Coherent light generators – Particular resonant cavity
Reexamination Certificate
2001-07-02
2004-03-16
Ip, Paul (Department: 2828)
Coherent light generators
Particular resonant cavity
C372S069000, C372S109000
Reexamination Certificate
active
06707837
ABSTRACT:
BACKGROUND OF THE INVENTION
The invention relates to a device and to a method for the controlled achieving of a photon flow between at least one selected resonance of an electromagnetic resonator and a selected target resonance of the resonator, wherein this photon flow in particular supports the redistribution of electromagnetic radiation between the resonances of the resonator in producing a Bose-Einstein photon condensate.
From WO 87/01503 there is known a method and a device for converting electromagnetic waves into monochromatic, coherent electromagnetic radiation with a predeterminable frequency and into heat radiation, wherein the predeterminable frequency lies at the lower edge of the Planck-distributed frequency spectrum of the heat radiation. Thereby, electromagnetic radiation is concentrated in a resonator in a manner such that the average radiation density in the resonator exceeds a critical value and the part of the radiation exceeding this critical value occupies the lowest electromagnetic energy mode of the resonator. The invention WO 87/01503 technically applies the Bose-Einstein condensation in the case of photons.
It is a disadvantage of this known device that the redistribution process for the number of photons exceeding the critical value of the electromagnetic radiation in the resonator is not exactly controllable. If for example the overcritical radiation density with respect to an average radiation temperature is produced by a stationary flow equilibrium, wherein the frequency spectrum of an essential part of the electromagnetic radiation radiated into the resonator lies in the vicinity of a certain resonator resonance, there arises the question of how the photons may flow out of the vicinity of this initial frequency into the fundamental mode of the resonator, wherein the frequency and the number of the net flowing photons adjust to the resonance frequencies which take part. The typical thermalisation processes for electromagnetic radiation in interaction with the resonator walls usually present only a small conversion potential for the Bose-Einstein condensation of photons.
SUMMARY OF THE INVENTION
The object of the present invention lies in avoiding the disadvantages of that which is known, in particular to provide a device where photons in the vicinity of a certain frequency, to a greatest extent as possible convert into photons in the region of a predetermined target frequency, wherein the target frequency is smaller than the initial frequency and the photons may be subjected to the modes of a Bose-Einstein condensation so that they may spontaneously flow from the vicinity of the initial frequency into the region of the target frequency which is the fundamental frequency of an electromagnetic resonator.
According to the invention this object is achieved with a device as described below.
Quantum statistical fundamentals
In a stationary flow equilibrium in a photon gas one may create a thermodynamic equilibrium in that the average photon energy density and the average photon number density are fixed independently of one another. This may for example be effected in that with an electromagnetic resonator the wall temperature is fixed whilst with a laser, photons are radiated in. By way of the mutually independent variation of the power and of the frequency of the laser a stationary photon accumulation may be built up whose parameters—average photon number and average photon energy—in pairs, may be set independently of one another. Also the free manipulation of the wall temperature and of the laser power or of the wall temperature and of the laser frequency for this are considered. In place of a laser, also by way of a heat radiation of a suitable temperature which is radiated into the resonator through a long-wave pass filter, there may be created a desired stationary deviation from Plancks's heat radiation.
Mathematically such a photon gas may be described by way of a so-called “grand canonical ensemble” with an indefinite particle number with the two Lagrange parameters &bgr;=1/(kT) and &mgr;, with inverse temperature and chemical potential. For the energy density of this free boson gas there applies:
u
(
β
,
μ
)
=
V
-
1
∑
k
=
1
,
2
,
…
ϵ
k
(
ⅇ
β
(
ϵ
k
-
μ
)
-
1
)
-
1
=
V
-
1
ϵ
1
(
ⅇ
β
(
ϵ
1
-
μ
)
-
1
)
-
1
+
V
-
1
∑
k
=
2
,
3
,
…
ϵ
k
(
ⅇ
β
(
ϵ
k
-
μ
)
-
1
)
-
1
(
1
)
&egr;k, k=1,2. . . stands for the energy value of the resonator, V for the volume. The second term of the second fine which sums up the energy of all excited modes, for sufficiently “large” cavities tends towards
u
e
(
β
,
μ
)
=
6
β
-
4
(
′
hc
)
-
3
π
-
2
g
4
(
ⅇ
β
μ
)
,
g
α
(
z
)
=
∑
n
=
1
,
2
,
…
z
n
n
-
α
,
(
2
)
′h stands for Planck's constant h divided by 2&pgr;. The parameters &bgr; and &mgr; are solutions to the equation system
u
(&bgr;,&mgr;)=
u
&rgr;(&bgr;,&mgr;)=
n
(3)
wherein u indicates the value of the set energy density and n the value of the set photon number. The photon number density &rgr; as a function of &bgr; and &mgr; is given by
ρ
(
β
,
μ
)
=
V
-
1
∑
k
=
1
,
2
,
…
(
ⅇ
β
(
ϵ
k
-
μ
)
-
1
)
-
1
=
V
-
1
(
ⅇ
β
(
ϵ
1
-
μ
)
-
1
)
-
1
+
V
-
1
∑
k
=
2
,
3
,
…
(
ⅇ
β
(
ϵ
k
-
μ
)
-
1
)
-
1
(
4
)
For sufficiently large cavities the second term in (4), the term of the excited modes results in
&rgr;
e
(&bgr;,&mgr;)=2&bgr;
−3
(′
hc
)
−3
&pgr;
−2
g
3
(
e
&bgr;&mgr;
) (5)
For the chemical potential there applies
&mgr;≦&egr;1 (6)
so that the occupation probabilities occuring in (4) may not become negative.
With an increasing size of the cavity, &egr;1 reciprocally to the characteristic “diameter” tends to 0. In the limit case of infinitely large cavities &mgr; is negative or equal to 0. If &mgr; equal to 0 excited modes absorb the maximum energy density
uc
(&bgr;):=
ue
(&bgr;,0) (7)
This is the energy density of the black body radiation. If there is set an energy density u which exceeds this value, the energy excess must be taken up by the fundamental mode, the first term in (1). In the ideal case of an infinitely large cavity, i.e. for each sufficiently large cavity, in a good approximation there then applies
V
−1
&egr;1(
e
&bgr;(&egr;
1
−&mgr;)
−1)
−1
=
u
−u
c
(&bgr;) (8)
Expanding the exponential function on the left side in
V
−1
&egr;
1
(&egr;
1
−&mgr;)
−1
&bgr;
−1
(9)
then it is obvious that &mgr;−&egr;1 reciprocally to the fourth power of the characteristic diameter of the cavity tends to 0.
If the fundamental mode energy is different to 0, which means that, the fundamental mode is occupied macroscopically, the photon number in the fundamental mode, thus the first term in (4), becomes singular in that it increases proportionally to the diameter of the cavity. This is plausible since an infinitely large number of photons of infinitesimal energy gives a finite energy term. This is precisely the infrared singularity.
On the Mechanism of the Redistribution of Photons
Bose-Einstein condensation of photons means that the photons exceeding the critical energy density uc(&bgr;) transfer into the fundamental mode of the resonator. This is possible by the interaction of the photons with the wall of the resonator. Since the quality factor of the cavity has a finite value a broadening of the resonances and thus an overlapping of the resonance curves result. This implies non-zero transition probabilities between the resonances. A cavity which suits for photon condensation may be designed such that t
Al-Nazer Leith A
Ip Paul
Shoemaker and Mattare
LandOfFree
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