A theorists view on losses Javier Redondo Universidad de Zaragoza - - PowerPoint PPT Presentation

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A theorists view on losses Javier Redondo Universidad de Zaragoza - - PowerPoint PPT Presentation

Jan 12, 2023 267 likes •394 views

A theorists view on losses Javier Redondo Universidad de Zaragoza (Spain) Max Planck Institute fr Physik Concept of dielectric haloscope - monochromatic axion-induced EM radiation ' m a - large Area A 1 / 2 - constructive

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A theorist’s view on losses

Javier Redondo Universidad de Zaragoza (Spain) Max Planck Institute für Physik

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SLIDE 2

Concept of dielectric haloscope

  • monochromatic axion-induced EM radiation
  • large Area
  • constructive interference
  • cavity effects

P ∼ " 2.2 × 10−27 W m2 ✓ B 10T ◆2 C2

aγ

# × A × β2 ω ' ma A 1/ω2

  • with disks, # channels

ma ∼ 25 − 200 µeV A ∼ 1m2 β2 ∼ 104 O(80) δωβ ∝ N/β2

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Problems (small selection)

  • how do we calibrate the boost factor?
  • ur idea is correlate it with R/phase delay
  • diffraction losses
  • losses are different for axion DM induced

and externally induced modes!

  • difficult to excite cavity exactly as axionDM!
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SLIDE 4

Diffraction

  • “Easy” exercise: 2D emission of a finite mirror

E(0, y) = −Ea E(0, y) = 0

  • neglecting polarisation, but easy to take to cylindrical 3D
  • only radiation field, omega=k shell

e Eq = sin(qL/2) q E(x, y) ∼ e−iωt Z d2k (2π)2 e Ekeikyyeikxx = e−iωt Z dq 2π e Eqeiqyei√

ω2−q2x

boundary conditions cancel the axion induced E-field at the mirror but not outside (no radiation from -inf)

x y

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SLIDE 5

Diffraction

  • in dimensionless variables ....

x y qL ∼ 1 y/L ∼ 1 E(x, y) ∼ e−iωt Z dqL 2π sin(qL/2) qL eiqL y

L ei

q 1− q2

ω2 ωx

ωx ∼ π qL ωL ⌧ 1 ei

q 1− q2

ω2 ωx ∼ eiωxe−i qL2 2ωL2 ωx

simplest approximation Modes excited are those with We are interested in values of Here, diffraction as we increase x comes from the term ei

q 1− q2

ω2 ωx

typically we have and we hope to have

L ωL = (50µeV) (0.2m) ∼ 25 ωL = (100µeV) ✓ 2 √π m ◆ ∼ 500

note that phase correction is ∝

x (ωL)2

much larger than velocity effects (yet similar to understand)

δv ∼ 1/(ωL)

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SLIDE 6
  • in dimensionless variables ....

x y 2L E(x, y) ∼ e−iωt Z dqL 2π sin(qL/2) qL eiqL y

L ei

q 1− q2

ω2 ωx

y[1/ω]

Diffraction

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SLIDE 7
  • in dimensionless variables ....

x y 2L E(x, y) ∼ e−iωt Z dqL 2π sin(qL/2) qL eiqL y

L ei

q 1− q2

ω2 ωx

ωL = 500 y[1/ω]

50 100 150 200 250 0.2 0.4 0.6 0.8 1.0 1.2

ωx = 10π ωx = π ωx = 100π

Border effects always large, but for wL = 500 it is a small effect Even at 100 halfwavelengths, the field is 10% coherent,

Diffraction

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SLIDE 8
  • Not yet quite final, need to include short distance effects, reflections, etc
  • disk of diameter D,
  • emission characterised by a transverse momentum distribution and correlation
  • but are these modes affected by propagation through further finite-size disks?

Diffraction

e Eq ∼ sin(qD/2)/q δv ∼ 1/ωD

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 Eout ∼ sin(qD/2N)/q

and most likely

e Eout ∼ sin(qD/2 √ N)/q

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SLIDE 9

what about the calibration ?

  • It is quite a different calculation
  • probably makes sense (Olaf?) a Gaussian beam calculation
  • Gaussian beam optics available
  • Gaussian beams
  • ABCD transfer formalism “works”

w = w0 s 1 + x2 x2

R

q = x + ixR E ∼ H(y/w(x))e−iω

y2 2q(x) eiψ(x)

xR = ωw2

0/2

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SLIDE 10

Gaussian beams

  • It is quite a different calculation
  • plane parallel resonators are unstable (this applies to aDM boosting)

in the case for Gauss beams, reflexion is equivalent to a shift of the beam waist

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SLIDE 11

Eref ' t Eγ

∞

X

b=0

(r2ei2ωL)b ! t Eγ 1 1 r2ei2ωL

plane wave multireflection solution in the case for Gauss beams, reflexion is equivalent to a shift of the beam waist unfortunately this does not factorise

Eref ∼ X

b

(r2eiωL)b ✓ w0 w(x + 2bL) ◆1/2 H ✓ y w(x + 2bL) ◆ e−iω

y2 2q(x+2bL) eiψ(x+2bL)

  • easy to do numerics (?)
  • possible interpretation of Olaf’s results:
  • Guoy phase -> shift of peaks
  • beam clipping ...
  • pure diffraction

ωD ∼ 25, xR ∼ O(ωDD/2) ∼ 1m

Gaussian beams

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SLIDE 12

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 !