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Finite-size corrections to the radiation laws of a Kerr nonlinear blackbody
Optics Communications 283 (2010) 2929–2934
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Finite-size corrections to the radiation laws of a Kerr nonlinear blackbody Qi-Jun Zeng a,b, Ze Cheng a,⁎ a b
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China
a r t i c l e
i n f o
Article history: Received 27 January 2010 Received in revised form 13 March 2010 Accepted 15 March 2010 Keywords: Kerr nonlinear blackbody Nonpolariton Finite-size correction Radiation law
a b s t r a c t In a Kerr nonlinear blackbody (KNB), bare photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. The nonpolariton system can constitute free thermal radiation in the blackbody. The present paper investigates the radiation of a KNB in a cavity of finitesize. For a given geometry, we obtain explicit expressions of the modified Planck and Stefan–Boltzmann blackbody radiation laws as a function of the temperature, size and shape of the cavity by using semiclassical techniques. Moreover, we discuss these corrections of a spherical KNB and the range of parameters (temperature and size of the cavity) in which these effects are accessible to experimental detection. Finally, we calculate the finite-size corrections (FSCs) under these conditions, and this work may lay the foundation for the experimental verification of a KNB. © 2010 Elsevier B.V. All rights reserved.
1. Introduction By definition, a blackbody can absorb 100% of all thermal radiation falling on it. The electromagnetic field within a blackbody is in thermal equilibrium. A primary quantity of the radiation properties of a blackbody is the density of energy per unit of frequency uðωÞ =
1 ρðωÞεðωÞ; V
ð1Þ
where V is the volume of the blackbody cavity, ρ(ω)dω represents the number of stationary electromagnetic modes with frequencies between ω and ω + dω, and ɛ(ω) is the average energy per mode. Besides u(ω), another important quantity to characterize the blackbody radiation is the total energy emitted per unit of time and area, RðT Þ =
c ∞ ∫ uðωÞdω; 4 0
ð2Þ
objects. In sum, the theory of blackbody radiation still attracts many physicists' attention up to the present [1,2]. The foregoing blackbody contains only the thermal radiation in its interior. However, in the early time Cheng [3,4] studied a new blackbody, named a KNB, whose interior is filled with a Kerr nonlinear crystal. Such a KNB can be regarded as a cavity having perfectly conducting walls and kept at a constant temperature T. In the previous works, Cheng pointed out that in a KNB bare photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into a different kind of quasiparticle, the nonpolariton. Furthermore, Cheng et al [5–8] also studied the other important properties of a KNB, including the radiation properties, the condensation state of the photon system and so on. They found that a KNB has many special properties which are quite different from a normal blackbody and may have enormous values of theory and application. In this paper, we will discuss the FSCs to the radiation laws of a KNB. Before discussing these questions, we will introduce the theory of a KNB briefly. 2. Kerr nonlinear blackbody
where c is the velocity of light in vacuum. The investigation on the radiation property has played an important role in the origins of quantum theory. Up to now, the exploration of blackbody radiation has spread over most fields of physics including condensed matter physics, statistical physics, optics and so on. Moreover, the blackbody radiation also has lots of valuable applications such as the technique of thermal remote sensing of ⁎ Corresponding author. Tel.: + 86 02787542637. E-mail address: [email protected] (Z. Cheng). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.03.034
The crystal being filled in the blackbody is determined as a specific crystal with a diamond structure, such as C. In this kind of crystal, the zero-wave-vector frequency of the Raman-active mode is denoted by ωR [9]. In Refs. [3,4], referring to the BCS theory, Cheng drew a conclusion that the interaction between photons and phonons can result in an attractive effective interaction among the photons themselves which leads to bound photon pairs. In the standingwave configuration a photon pair is stable only if the two photons have opposite wave vectors and helicities.
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According to Ref. [4], the pair Hamiltonian of the photon system is given by ′ = Hem
1 þ þ ∑ ℏωk akσ akσ + a−k;−σ a−k;−σ 2 kσ þ þ + ∑ Vkσ;k′ σ ′ ak′ σ ′ a−k′ ;−σ ′ a−k;−σ akσ ;
ð3Þ
kσ;k′ σ ′
where ℏ is Planck constant, a+ kσ(t) and akσ(t) are the creation and annihilation operators of circularly polarized photons with wave vector k and helicity σ = ± 1, respectively. Vkσ,k′σ′ is the pair potential and it is a function of ωk [4]. We assume that the crystal has a dispersion-free refractive index n, so the photonic frequency is given by ωk = c|k|/n. The first term of Eq. (3) represents the contribution from the electromagnetic field and the second term stems from the interaction between the electromagnetic field and the crystal lattice. In the photon system, unpaired bare photons are transformed into a new kind of quasiparticle, the nonpolariton. A nonpolariton is the condensate of virtual nonpolar phonons with a bare photon acting as the nucleus of condensation. The diagonalization of the pair Hamiltonian can be performed by the Bogoliubov transformation: þ
þ
ckσ ðt Þ = Uakσ ðt ÞU = akσ ðt Þ cosh φkσ −a−k;−σ ðt Þ sinh φkσ ; ð4Þ þ
þ
þ
þ
strength between a bare photon and virtual nonpolar phonons, and so both transition temperature Tc and velocity v(T) depend on the parameter γ. In what follows, we regard γ as an adjustable parameter. It is convenient to define the number operators Nkσ = c+ kσ(t)ckσ(t) for nonpolaritons. The number operators have the eigenvalues nkσ = 0, 1, 2.... The eigenstates of number operators are given by [4]: " # 1 þ nkσ j fnkσ g〉 = ∏ pffiffiffiffiffiffiffiffiffiffi ckσ jG〉; nkσ ! kσ
where |G〉 = U|0〉 is the normalized state vector of photon pairs in the photon system and |0〉 is the vacuum state of the electromagnetic field. Similar to a normal blackbody, in a KNB the gas of free nonpolaritons constitutes a thermal radiation. We can conceive a grand canonical ensemble of nonpolaritons to characterize the squeezed thermal radiation state in a KNB. Some identical systems of the ensemble may be in an eigenstate of the Hamiltonian H′em given by Eq. (7). The distribution of the ensemble over the eigenstates is described by the density operator of the squeezed thermal radiation state ρ=
′ = k TÞ expð−Hem B ′ =k T ; Tr exp −Hem B
ckσ ðt Þ; cþ k′ σ ′ ðt Þ − = δk;k′ δσ;σ ′ ; ckσ ðt Þ; ck′ σ ′ ðt Þ − = 0:
ð5Þ
þ aþ kσ ðt Þ = ckσ ðt Þ cosh φkσ + c−k;−σ ðt Þ sinh φkσ :
ð6Þ
Under the mean-field approximation, after substituting Eq. (6) into Eq. (3), the pair Hamiltonian of the photon system can be diagonalized into ′ = E + ∑ ℏω ˜ k ðT Þcþ Hem p kσ ðt Þckσ ðt Þ;
ð7Þ
kσ
˜ k ðT Þ = vðT Þjkj, v ˜ k ðT Þ is the frequency of nonpolaritons with ω where ω (T) is the velocity of nonpolaritons and it is determined by the following equation 0
vðT Þ = 2ðc = nÞV0 ∑ ℏωk′ coth k′
ℏvðT Þjk′j ; 2kB T
〈Nkσ 〉 =
1 eℏ ω˜ k ðT Þ = kB T −1
ð11Þ
A main thermodynamic quantity in a KNB is the total energy U of the thermal radiation, as given by ˜ k ðT Þ〈Nkσ 〉: U = ∑ ℏω
