = W = π 1 # & % d $ R ' ( + # % d &. 4 - $ R ( 05 3, '/ See chapter I0 for details (symmetrical resonator): R 1 = R = R z 1 = d /, z = d / ) # R( z ) = z 1+ z & 0 ) % ( * $ z ' - # W( z ) = W 0 ) 1+ % z *) $ z 0 + +, & ( '. +, + 1

23 3.3 Gaussian Modes and resonance frequencies 3

24 3.3 Gaussian Modes and resonance frequencies z 0 = d # % R $ d 1 & ( ' 1 W 0 = λd # % π R $ d 1 & ( ' λd W 1 = W = π 1 # & % d $ R ' ( + # % d &. 4 - $ R ( 05 3, '/ See chapter I0 for details (symmetrical resonator): R 1 = R = R z 1 = d /, z = d / ) # R( z ) = z 1+ z & 0 ) % ( * $ z ' - # W( z ) = W 0 ) 1+ % z *) 4$ z 0 + +, & ( '. +, + 1

25 3.3 Gaussian modes and resonance frequencies Resonance frequencies can be calculated from the resonance condition (round trip phase change is exactly π): At mirrors location: ( ) = kz 1 ζ ( z 1 ) and ϕ( 0,z ) = kz ζ ( z ) ϕ 0,z 1 Phase change from z1 to z: Δϕ =ϕ( 0,z ) ϕ( 0,z 1 ) = k( z z 1 ) ζ ( z ) ζ z 1 [ ( )] = kd Δζ For one round trip + phase matching condition: ( ) = qπ ( q = ±1,±,... ) Δϕ = kd Δζ ν q = qν F + Δζ π ν, frequency spacing: & ν = c ) F ( F + ' d * 5

26 3.3 Gaussian modes and resonance frequencies All the Hermite-Gaussian beams of order (l,m) are also good solutions. All (l,m) modes have same wavefronts as (0,0) but different amplitudes: Conditions for wavefront matching are identical. The entire family of A l,m G l G m are also modes of the spherical mirror resonator The resonance frequencies depend on (l,m) 6

27 3.3 Gaussian modes and resonance frequencies Phase matching conditions provide resonance frequencies: Phase of the axial modes: ϕ ( 0, z) = kz ( l + m + 1)ζ ( z) After a round trip + phase matching condition ( ) Δζ = qπ ( q = ±1, ±,... ) kd l + m + 1 Resonance frequencies: ν q = qν F + l + m + 1 ( ) Δζ π ν F Modes of different q but same (l,m) are called longitudinal (axial) modes Modes with different (l,m) represent different transverse modes 7

28 3.4 Unstable Resonator Close to regions of unconfinement, beam size increases Light losses due to missing the mirror become important (diffraction losses). For high power applications, large volume modes and diffraction losses are desirable High diffraction losses are good for a high gain situation (see later). Output beam has large aperture: optics are simplified Losses depend only on mirrors radii of curvature and separation distance. 8

29 3.4 Unstable Resonator Spherical wave picture of the mode in an unstable resonator. Points P 1 and P are the virtual centres of the spherical waves. 9

S. Blair September 27,

S. Blair September 27, S. Blair September 7, 010 54 4.3. Optical Resonators With Spherical Mirrors Laser resonators have the same characteristics as Fabry-Perot etalons. A laser resonator supports longitudinal modes of a discrete