Optical Microcavities Vahala, Nature 424, 839 (2003) Microcavity characteristics: Quality factor Q, mode volume V
Simplest cavity: Fabry-Perot etalon Transmission peaks: constructive interference between multiple reflections between the two reflecting surfaces (wavelength fits an integer number of times in cavity). Next few slides: definition and interpretation of free spectral range , quality factor Q, and finesse F.
Free spectral range Free spectral range (FSR) is frequency (or wavelength) spacing between adjacent resonances. n T d R 1 2 The smaller d, the larger the free spectral range !! m: integer; n: refractive index depends on cavity length: Eq. 1: Eq. 2:
Consider traveling wave in the cavity: Look at phase front that is at x = 0 at t = 0 : k 0 x 0 t = 0 The time t to travel a distance x is: The time t RT to make 1 round trip 2d is then: Free spectral range (divided by ) is a measure for the optical cycle time compared to the round trip time Optical cycle time Free-space wavelength Interpretation of free spectral range in the time domain:
Quality factor Q 1 1/e 2/ Consider the ‘ring-down’ of a microcavity: Optical period T = 1/f 0 = 2 / 0 0 E =Electric field at a certain position u =Energy density 1. Definition of Q via energy storage: Energy density decay:
1 1/e 2/ Time domainFrequency domain Fourier The two definitions for Q are equivalent ! Lorentzian 2. Definition of Q via resonance bandwidth:
Finesse F 1 2 F Definition of F via resonance bandwidth: F This can be rewritten as: F is similar to Q except that optical cycle time T is replaced by round trip time t RT See slide on FSR See slide on Q
Quality factor vs. Finesse Quality factor: number of optical cycles (times 2 ) before stored energy decays to 1/e of original value. Finesse: number of round trips (times 2 ) before stored energy decays to 1/e of original value. Suppose mirror losses dominate cavity losses, then: Q can be increased by increasing cavity length F is independent of cavity length !! This shows that Q and F are different figures of merit for the light circulation capabilities of a microcavity
On threshold: P in = 16 W. If all light is coupled into the cavity, then in steady state: 1.Ultra-high F leads to an extremely high circulating power relative to the input power ! APL 84, 1037 (2004) with D = 40 m Q = 4 10 7 Application: Low-threshold lasing P in = 16 W P circ = 800 mW !!!
Application: Low-threshold lasing 2.A small mode volume V mode leads to strong confinement of the circulating power, and thus to a high circulating intensity: The light circulation concept is not only useful for lasing, but also for: Nonlinear optics (e.g. Raman scattering) Purcell effect Strong coupling between light and matter … See also: Vahala, Nature 424, 839 (2003), and www.vahala.caltech.edu
Differences between microcavities Practical differences are related to: Ease of fabrication Connectivity to waveguides Integration in larger circuits Principle differences are related to the figure of merits: Free spectral range (= spectral mode separation) Quality factor (= temporal time) Mode volume (= spatial confinement) One example: the cavity build-up factor See next slide…
Differences between cavities Q/V = 10 2 Q/V = 10 3 Q/V = 10 4 Q/V = 10 6 Q/V = 10 5 Q/V in units ( /n) 3 Highest Q/V: geometries useful for fundamental research on QED (Kimble, Caltech) but not practical for devices Vahala, Nature 424, 839 (2003)
Critical coupling For derivation, see: Kippenberg, Ph.D. Thesis, section 3.3.2 (http://www.mpq.mpg.de/~tkippenb/TJKippenbergThesis.pdf) Decay rates (s -1 ): 1/ ex : coupling to waveguide 1/ 0 : internal losses If = 0 and ex = 0, then T = 0 !! If the intrinsic damping rate equals the coupling rate, then 100 % of the incoming light is transferred into the cavity (perfect destructive interference at output waveguide) ex 00
Sensing example: D 2 O detection Subtle difference in optical absorptions between D 2 O and H 2 O is magnified due to light circulation in cavity. Sensitivity: 1 part per million !!! Armani and Vahala, Opt. Lett. 31, 1896 (2006) Evanescent waves are essential for both sensing and fiber coupling
Summary Microcavities: Confinement of light to small volumes by resonant recirculation. Applications: lasing, nonlinear optics, QED, sensing, etc. FSR, Q, V mode, and F characterize different aspects of the light recirculation capabilities of a microcavity. Different microcavity realizations (e.g. micropost, microsphere) differ in FSR, Q, V mode, and F.