ð12Þ
kσ
After putting Eq. (11) into Eq. (12) and in the usual way of altering the summation to an integration, we can obtain the following expressions:
Transformation (4) can easily be inverted into akσ ðt Þ = ckσ ðt Þ cosh φkσ + cþ −k;−σ ðt Þ sinh φkσ ;
ð10Þ
and the ensemble average 〈Nkσ〉 of the number operators of nonpolaritons in a mode kσ is given by
ckσ ðt Þ = Uakσ ðt ÞU = akσ ðt Þ cosh φkσ −a−k;−σ ðt Þ sinh φkσ ; where U is a function of φkσ and the parameter φkσ is assumed to be real and spherically symmetric. c+ kσ(t) and ckσ(t) are the creation and annihilation operators, respectively, of nonpolaritons in the photon system. They obey the Bose equal-time commutation relations:
ð9Þ
ð8Þ
where the prime on the summation means that ωk′ b ωR. In addition, we ˜ k ðT Þ = nvðT Þ. The velocity v(T) can also get the equation ℏωk = cℏω determined by Eq. (8) is a monotonically increasing function of temperature T, which is equal to c/n at the transition temperature Tc. In Ref. [4], it can be found that below Tc the photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. At Tc, both photon pairs and nonpolaritons are all transformed into individual bare photons. Above Tc, a KNB behaves like a normal blackbody. Moreover, a dimensionless constant γ can be defined by γ = nv(0)/c, where v(0) is the velocity of nonpolaritons at the absolute zero of temperature. The constant γ is characteristic of a crystal and is meaningful only if 0 b γ b 1. It signifies the coupling
∞ ˜ dω; ˜ U = V∫0 u ω
ð13Þ
˜ 3 ðT Þ 1 ℏ ω ˜ ε ω ˜ = ˜ = ρ ω ; u ω 2 3 ℏ ω ˜ ð V π v ðT Þ e T Þ = kB T −1
ð14Þ
˜ is the density of energy per unit of frequency ω. ˜ In where u ω ˜ and ε ω ˜ , respectively, are determined Eq. (14) the parameters ρ ω by ˜ 2 ðT Þ ω ˜ =V ; ρ ω π2 v3 ðT Þ
ð15Þ
˜ = ε ω
ð16Þ
˜ ðT Þ ℏω ; eℏ ω˜ ðT Þ = kB T −1
˜ dω ˜ represents the number of stationary electromagnetic where ρ ω ˜ and ω ˜ + dω, ˜ ε ω ˜ is the average modes with frequencies between ω energy per mode. Similar to a normal blackbody, according to Eq. (2) we can get the total energy of the nonpolariton system emitted per unit of time and area as follows: RðT Þ =
vðT Þ ∞ ˜ dω ˜ = σ ′ ðT ÞT 4 ; ∫ u ω 4 0
ð17Þ
where σ′(T) = π2k4B/60ℏ3v2(T). Eq. (17) is the Stefan–Boltzmann law of a KNB. Eqs. (14) and (17) are the main radiation laws of a KNB and next we will discuss FSCs to the two laws.
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˜ into ˜ , ρC and ρ˜ ω After substituting the expressions of ρV ω Eq. (23), we can get
3. Finite-size corrections to the radiation laws In fact Eqs. (14) and (17) are applicable only under the condition ˜ ,ρ ω ˜ and R(T) that V → ∞, this is to say , the previous quantity u ω ˜ , ρV ω ˜ and RV(T) in the V → ∞ limit, should be rewritten as uV ω ˜ is respectively. FSCs to Eqs. (19) and (22) are expected since ρV ω ˜ in powers of a typical only the leading term of a full expansion of ρ ω ˜ can be regarded as a volume term length L of the cavity, and ρV ω 3 which is proportional to L . A comprehensive theoretical study about ρ(ω) of a normal blackbody was carried out in Refs. [10,11] by using the Green's function formalism. The Green's function formalism was previously developed for scalar waves in Refs. [12,13]. Because the radiation properties of free nonpolariton gas are quite similar to those of photon gas, referring to Ref. [14] we can obtain the following expression in the special case of three-dimensional (3D) cavities with smooth surfaces and L NN λ, ˜ = ∑ δ ω− ˜ ω ˜ i = ρV ω ˜ ; ˜ + ρC + ρ˜ ω ρ ω
2931
ð18Þ
˜ =V ρ ω
˜2 −2 C ω + ⋅ 3π2 vðT Þ π2 v3 ðT Þ h i ˜ L cos ω⋅l ˜ p = vðT Þ + μp : + ∑ DC AP ω;
ð21Þ
p
We note the following two points. (a) The expression C = ∬s ð1 = R1 + 1 = R2 Þdσ only applies to cavities with smooth surfaces, but doesn't fit cavities with edges such as the rectangular box. In the later case, we should regard C as a equivalent parameter in the ˜ . (b) In this approach ρðωÞ ˜ is effectively written as expression of ρC ω an expansion in power of the parameter λ = L, so the limits of applicability of this formalism are restricted to a range of sizes L and ˜ such that L NN λ. frequencies ω 3.1. Corrections to the Planck and Stefan–Boltzmann laws
i
˜ i stands for the ith where λ = 2π = jkj represents the wavelength, ω ˜ satisfies Eq. (15). The second natural mode of the cavity and ρV ω −2 C term ρC = 3π 2 ⋅ vðT Þ is a curvature term, where C is the mean curvature and it satisfies C = ∬s ð1 = R1 + 1 = R2 Þdσ, R1 and R1 are the two principal radii of curvature of the cavity [12]. For a sphere of radius r0, it can be gotten easily that C = 4πr0. The third term is an oscillatory part, which is given by h i ˜ L cos ω⋅l ˜ p = vðT Þ + μp ; ˜ = ∑ DC Ap ω; ρ˜ ω p
! h i v2 ðT Þ v2−m = 2 ðT Þ ˜ P cos ω⋅l ˜ = ρV ω ˜ p = vðT Þ + μp ; ˜ 1 + a1 ρ ω + ∑ D A C ˜2 ˜ 2−m = 2 p L2 ω ωL
ð22Þ −2C 3L
where a1 = ð19Þ
is a dimensionless coefficient thath depends on the meani ˜ L = ðL =vðT ÞÞ1 + m = 2 ω ˜ m=2 curvature C∝L of the cavity, A˜ P = AP ω;
is the dimensionless amplitude corresponding to a single periodic
where the sum runs over the periodic orbits p of length of lp of the classical counterpart, and DC is a convergence factor with DC = zp, where z is a positive dimensionless quantity determined by the condition of a cavity and it satisfies z b 1, p is the number of vertices of a periodic orbit [13]. For electromagnetic waves the periodic orbits are the trajectories of the light rays inside the cavity as dictated by geometrical optics, and mirror reflection on surface S of a cavity will take once at each vertex of a periodic orbit. The amplitude place ˜ L and the Maslov index μp can be evaluated explicitly from the Ap ω; knowledge of the classical periodic orbits (refer to Ref. [15]). In 3D cavities, we can reach the following conclusion by the method of dimensionality analysis (refer to Ref. [13]) ˜ L ∝½L =vðT Þ1 + m = 2 ω ˜ m = 2; AP ω;
According to Eq. (18), it's clear that the finite deviations from the ˜ , then we can modify Planck law Eq. (15) only come from ρC and ρ˜ ω Eq. (21) as follows:
ð20Þ
where m is the degeneracy of the periodic orbits and it is associated with the symmetry of the cavity, for example, m = 3 for a sphere. ˜ L can also be influenced by the Moreover, the amplitude Ap ω; polarization effects and it picks up an additional factor of 2 for planar orbits [10]. In terms of classical optics, the existence of oscillatory term ˜ is easy to be understood as follows: Considering a periodic orbit ρ˜ ω which starts from r and comes back to r after one loop with multiple reflection, according to the value of frequency, one wave may enter in resonance with itself, and this situation thus corresponds to maxima of the density of modes. On the contrary another wave may interfere destructively with itself and such a situation leads to minima of the ˜ will arise for density of modes. Hence the oscillatory term of ρ ω different frequencies. Moreover for plane trajectories, in terms of polarizations of the transverse electromagnetic wave, it can be proved that only orbits with an even number of vertices contribute remarkably to the oscillatory term [10].
orbit of length lp. Consequently, by inserting Eqs. (16) and (22) ˜ =ρ ω ˜ ⋅ε ω ˜ = V, we can get the modified Planck law as into u ω follows: 1 ˜ ⋅ε ω ˜ ˜ + ρC + ρ˜ ω ˜ = ρ ω u ω V V =
1 + a1 ×
h i v2 ðT Þ v2−m = 2 ðT Þ ˜ p = vðT Þ + μp + 2−m = 2 ∑ DC A˜ P cos ω⋅l 2˜2 p ˜ L ω ωL
˜ 3 ðT Þ ℏ ω : ℏω ˜ ðT Þ = k B T π v ðT Þ e −1 2 3
!
