A theorist’s view on losses Javier Redondo Universidad de Zaragoza (Spain) Max Planck Institute für Physik
Concept of dielectric haloscope - monochromatic axion-induced EM radiation ω ' m a - large Area A � 1 / ω 2 - constructive interference - cavity effects ✓ B " # ◆ 2 2 . 2 × 10 − 27 W C 2 × A × β 2 P ∼ m 2 a γ 10T - m a ∼ 25 − 200 µ eV - A ∼ 1m 2 - with disks, # channels δω β ∝ N/ β 2 β 2 ∼ 10 4 O (80)
Problems (small selection) - diffraction losses - how do we calibrate the boost factor? our idea is correlate it with R/phase delay - losses are different for axion DM induced and externally induced modes! - difficult to excite cavity exactly as axionDM!
Diffraction - “Easy” exercise: 2D emission of a finite mirror Z Z dq d 2 k E q e iqy e i √ E k e ik y y e ik x x = e − i ω t (2 π ) 2 e e ω 2 − q 2 x E ( x, y ) ∼ e − i ω t 2 π y x boundary conditions cancel the axion induced E-field at the mirror E (0 , y ) = − E a E q = sin( qL/ 2) e q but not outside (no radiation from -inf) E (0 , y ) = 0 - neglecting polarisation, but easy to take to cylindrical 3D - only radiation field, omega=k shell
Diffraction - in dimensionless variables .... Z dqL sin( qL/ 2) q 1 − q 2 e iqL y L e i ω 2 ω x E ( x, y ) ∼ e − i ω t 2 π qL y x Modes excited are those with qL ∼ 1 y/L ∼ 1 We are interested in values of q 1 − q 2 Here, diffraction as we increase x comes from the term e i ω 2 ω x qL L typically we have and we hope to have ω L ⌧ 1 ω x ∼ π ✓ 2 ◆ ω L = (100 µ eV) ∼ 500 √ π m ω L = (50 µ eV) (0 . 2 m ) ∼ 25 q ω 2 ω x ∼ e i ω x e − i qL 2 1 − q 2 e i 2 ω L 2 ω x simplest approximation x note that phase correction is ∝ ( ω L ) 2 δ v ∼ 1 / ( ω L ) much larger than velocity effects (yet similar to understand)
Diffraction - in dimensionless variables .... Z dqL sin( qL/ 2) q 1 − q 2 e iqL y L e i ω 2 ω x E ( x, y ) ∼ e − i ω t 2 π qL y x 2 L y [1 / ω ]
Diffraction - in dimensionless variables .... Z dqL sin( qL/ 2) q 1 − q 2 e iqL y L e i ω 2 ω x E ( x, y ) ∼ e − i ω t 2 π qL y 1.2 x 1.0 ω L = 500 0.8 ω x = π ω x = 10 π 2 L 0.6 ω x = 100 π 0.4 0.2 50 100 150 200 250 y [1 / ω ] Border effects always large, but for wL = 500 it is a small effect Even at 100 halfwavelengths, the field is 10% coherent,
Diffraction - Not yet quite final, need to include short distance effects, reflections, etc - disk of diameter D, e E q ∼ sin( qD/ 2) /q - emission characterised by a transverse momentum distribution and correlation δ v ∼ 1 / ω D - but are these modes affected by propagation through further finite-size disks? matching at each boundary (y-dependent) some ideas difficult to solve self consistently the basic idea is that every disk implies one more convolution so naively .... e E out ∼ sin( qD/ 2 N ) /q and most likely √ e E out ∼ sin( qD/ 2 N ) /q
what about the calibration ? - It is quite a different calculation - probably makes sense (Olaf?) a Gaussian beam calculation - Gaussian beam optics available s 1 + x 2 y 2 - Gaussian beams E ∼ H ( y/w ( x )) e − i ω 2 q ( x ) e i ψ ( x ) q = x + ix R w = w 0 x 2 R - ABCD transfer formalism “works” x R = ω w 2 0 / 2
Gaussian beams - It is quite a different calculation in the case for Gauss beams, reflexion is equivalent to a shift of the beam waist - plane parallel resonators are unstable (this applies to aDM boosting)
Gaussian beams plane wave multireflection solution ∞ 1 X ( r 2 e i 2 ω L ) b ! t E γ E ref ' t E γ 1 � r 2 e i 2 ω L b =0 in the case for Gauss beams, reflexion is equivalent to a shift of the beam waist ◆ 1 / 2 ✓ ✓ ◆ w 0 y y 2 X e − i ω ( r 2 e i ω L ) b 2 q ( x +2 bL ) e i ψ ( x +2 bL ) E ref ∼ H w ( x + 2 bL ) w ( x + 2 bL ) b unfortunately this does not factorise - easy to do numerics (?) - possible interpretation of Olaf’s results: ω D ∼ 25 , x R ∼ O ( ω DD/ 2) ∼ 1 m - Guoy phase -> shift of peaks - beam clipping ... - pure diffraction
Conclusions - Difraction is small if wL is large, effects ~ (wL)^2 - Some ideas how to solve the booster equations by matching multimodes across boundaries - Gaussian beam analysis can help understanding calibration and Olaf’s 20 cm results - preliminary estimates are compatible with O( few %) losses / disk not good reason to be worried, plenty of reasons/ideas to sit down and compute !
Recommend
More recommend
