124
R. Röhlsberger and J. Evers
below the critical angle φ c =
√
2δ. Since δ = 10
−6
. . . 10
−5 for hard X-rays with
energies between about 10 and 20 keV, the critical angle ϕ C is typically a few mrad. In
the regime of total reflection, the radiation penetrates only a few nm into the material
via the evanescent wave. In the example shown in Fig. 3.6, the top Pt layer is thin
enough (2.2 nm) so that x-rays impinging under grazing angles can evanescently
couple into the cavity.
Constructive superposition of the partial waves inside the cavity occurs at certain
angles when the thickness of the guiding layer equals an integer multiple of the
standing wave period that is given by (λ/2)/
ϕ 2 − ϕ
2
C , where ϕ C is the critical angle
of total reflection of the guiding layer material. This leads to a strong amplification of
the local photonic density of states, limited only by the photoabsorption in the guiding
layer material. In the first-order mode excited at about ϕ = 2.5 mrad, illustrated in
Fig. 3.6, one obtains a 25-fold enhancement of the normalized intensity in the center
of the cavity.
In the following we calculate the spectral response of this system around the
nuclear resonance energy to determine the collective decay width and the collective
Lamb shift of the nuclei in the cavity. This can be accomplished via a perturbation
expansion of the resonant reflectivity R of the cavity in powers of the nuclear scattering amplitude f n at the angular position ϕ = ϕ 1 of the first-order mode [27]. Each
order of the perturbation series of R corresponds to one of the outgoing partial waves
A i that are emitted from the nuclear ensemble at the ‘vertices’ (denoted by the black
dots) in the diagram. In order to sum up all the partial waves A i , we note that the
scattered amplitude in the nth outgoing wave is related to the (n − 1)th amplitude
via
A n = (i d f n ) p q A n−1
(3.36)
Here d is the thickness of the
57 Fe layer and p and q are the amplitudes of the
wavefields (at the position of the resonant nuclei) propagating in the directions of the
incident and the reflected beams, respectively. The depth dependence of the relevant
product p q for the first-order mode of the cavity used here is shown in Fig. 3.6b.
For the first vertex we have A 1 = (i d f n ) p
2 A 0 that also includes the coupling of
the radiation into the cavity. Finally, the sum over all orders results in
R = i d f n p
2
∞
k=0
(i d p q f n )
k
=
i dp
2 f n
1 − i d p q f n
.
(3.37)
Inserting f n (ω) as defined in Eq. (3.31) we obtain a spectral response that is again a
Lorentzian resonance line
R(ω) =
C d p
2
(( 0 /2)
ω − ω 0 + L C + i(( 0 + C )/2
with C =
2πρ c n
k 0 k 0z
(3.38)
that exhibits a decay width of C = C d |Re( p q)| 0 =: χ χ 0 and an energy shift
of L C = −C d Im( p q) ) 0 /2. Combining these results into one expression for the
complex-valued frequency shift C , we obtain
R. Röhlsberger and J. Evers
below the critical angle φ c =
√
2δ. Since δ = 10
−6
. . . 10
−5 for hard X-rays with
energies between about 10 and 20 keV, the critical angle ϕ C is typically a few mrad. In
the regime of total reflection, the radiation penetrates only a few nm into the material
via the evanescent wave. In the example shown in Fig. 3.6, the top Pt layer is thin
enough (2.2 nm) so that x-rays impinging under grazing angles can evanescently
couple into the cavity.
Constructive superposition of the partial waves inside the cavity occurs at certain
angles when the thickness of the guiding layer equals an integer multiple of the
standing wave period that is given by (λ/2)/
ϕ 2 − ϕ
2
C , where ϕ C is the critical angle
of total reflection of the guiding layer material. This leads to a strong amplification of
the local photonic density of states, limited only by the photoabsorption in the guiding
layer material. In the first-order mode excited at about ϕ = 2.5 mrad, illustrated in
Fig. 3.6, one obtains a 25-fold enhancement of the normalized intensity in the center
of the cavity.
In the following we calculate the spectral response of this system around the
nuclear resonance energy to determine the collective decay width and the collective
Lamb shift of the nuclei in the cavity. This can be accomplished via a perturbation
expansion of the resonant reflectivity R of the cavity in powers of the nuclear scattering amplitude f n at the angular position ϕ = ϕ 1 of the first-order mode [27]. Each
order of the perturbation series of R corresponds to one of the outgoing partial waves
A i that are emitted from the nuclear ensemble at the ‘vertices’ (denoted by the black
dots) in the diagram. In order to sum up all the partial waves A i , we note that the
scattered amplitude in the nth outgoing wave is related to the (n − 1)th amplitude
via
A n = (i d f n ) p q A n−1
(3.36)
Here d is the thickness of the
57 Fe layer and p and q are the amplitudes of the
wavefields (at the position of the resonant nuclei) propagating in the directions of the
incident and the reflected beams, respectively. The depth dependence of the relevant
product p q for the first-order mode of the cavity used here is shown in Fig. 3.6b.
For the first vertex we have A 1 = (i d f n ) p
2 A 0 that also includes the coupling of
the radiation into the cavity. Finally, the sum over all orders results in
R = i d f n p
2
∞
k=0
(i d p q f n )
k
=
i dp
2 f n
1 − i d p q f n
.
(3.37)
Inserting f n (ω) as defined in Eq. (3.31) we obtain a spectral response that is again a
Lorentzian resonance line
R(ω) =
C d p
2
(( 0 /2)
ω − ω 0 + L C + i(( 0 + C )/2
with C =
2πρ c n
k 0 k 0z
(3.38)
that exhibits a decay width of C = C d |Re( p q)| 0 =: χ χ 0 and an energy shift
of L C = −C d Im( p q) ) 0 /2. Combining these results into one expression for the
complex-valued frequency shift C , we obtain