ð23Þ
˜ firstly we must get these In order to fully determine uðωÞ, coefficients such as Ap and μp. In the case of symmetric cavities like spheres (m = 3) and rectangles (m = 2), these coefficients can be gotten easily (see below and Ref. [15] ). But for chaotic cavities (m = 0), these coefficients are difficult to be obtained [16]. In this paper we should only study the case of symmetric cavities. Secondly, we have yet to set a cutoff in the sum of Eq. (28). We define that lesc represents the length of a ray's orbit before the ray escapes through the aperture. From a physical point of view, only the periodic orbits which are shorter than the length lesc can make remarkable ˜ The explicit expression of the cutoff function contributions to uðωÞ. strongly depends on the symmetries of the cavity. For symmetric cavities (m = 3, 2), the cutoff function has power-law tails but for chaotic cavities it's quite complicated [17]. Accordingly, after inserting Eq. (23) into Eq. (17), the Stefan– Boltzman law can also be modified as follows:
∞
RðT Þ = ∫0 1 +
" #! 2 2−m = 2 ˜ p ω⋅l ˜ 3 ðT Þ a1 v ðT Þ v ðT Þ ℏ ω ˜ + μp dω: + 2−m = 2 ∑ DC A˜ P cos 2˜2 vðTÞ ˜ p 4π2 v2 ðT Þ eℏ ω˜ ðT Þ = k B T −1 L ω ωL
ð24Þ
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˜ and considering the cutoff function By completing the integral of ω of the sum, we can get
about the parameters such as lp,t and m, we can expand the oscillatory ˜ as follows: term ρ˜ ω
RðT Þ = RV ðT Þ + RC ðT Þ + R˜ ðT Þ;
˜ = − ∑ DC′ ρ˜ ω
ð25Þ
˜ ˜ 3=2 3V ω 3V ω t ˜ = vðT Þ + sin 4r0 ω⋅t ∑ DC ð−1Þ 2 2 v ð T Þ v ð T Þ 2v ð T Þr tπ t =1 t;p N 2t 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sinðπt = pÞ 3 π 2ωpr ˜ 0 × sinð2πt = pÞ⋅ sinðπt = pÞ ; ð29Þ sin p + + vðT Þ 2 2 r0 pπ3
2
ðTkB Þ 1 where RV(T) = σ ’(T)T 4, RC ðT Þ = − C36L 3 ℏ ∝ L2 and it comes from the ˜ curvature term ρC in the spectral density. The fluctuating term RðTÞ can be approximatively written as follows in the limit lmin N N ℏvðTÞ , kB T where lmin is the length of the shortest periodic orbit, 2
vðT ÞkB T ℏv ðT Þ + b0 ; R˜ ðT Þ≈b1 L3 L4
ð26Þ
where D’C = z2t is a convergence factor. In addition, we can get ρC = −8r0 ˜ by Eq. 3πvðT Þ, and ρV ω is still (15). Then after substituting denoted ˜ into Eq. (23), we can get the ˜ , ρC and ρ˜ ω the expressions of ρV ω modified Planck law of a SKNB as follows: ˜ = u ω
where b0 and b1 are the dimensionless coefficients which depend on the number m. We will discuss the two coefficients of a spherical cavity (m = 3) separately in the next section. For m = 0, 2, we can get (refer to Ref. [14])
Ap;t ðLÞf μp ðm + 2Þπ −2πL 1 + m = 2 ∑ 2 + m = 2 ; b0 = − 4 vðTÞ p;t τp t
ð27Þ
Ap;t ðLÞg μp 2πL m = 2 ∑ 1 + m = 2 ; b1 = π vðT Þ p;t τ t
ð28Þ
(
˜ ˜ 2 ðT Þ 8r0 3ω ω ˜ = v ðT Þ − − ∑ D′ sin 4r0 ω⋅t π2 v3 ðT Þ 3VπvðT Þ t = 1 C 2v2 ðT Þr0 tπ2
˜ 3=2 ω 3 t ∑ D ð−1Þ + vðTÞ vðT Þ t;p N 2t C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
˜ 0 ˜ ðT Þ sinðπt = pÞ 3 π 2ωpr ℏω + sinðπt = pÞ ⋅ ˜ sin p + × sinð2πt = pÞ : 2 2 vðT Þ r0 pπ3 eℏ ωðT Þ = kB T −1
g
ð30Þ
Secondly, in the limit eℏ ω˜ ðT Þ = kB T ≫1 and lmin N N ℏvkBðTT Þ, by inserting ˜ the modified Stefan– Eq. (30) into Eq. (17) and integrating over ω, Boltzmann law can be obtained as, 4
RðT Þ = σ ′ ðT ÞT −
p
1 2 3 2 4 ðTkB Þ + b1 kB Tv = r0 + b0 ℏv = r0 ; 12r02 ℏ
ð31Þ
where
1 −3 3 . So Eq. (31) can be where b1 ≈ 0 and b0 ≈ 256π 2 ∑ 4 ≈1:2 × 10 t t reduced to
h i f μp = cos μp Þ + sin μp − cos μp m = 2;
RðT Þ = σ ′ ðT ÞT −
h i g μp = − sin μp − sin μp − cos μp m = 2:
The amplitude Ap,t describes the contribution of a single periodic orbit, and the single periodic orbit has a period τp = lp/v(T) which is repeated t times. Here we note that (a) the largest contribution to the sum comes from the shortest periodic orbits, and (b) for the shortest periodic orbits, the Maslov indices μp and amplitude Ap,t can be computed analytically [15].
4
1 2 −3 2 4 ðTkB Þ + 1:2 × 10 ℏv = r0 : 12r02 ℏ
ð32Þ
The second term of Eq. (32) corresponds to the curvature contribution RC(T) and the third term is the oscillating part R˜ ðT Þ. Eqs. (30) and (32) are the modified Planck law and Stefan– Boltzmann law, respectively, for a KNB. Here we note that since the sum in Eq. (30) is dominated by the contribution of the shortest periodic orbits, consequently the corrections above are quite insensitive to the cutoff lesc ≫ lmin. 3.3. Suitable conditions for experimental detection and numerical calculation
3.2. Corrections to laws of a spherical kerr nonlinear blackbody The case of a spherical Kerr nonlinear blackbody (SKNB) can be solved exactly because a spherical cavity has a high degree of symmetry. Similar studies of other kinds of the normal blackbody have been carried out in Ref. [14,18]. Here we shall only discuss the ideal spherical blackbody with a smooth interior surface. Under such a condition we can ignore the effects of small, nonoverlapping bumps on the surface, these effects will influence the corrections by suppressing periodic orbits which are longer than a certain cutoff, and the cutoff is determined by the bumps' size [19]. For this case the concerned closed periodic orbits are given by planar regular polygons, and these polygons have an even number of vertices [13]. Moreover, since nonpolarations and photons abide by the same reflection law, an identical polygon should lie in a plane which contains the diameter of the sphere. The length lp,t of the trajectories can be given by lp,t = 2pr0 sin (πt/p), where r0 represents the radius of the sphere, p is the number of vertices of a polygon and t is the number of turns around the origin of a specific periodic orbit. With these definitions, we can investigate the corrections to Planck and Stefan–Boltzmann laws. Firstly, in terms of the above discussions
In order to understand the significance of FSCs to the radiation laws ˜ and R(T). more clearly, we should make numerical calculation of u ω For this aim, we must determine some parameters such as the temperature T, the radius of a spherical cavity r0 and so on, these parameters have close relations to the experimental detection. Firstly, we discuss the optimal radius for experimental verification. In general, the larger the radius, the smaller the corrections are. In addition, it will be difficult to keep the temperature stable if the radius becomes larger. Whereas a cavity with too small radius is hard to be made with perfect geometrical shape, and the flux of energy escaping through the aperture of a cavity is too small to be measured with the necessary accuracy. Hence the optimal radius can be estimated to be r0 = 10– 70 mm [14] and we choose r0 = 25 mm as a tentative radius. Secondly, the suitable temperature T should be below the transition temperature Tc and easy to be maintained. Forγ = 0.9, the transition temperature Tc of a KNB is equal to 464.9 K. We pick up T ≤ 5 K, and can obtain that v(T) ≈ 0.9c and λm = 0.92 mm, where λm is the wavelength at which the ˜ is a maximum. Besides, this condition satisfies the energy density uðωÞ limit L ≫ λ well and the semiclassical formalism is accurate enough. Thirdly, in the actual numerical calculation of the summation in Eq. (30), we can only consider a finite number of values of indices t and p due to
Q.-J. Zeng, Z. Cheng / Optics Communications 283 (2010) 2929–2934
2933
the existence of convergence factor DC′ and DC, and this method is in agreement with the method in Ref. [13]. Finally, for the experimental ˜ for ω ˜ in the formulas detection of FSCs, we should substitute ω = nvcðT Þ ω ˜ and R(T) before the numerical calculation. of uðωÞ For the convenience of numerical calculation, we take the diamond crystal whose zero-wave-vector frequency of the Raman-active mode is ωR = 2.51 × 1014 s− 1 and assume its refractive index n = 1. Then the variation with ω of u(ω) can be shown in Fig. 1, where we let T = 5 K and γ = 0.9. From this figure it is clear that FSC to Planck law can't be neglected, especially at the frequency ωm = 2.05 × 1012(Hz) where the energy density u(ωm) is a maximum. Furthermore, the variation with ω of u(ω) is also shown in Fig. 2 under the condition that T = 5 K and γ = 0.6. By contrast these two figures, we can see that the variation of parameter γ will influence FSC to the Planck law, exactly speaking, the smaller the parameter γ, the larger the deviations are. Next the variation with T and γ of RS(T) can be shown in Fig. 3, h i where RS ðT Þ = RðT Þ = RV ðT Þ = 1 + RC ðT Þ + R˜ ðT Þ = RV ðT Þ is the relative total energy emitted per unit of time and area. In addition, RS(T) of a normal blackbody is also shown in Fig. 3 in contrast with a KNB. From Fig. 3 we can find two features: (a) For fixed γ, RS(T) increases continuously as the temperature T increases from 0.5 K to 2.5 K. (b) For fixed T, RS(T) is a monotonically decreasing function of γ. 4. Discussions In this paper we first introduce the theory of a KNB which is brought forward and developed by Cheng. Next considering the fact that nonpolaritons behave similarly to the photons in some aspects such as propagation, reflection, interference and diffraction, we study the corrections of a finite-size cavity to the radiation laws of a KNB by using semiclassical techniques. We find the dominant corrections include two terms. One term, which is represented by ρc, is dominated by C which is the mean curvature of the cavity. Another term, which is ˜ , is an oscillatory part which is associated with the denoted by ρ˜ ω symmetry of the cavity and the closed classical trajectories. To sum up, both of the two terms come from the fact the distribution of wave vector k in a finite-size cavity is quite different from that in an infinitesize cavity. For the former case, the electromagnetic waves will experience multiple reflection, scattering, interference and diffraction ˜ . In which yield additional corrections to the density of energy u ω short, the corrections are determined by the size and shape of a cavity. In the limit L ≫ λ, by using semiclassical techniques, we get the universal expressions of FSCs to the radiation laws of a KNB. Espe-
Fig. 1. For fixed γ = 0.9, radius r0 = 25 mm and temperature T = 5 K, variation of u(ω) with ω, where u(ω) is the energy density per unit of frequency. The dashed and solid lines correspond to an ideal and an actual SKNB, respectively.
Fig. 2. For fixedγ = 0.6, radius r0 = 25 mm and temperature T = 5 K, variation of u(ω) with ω, where u(ω) is the energy density per unit of frequency. The dashed and solid lines correspond to an ideal and an actual SKNB, respectively.
cially, we have deduced the explicit expressions of a SKNB and made a numerical calculation. From these investigations we can find three features as follows: (a) For fixed temperature T, according to the numerical calculation of u(ω) and the discussions about Figs. 1 and 2, we can find that the dominant term of FSCs to Planck law is the oscillatory part ρ˜ðωÞ, and this term is a decreasing function of v(T) and γ (due to the fact that γ signifies the coupling strength between a bare photon and virtual nonpolar phonons, and v(T) is an increasing function of γ [4]). The decreasing of γ means that the coupling strength between a photon and phonons will be enforced and so more nonpolaritons will arise. This effect will lead to a result that both the density of energy uV(ω) and the fluctuation of u(ω) increase simultaneously, so ρ˜ðωÞ will increase with decreasing of γ. Besides, it also can be found that the relative fluctuation ΔuðωÞ = juðωÞ−uV ðωÞj = uV ðωÞ almost remains unchanged with decreasing of γ. (b) For fixed γ, RS(T) is an increasing function of temperature T. The basic physical mechanism about this phenomenon can be explained as follows: as the temperature T is increased, the velocity of a nonpolariton increases correspondingly, ˜ and so RV(T), RðTÞ and the absolute value of RC(T) will become larger simultaneously. Due to the fact that RC(T) is negative, the sum ˜ of RC(T) and RðTÞ will approach zero with increasing of T, and consequently RS(T) will increase and finally approach one. (c) For fixed T,
Fig. 3. For three values of γ of an actual SKNB and a normal spherical blackbody, variation of RS(T) with temperature T, where RS(T) is the relative total energy emitted per unit of time and area. The solid line corresponds to the normal blackbody.
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RS(T) is a decreasing function of γ. The reason is that with increasing of parameter γ, the nonpolar phonon weight in a nonpolariton is decreased but the velocity of a nonpolariton increases, so that the energy of the nonpolariton system and RV(T) becomes smaller, and R˜ ðT Þ becomes larger. At the same time, considering that RC(T) is a constant with a fixed temperature and RC ðT Þ + R˜ ðT Þ≤0, so RS(T) will become smaller with increasing of parameter γ. In addition, we can find that RS(T) of a KNB is larger than RS(T) of a normal blackbody, and the difference of RS(T) between a KNB and a normal blackbody becomes larger with decreasing of parameter γ, whereas the differences of RS(T) among all these blackbodies approach zero with increasing of the temperature T. The reason is that more and more virtual nonpolar phonons will evaporate when the temperature T approaches the transition temperature TC, and so the behavior of a nonpolariton is more and more like that of a bare photon. At T ≥ TC, the KNB behaves just as a normal blackbody. In conclusion, by using semiclassical techniques, we have obtained explicit expressions for FSCs of a KNB to the radiation laws, these corrections can be denoted as functions of the temperature, size and shape of the cavity. Moreover, we have investigated a SKNB in detail and discussed the suitable parameters which may be suitable to detect these corrections. This work may lay the foundation for experimental verification of a KNB and have a widely prospect of application.
Acknowledgments This research was supported by the National Natural Science Foundation of China under Grants No. 10174024 and No. 10474025. References [1] S. Hoekstra, J.J. Gilijamse, B. Sartakov, N. Vanhaecke, L. Scharfenberg, S.Y.T. van de Meerakker, G. Meijer, Phys. Rev. lett. 98 (2007) 133001. [2] E.J. Angstmann, V.A. Dzuba, V.V. Flambaum, Phys. Rev. lett. 97 (2006) 040802. [3] Z. Cheng, Phys. Rev. Lett. 67 (1991) 2788. [4] Z. Cheng, J. Opt. Soc. Am. B 19 (2002) 1692. [5] Z. Cheng, Phys. Rev. A 71 (2005) 033808. [6] M. Yin, Z. Cheng, Phys. Rev. A 78 (2008) 063829. [7] M. Yin, Z. Cheng, Opt. Commun. 283 (2010) 60. [8] Z. Cheng, Phys. Rev. A 80 (2009) 033826. [9] W. Hayes, R. Loudon, Scattering of Light by Crystals, Wiley, New York, 1978. [10] R. Balian, B. Duplantier, Ann. Phys. (N.Y.) 104 (1977) 300. [11] R. Balian, C. Bloch, Ann. Phys. (N.Y.) 64 (1971) 271. [12] R. Balian, C. Bloch, Ann. Phys. (N.Y.) 60 (1970) 401. [13] R. Balian, C. Bloch, Ann. Phys. (N.Y.) 69 (1972) 76. [14] A.M. García-García, Phys. Rev. A 78 (2008) 023806. [15] M. Brack, R.K. Bhaduri, Semiclassical Physics, Addison-Wesley, New York, 1997. [16] C. Dembowski, B. Dietz, H.D. Graf, A. Heine, T. Papenbrock, A. Richter, C. Richter, Phys. Rev. Lett. 89 (2002) 064101. [17] R. A. Jalabert, e-print arXiv: con-mat/9912038. [18] K.M. Case, S.C. Chiu, Phys. Rev. A 1 (1970) 1170. [19] N. Pavloff, C. Schmit, Phys. Rev. B 58 (1998) 4942.
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Finite-size corrections to the radiation laws of a Kerr nonlinear blackbody Qi-Jun Zeng a,b, Ze Cheng a,⁎ a b
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China
a r t i c l e
i n f o
Article history: Received 27 January 2010 Received in revised form 13 March 2010 Accepted 15 March 2010 Keywords: Kerr nonlinear blackbody Nonpolariton Finite-size correction Radiation law
a b s t r a c t In a Kerr nonlinear blackbody (KNB), bare photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. The nonpolariton system can constitute free thermal radiation in the blackbody. The present paper investigates the radiation of a KNB in a cavity of finitesize. For a given geometry, we obtain explicit expressions of the modified Planck and Stefan–Boltzmann blackbody radiation laws as a function of the temperature, size and shape of the cavity by using semiclassical techniques. Moreover, we discuss these corrections of a spherical KNB and the range of parameters (temperature and size of the cavity) in which these effects are accessible to experimental detection. Finally, we calculate the finite-size corrections (FSCs) under these conditions, and this work may lay the foundation for the experimental verification of a KNB. © 2010 Elsevier B.V. All rights reserved.
1. Introduction By definition, a blackbody can absorb 100% of all thermal radiation falling on it. The electromagnetic field within a blackbody is in thermal equilibrium. A primary quantity of the radiation properties of a blackbody is the density of energy per unit of frequency uðωÞ =
1 ρðωÞεðωÞ; V
ð1Þ
where V is the volume of the blackbody cavity, ρ(ω)dω represents the number of stationary electromagnetic modes with frequencies between ω and ω + dω, and ɛ(ω) is the average energy per mode. Besides u(ω), another important quantity to characterize the blackbody radiation is the total energy emitted per unit of time and area, RðT Þ =
c ∞ ∫ uðωÞdω; 4 0
ð2Þ
objects. In sum, the theory of blackbody radiation still attracts many physicists' attention up to the present [1,2]. The foregoing blackbody contains only the thermal radiation in its interior. However, in the early time Cheng [3,4] studied a new blackbody, named a KNB, whose interior is filled with a Kerr nonlinear crystal. Such a KNB can be regarded as a cavity having perfectly conducting walls and kept at a constant temperature T. In the previous works, Cheng pointed out that in a KNB bare photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into a different kind of quasiparticle, the nonpolariton. Furthermore, Cheng et al [5–8] also studied the other important properties of a KNB, including the radiation properties, the condensation state of the photon system and so on. They found that a KNB has many special properties which are quite different from a normal blackbody and may have enormous values of theory and application. In this paper, we will discuss the FSCs to the radiation laws of a KNB. Before discussing these questions, we will introduce the theory of a KNB briefly. 2. Kerr nonlinear blackbody
where c is the velocity of light in vacuum. The investigation on the radiation property has played an important role in the origins of quantum theory. Up to now, the exploration of blackbody radiation has spread over most fields of physics including condensed matter physics, statistical physics, optics and so on. Moreover, the blackbody radiation also has lots of valuable applications such as the technique of thermal remote sensing of ⁎ Corresponding author. Tel.: + 86 02787542637. E-mail address: [email protected] (Z. Cheng). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.03.034
The crystal being filled in the blackbody is determined as a specific crystal with a diamond structure, such as C. In this kind of crystal, the zero-wave-vector frequency of the Raman-active mode is denoted by ωR [9]. In Refs. [3,4], referring to the BCS theory, Cheng drew a conclusion that the interaction between photons and phonons can result in an attractive effective interaction among the photons themselves which leads to bound photon pairs. In the standingwave configuration a photon pair is stable only if the two photons have opposite wave vectors and helicities.
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According to Ref. [4], the pair Hamiltonian of the photon system is given by ′ = Hem
1 þ þ ∑ ℏωk akσ akσ + a−k;−σ a−k;−σ 2 kσ þ þ + ∑ Vkσ;k′ σ ′ ak′ σ ′ a−k′ ;−σ ′ a−k;−σ akσ ;
ð3Þ
kσ;k′ σ ′
where ℏ is Planck constant, a+ kσ(t) and akσ(t) are the creation and annihilation operators of circularly polarized photons with wave vector k and helicity σ = ± 1, respectively. Vkσ,k′σ′ is the pair potential and it is a function of ωk [4]. We assume that the crystal has a dispersion-free refractive index n, so the photonic frequency is given by ωk = c|k|/n. The first term of Eq. (3) represents the contribution from the electromagnetic field and the second term stems from the interaction between the electromagnetic field and the crystal lattice. In the photon system, unpaired bare photons are transformed into a new kind of quasiparticle, the nonpolariton. A nonpolariton is the condensate of virtual nonpolar phonons with a bare photon acting as the nucleus of condensation. The diagonalization of the pair Hamiltonian can be performed by the Bogoliubov transformation: þ
þ
ckσ ðt Þ = Uakσ ðt ÞU = akσ ðt Þ cosh φkσ −a−k;−σ ðt Þ sinh φkσ ; ð4Þ þ
þ
þ
þ
strength between a bare photon and virtual nonpolar phonons, and so both transition temperature Tc and velocity v(T) depend on the parameter γ. In what follows, we regard γ as an adjustable parameter. It is convenient to define the number operators Nkσ = c+ kσ(t)ckσ(t) for nonpolaritons. The number operators have the eigenvalues nkσ = 0, 1, 2.... The eigenstates of number operators are given by [4]: " # 1 þ nkσ j fnkσ g〉 = ∏ pffiffiffiffiffiffiffiffiffiffi ckσ jG〉; nkσ ! kσ
where |G〉 = U|0〉 is the normalized state vector of photon pairs in the photon system and |0〉 is the vacuum state of the electromagnetic field. Similar to a normal blackbody, in a KNB the gas of free nonpolaritons constitutes a thermal radiation. We can conceive a grand canonical ensemble of nonpolaritons to characterize the squeezed thermal radiation state in a KNB. Some identical systems of the ensemble may be in an eigenstate of the Hamiltonian H′em given by Eq. (7). The distribution of the ensemble over the eigenstates is described by the density operator of the squeezed thermal radiation state ρ=
′ = k TÞ expð−Hem B ′ =k T ; Tr exp −Hem B
ckσ ðt Þ; cþ k′ σ ′ ðt Þ − = δk;k′ δσ;σ ′ ; ckσ ðt Þ; ck′ σ ′ ðt Þ − = 0:
ð5Þ
þ aþ kσ ðt Þ = ckσ ðt Þ cosh φkσ + c−k;−σ ðt Þ sinh φkσ :
ð6Þ
Under the mean-field approximation, after substituting Eq. (6) into Eq. (3), the pair Hamiltonian of the photon system can be diagonalized into ′ = E + ∑ ℏω ˜ k ðT Þcþ Hem p kσ ðt Þckσ ðt Þ;
ð7Þ
kσ
˜ k ðT Þ = vðT Þjkj, v ˜ k ðT Þ is the frequency of nonpolaritons with ω where ω (T) is the velocity of nonpolaritons and it is determined by the following equation 0
vðT Þ = 2ðc = nÞV0 ∑ ℏωk′ coth k′
ℏvðT Þjk′j ; 2kB T
〈Nkσ 〉 =
1 eℏ ω˜ k ðT Þ = kB T −1
ð11Þ
A main thermodynamic quantity in a KNB is the total energy U of the thermal radiation, as given by ˜ k ðT Þ〈Nkσ 〉: U = ∑ ℏω
ð12Þ
kσ
After putting Eq. (11) into Eq. (12) and in the usual way of altering the summation to an integration, we can obtain the following expressions:
Transformation (4) can easily be inverted into akσ ðt Þ = ckσ ðt Þ cosh φkσ + cþ −k;−σ ðt Þ sinh φkσ ;
ð10Þ
and the ensemble average 〈Nkσ〉 of the number operators of nonpolaritons in a mode kσ is given by
ckσ ðt Þ = Uakσ ðt ÞU = akσ ðt Þ cosh φkσ −a−k;−σ ðt Þ sinh φkσ ; where U is a function of φkσ and the parameter φkσ is assumed to be real and spherically symmetric. c+ kσ(t) and ckσ(t) are the creation and annihilation operators, respectively, of nonpolaritons in the photon system. They obey the Bose equal-time commutation relations:
ð9Þ
ð8Þ
where the prime on the summation means that ωk′ b ωR. In addition, we ˜ k ðT Þ = nvðT Þ. The velocity v(T) can also get the equation ℏωk = cℏω determined by Eq. (8) is a monotonically increasing function of temperature T, which is equal to c/n at the transition temperature Tc. In Ref. [4], it can be found that below Tc the photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. At Tc, both photon pairs and nonpolaritons are all transformed into individual bare photons. Above Tc, a KNB behaves like a normal blackbody. Moreover, a dimensionless constant γ can be defined by γ = nv(0)/c, where v(0) is the velocity of nonpolaritons at the absolute zero of temperature. The constant γ is characteristic of a crystal and is meaningful only if 0 b γ b 1. It signifies the coupling
∞ ˜ dω; ˜ U = V∫0 u ω
ð13Þ
˜ 3 ðT Þ 1 ℏ ω ˜ ε ω ˜ = ˜ = ρ ω ; u ω 2 3 ℏ ω ˜ ð V π v ðT Þ e T Þ = kB T −1
ð14Þ
˜ is the density of energy per unit of frequency ω. ˜ In where u ω ˜ and ε ω ˜ , respectively, are determined Eq. (14) the parameters ρ ω by ˜ 2 ðT Þ ω ˜ =V ; ρ ω π2 v3 ðT Þ
ð15Þ
˜ = ε ω
ð16Þ
˜ ðT Þ ℏω ; eℏ ω˜ ðT Þ = kB T −1
˜ dω ˜ represents the number of stationary electromagnetic where ρ ω ˜ and ω ˜ + dω, ˜ ε ω ˜ is the average modes with frequencies between ω energy per mode. Similar to a normal blackbody, according to Eq. (2) we can get the total energy of the nonpolariton system emitted per unit of time and area as follows: RðT Þ =
vðT Þ ∞ ˜ dω ˜ = σ ′ ðT ÞT 4 ; ∫ u ω 4 0
ð17Þ
where σ′(T) = π2k4B/60ℏ3v2(T). Eq. (17) is the Stefan–Boltzmann law of a KNB. Eqs. (14) and (17) are the main radiation laws of a KNB and next we will discuss FSCs to the two laws.
Q.-J. Zeng, Z. Cheng / Optics Communications 283 (2010) 2929–2934
˜ into ˜ , ρC and ρ˜ ω After substituting the expressions of ρV ω Eq. (23), we can get
3. Finite-size corrections to the radiation laws In fact Eqs. (14) and (17) are applicable only under the condition ˜ ,ρ ω ˜ and R(T) that V → ∞, this is to say , the previous quantity u ω ˜ , ρV ω ˜ and RV(T) in the V → ∞ limit, should be rewritten as uV ω ˜ is respectively. FSCs to Eqs. (19) and (22) are expected since ρV ω ˜ in powers of a typical only the leading term of a full expansion of ρ ω ˜ can be regarded as a volume term length L of the cavity, and ρV ω 3 which is proportional to L . A comprehensive theoretical study about ρ(ω) of a normal blackbody was carried out in Refs. [10,11] by using the Green's function formalism. The Green's function formalism was previously developed for scalar waves in Refs. [12,13]. Because the radiation properties of free nonpolariton gas are quite similar to those of photon gas, referring to Ref. [14] we can obtain the following expression in the special case of three-dimensional (3D) cavities with smooth surfaces and L NN λ, ˜ = ∑ δ ω− ˜ ω ˜ i = ρV ω ˜ ; ˜ + ρC + ρ˜ ω ρ ω
2931
ð18Þ
˜ =V ρ ω
˜2 −2 C ω + ⋅ 3π2 vðT Þ π2 v3 ðT Þ h i ˜ L cos ω⋅l ˜ p = vðT Þ + μp : + ∑ DC AP ω;
ð21Þ
p
We note the following two points. (a) The expression C = ∬s ð1 = R1 + 1 = R2 Þdσ only applies to cavities with smooth surfaces, but doesn't fit cavities with edges such as the rectangular box. In the later case, we should regard C as a equivalent parameter in the ˜ . (b) In this approach ρðωÞ ˜ is effectively written as expression of ρC ω an expansion in power of the parameter λ = L, so the limits of applicability of this formalism are restricted to a range of sizes L and ˜ such that L NN λ. frequencies ω 3.1. Corrections to the Planck and Stefan–Boltzmann laws
i
˜ i stands for the ith where λ = 2π = jkj represents the wavelength, ω ˜ satisfies Eq. (15). The second natural mode of the cavity and ρV ω −2 C term ρC = 3π 2 ⋅ vðT Þ is a curvature term, where C is the mean curvature and it satisfies C = ∬s ð1 = R1 + 1 = R2 Þdσ, R1 and R1 are the two principal radii of curvature of the cavity [12]. For a sphere of radius r0, it can be gotten easily that C = 4πr0. The third term is an oscillatory part, which is given by h i ˜ L cos ω⋅l ˜ p = vðT Þ + μp ; ˜ = ∑ DC Ap ω; ρ˜ ω p
! h i v2 ðT Þ v2−m = 2 ðT Þ ˜ P cos ω⋅l ˜ = ρV ω ˜ p = vðT Þ + μp ; ˜ 1 + a1 ρ ω + ∑ D A C ˜2 ˜ 2−m = 2 p L2 ω ωL
ð22Þ −2C 3L
where a1 = ð19Þ
is a dimensionless coefficient thath depends on the meani ˜ L = ðL =vðT ÞÞ1 + m = 2 ω ˜ m=2 curvature C∝L of the cavity, A˜ P = AP ω;
is the dimensionless amplitude corresponding to a single periodic
where the sum runs over the periodic orbits p of length of lp of the classical counterpart, and DC is a convergence factor with DC = zp, where z is a positive dimensionless quantity determined by the condition of a cavity and it satisfies z b 1, p is the number of vertices of a periodic orbit [13]. For electromagnetic waves the periodic orbits are the trajectories of the light rays inside the cavity as dictated by geometrical optics, and mirror reflection on surface S of a cavity will take once at each vertex of a periodic orbit. The amplitude place ˜ L and the Maslov index μp can be evaluated explicitly from the Ap ω; knowledge of the classical periodic orbits (refer to Ref. [15]). In 3D cavities, we can reach the following conclusion by the method of dimensionality analysis (refer to Ref. [13]) ˜ L ∝½L =vðT Þ1 + m = 2 ω ˜ m = 2; AP ω;
According to Eq. (18), it's clear that the finite deviations from the ˜ , then we can modify Planck law Eq. (15) only come from ρC and ρ˜ ω Eq. (21) as follows:
ð20Þ
where m is the degeneracy of the periodic orbits and it is associated with the symmetry of the cavity, for example, m = 3 for a sphere. ˜ L can also be influenced by the Moreover, the amplitude Ap ω; polarization effects and it picks up an additional factor of 2 for planar orbits [10]. In terms of classical optics, the existence of oscillatory term ˜ is easy to be understood as follows: Considering a periodic orbit ρ˜ ω which starts from r and comes back to r after one loop with multiple reflection, according to the value of frequency, one wave may enter in resonance with itself, and this situation thus corresponds to maxima of the density of modes. On the contrary another wave may interfere destructively with itself and such a situation leads to minima of the ˜ will arise for density of modes. Hence the oscillatory term of ρ ω different frequencies. Moreover for plane trajectories, in terms of polarizations of the transverse electromagnetic wave, it can be proved that only orbits with an even number of vertices contribute remarkably to the oscillatory term [10].
orbit of length lp. Consequently, by inserting Eqs. (16) and (22) ˜ =ρ ω ˜ ⋅ε ω ˜ = V, we can get the modified Planck law as into u ω follows: 1 ˜ ⋅ε ω ˜ ˜ + ρC + ρ˜ ω ˜ = ρ ω u ω V V =
1 + a1 ×
h i v2 ðT Þ v2−m = 2 ðT Þ ˜ p = vðT Þ + μp + 2−m = 2 ∑ DC A˜ P cos ω⋅l 2˜2 p ˜ L ω ωL
˜ 3 ðT Þ ℏ ω : ℏω ˜ ðT Þ = k B T π v ðT Þ e −1 2 3
!
ð23Þ
˜ firstly we must get these In order to fully determine uðωÞ, coefficients such as Ap and μp. In the case of symmetric cavities like spheres (m = 3) and rectangles (m = 2), these coefficients can be gotten easily (see below and Ref. [15] ). But for chaotic cavities (m = 0), these coefficients are difficult to be obtained [16]. In this paper we should only study the case of symmetric cavities. Secondly, we have yet to set a cutoff in the sum of Eq. (28). We define that lesc represents the length of a ray's orbit before the ray escapes through the aperture. From a physical point of view, only the periodic orbits which are shorter than the length lesc can make remarkable ˜ The explicit expression of the cutoff function contributions to uðωÞ. strongly depends on the symmetries of the cavity. For symmetric cavities (m = 3, 2), the cutoff function has power-law tails but for chaotic cavities it's quite complicated [17]. Accordingly, after inserting Eq. (23) into Eq. (17), the Stefan– Boltzman law can also be modified as follows:
∞
RðT Þ = ∫0 1 +
" #! 2 2−m = 2 ˜ p ω⋅l ˜ 3 ðT Þ a1 v ðT Þ v ðT Þ ℏ ω ˜ + μp dω: + 2−m = 2 ∑ DC A˜ P cos 2˜2 vðTÞ ˜ p 4π2 v2 ðT Þ eℏ ω˜ ðT Þ = k B T −1 L ω ωL
ð24Þ
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Q.-J. Zeng, Z. Cheng / Optics Communications 283 (2010) 2929–2934
˜ and considering the cutoff function By completing the integral of ω of the sum, we can get
about the parameters such as lp,t and m, we can expand the oscillatory ˜ as follows: term ρ˜ ω
RðT Þ = RV ðT Þ + RC ðT Þ + R˜ ðT Þ;
˜ = − ∑ DC′ ρ˜ ω
ð25Þ
˜ ˜ 3=2 3V ω 3V ω t ˜ = vðT Þ + sin 4r0 ω⋅t ∑ DC ð−1Þ 2 2 v ð T Þ v ð T Þ 2v ð T Þr tπ t =1 t;p N 2t 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sinðπt = pÞ 3 π 2ωpr ˜ 0 × sinð2πt = pÞ⋅ sinðπt = pÞ ; ð29Þ sin p + + vðT Þ 2 2 r0 pπ3
2
ðTkB Þ 1 where RV(T) = σ ’(T)T 4, RC ðT Þ = − C36L 3 ℏ ∝ L2 and it comes from the ˜ curvature term ρC in the spectral density. The fluctuating term RðTÞ can be approximatively written as follows in the limit lmin N N ℏvðTÞ , kB T where lmin is the length of the shortest periodic orbit, 2
vðT ÞkB T ℏv ðT Þ + b0 ; R˜ ðT Þ≈b1 L3 L4
ð26Þ
where D’C = z2t is a convergence factor. In addition, we can get ρC = −8r0 ˜ by Eq. 3πvðT Þ, and ρV ω is still (15). Then after substituting denoted ˜ into Eq. (23), we can get the ˜ , ρC and ρ˜ ω the expressions of ρV ω modified Planck law of a SKNB as follows: ˜ = u ω
where b0 and b1 are the dimensionless coefficients which depend on the number m. We will discuss the two coefficients of a spherical cavity (m = 3) separately in the next section. For m = 0, 2, we can get (refer to Ref. [14])
Ap;t ðLÞf μp ðm + 2Þπ −2πL 1 + m = 2 ∑ 2 + m = 2 ; b0 = − 4 vðTÞ p;t τp t
ð27Þ
Ap;t ðLÞg μp 2πL m = 2 ∑ 1 + m = 2 ; b1 = π vðT Þ p;t τ t
ð28Þ
(
˜ ˜ 2 ðT Þ 8r0 3ω ω ˜ = v ðT Þ − − ∑ D′ sin 4r0 ω⋅t π2 v3 ðT Þ 3VπvðT Þ t = 1 C 2v2 ðT Þr0 tπ2
˜ 3=2 ω 3 t ∑ D ð−1Þ + vðTÞ vðT Þ t;p N 2t C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
˜ 0 ˜ ðT Þ sinðπt = pÞ 3 π 2ωpr ℏω + sinðπt = pÞ ⋅ ˜ sin p + × sinð2πt = pÞ : 2 2 vðT Þ r0 pπ3 eℏ ωðT Þ = kB T −1
g
ð30Þ
Secondly, in the limit eℏ ω˜ ðT Þ = kB T ≫1 and lmin N N ℏvkBðTT Þ, by inserting ˜ the modified Stefan– Eq. (30) into Eq. (17) and integrating over ω, Boltzmann law can be obtained as, 4
RðT Þ = σ ′ ðT ÞT −
p
1 2 3 2 4 ðTkB Þ + b1 kB Tv = r0 + b0 ℏv = r0 ; 12r02 ℏ
ð31Þ
where
1 −3 3 . So Eq. (31) can be where b1 ≈ 0 and b0 ≈ 256π 2 ∑ 4 ≈1:2 × 10 t t reduced to
h i f μp = cos μp Þ + sin μp − cos μp m = 2;
RðT Þ = σ ′ ðT ÞT −
h i g μp = − sin μp − sin μp − cos μp m = 2:
The amplitude Ap,t describes the contribution of a single periodic orbit, and the single periodic orbit has a period τp = lp/v(T) which is repeated t times. Here we note that (a) the largest contribution to the sum comes from the shortest periodic orbits, and (b) for the shortest periodic orbits, the Maslov indices μp and amplitude Ap,t can be computed analytically [15].
4
1 2 −3 2 4 ðTkB Þ + 1:2 × 10 ℏv = r0 : 12r02 ℏ
ð32Þ
The second term of Eq. (32) corresponds to the curvature contribution RC(T) and the third term is the oscillating part R˜ ðT Þ. Eqs. (30) and (32) are the modified Planck law and Stefan– Boltzmann law, respectively, for a KNB. Here we note that since the sum in Eq. (30) is dominated by the contribution of the shortest periodic orbits, consequently the corrections above are quite insensitive to the cutoff lesc ≫ lmin. 3.3. Suitable conditions for experimental detection and numerical calculation
3.2. Corrections to laws of a spherical kerr nonlinear blackbody The case of a spherical Kerr nonlinear blackbody (SKNB) can be solved exactly because a spherical cavity has a high degree of symmetry. Similar studies of other kinds of the normal blackbody have been carried out in Ref. [14,18]. Here we shall only discuss the ideal spherical blackbody with a smooth interior surface. Under such a condition we can ignore the effects of small, nonoverlapping bumps on the surface, these effects will influence the corrections by suppressing periodic orbits which are longer than a certain cutoff, and the cutoff is determined by the bumps' size [19]. For this case the concerned closed periodic orbits are given by planar regular polygons, and these polygons have an even number of vertices [13]. Moreover, since nonpolarations and photons abide by the same reflection law, an identical polygon should lie in a plane which contains the diameter of the sphere. The length lp,t of the trajectories can be given by lp,t = 2pr0 sin (πt/p), where r0 represents the radius of the sphere, p is the number of vertices of a polygon and t is the number of turns around the origin of a specific periodic orbit. With these definitions, we can investigate the corrections to Planck and Stefan–Boltzmann laws. Firstly, in terms of the above discussions
In order to understand the significance of FSCs to the radiation laws ˜ and R(T). more clearly, we should make numerical calculation of u ω For this aim, we must determine some parameters such as the temperature T, the radius of a spherical cavity r0 and so on, these parameters have close relations to the experimental detection. Firstly, we discuss the optimal radius for experimental verification. In general, the larger the radius, the smaller the corrections are. In addition, it will be difficult to keep the temperature stable if the radius becomes larger. Whereas a cavity with too small radius is hard to be made with perfect geometrical shape, and the flux of energy escaping through the aperture of a cavity is too small to be measured with the necessary accuracy. Hence the optimal radius can be estimated to be r0 = 10– 70 mm [14] and we choose r0 = 25 mm as a tentative radius. Secondly, the suitable temperature T should be below the transition temperature Tc and easy to be maintained. Forγ = 0.9, the transition temperature Tc of a KNB is equal to 464.9 K. We pick up T ≤ 5 K, and can obtain that v(T) ≈ 0.9c and λm = 0.92 mm, where λm is the wavelength at which the ˜ is a maximum. Besides, this condition satisfies the energy density uðωÞ limit L ≫ λ well and the semiclassical formalism is accurate enough. Thirdly, in the actual numerical calculation of the summation in Eq. (30), we can only consider a finite number of values of indices t and p due to
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the existence of convergence factor DC′ and DC, and this method is in agreement with the method in Ref. [13]. Finally, for the experimental ˜ for ω ˜ in the formulas detection of FSCs, we should substitute ω = nvcðT Þ ω ˜ and R(T) before the numerical calculation. of uðωÞ For the convenience of numerical calculation, we take the diamond crystal whose zero-wave-vector frequency of the Raman-active mode is ωR = 2.51 × 1014 s− 1 and assume its refractive index n = 1. Then the variation with ω of u(ω) can be shown in Fig. 1, where we let T = 5 K and γ = 0.9. From this figure it is clear that FSC to Planck law can't be neglected, especially at the frequency ωm = 2.05 × 1012(Hz) where the energy density u(ωm) is a maximum. Furthermore, the variation with ω of u(ω) is also shown in Fig. 2 under the condition that T = 5 K and γ = 0.6. By contrast these two figures, we can see that the variation of parameter γ will influence FSC to the Planck law, exactly speaking, the smaller the parameter γ, the larger the deviations are. Next the variation with T and γ of RS(T) can be shown in Fig. 3, h i where RS ðT Þ = RðT Þ = RV ðT Þ = 1 + RC ðT Þ + R˜ ðT Þ = RV ðT Þ is the relative total energy emitted per unit of time and area. In addition, RS(T) of a normal blackbody is also shown in Fig. 3 in contrast with a KNB. From Fig. 3 we can find two features: (a) For fixed γ, RS(T) increases continuously as the temperature T increases from 0.5 K to 2.5 K. (b) For fixed T, RS(T) is a monotonically decreasing function of γ. 4. Discussions In this paper we first introduce the theory of a KNB which is brought forward and developed by Cheng. Next considering the fact that nonpolaritons behave similarly to the photons in some aspects such as propagation, reflection, interference and diffraction, we study the corrections of a finite-size cavity to the radiation laws of a KNB by using semiclassical techniques. We find the dominant corrections include two terms. One term, which is represented by ρc, is dominated by C which is the mean curvature of the cavity. Another term, which is ˜ , is an oscillatory part which is associated with the denoted by ρ˜ ω symmetry of the cavity and the closed classical trajectories. To sum up, both of the two terms come from the fact the distribution of wave vector k in a finite-size cavity is quite different from that in an infinitesize cavity. For the former case, the electromagnetic waves will experience multiple reflection, scattering, interference and diffraction ˜ . In which yield additional corrections to the density of energy u ω short, the corrections are determined by the size and shape of a cavity. In the limit L ≫ λ, by using semiclassical techniques, we get the universal expressions of FSCs to the radiation laws of a KNB. Espe-
Fig. 1. For fixed γ = 0.9, radius r0 = 25 mm and temperature T = 5 K, variation of u(ω) with ω, where u(ω) is the energy density per unit of frequency. The dashed and solid lines correspond to an ideal and an actual SKNB, respectively.
Fig. 2. For fixedγ = 0.6, radius r0 = 25 mm and temperature T = 5 K, variation of u(ω) with ω, where u(ω) is the energy density per unit of frequency. The dashed and solid lines correspond to an ideal and an actual SKNB, respectively.
cially, we have deduced the explicit expressions of a SKNB and made a numerical calculation. From these investigations we can find three features as follows: (a) For fixed temperature T, according to the numerical calculation of u(ω) and the discussions about Figs. 1 and 2, we can find that the dominant term of FSCs to Planck law is the oscillatory part ρ˜ðωÞ, and this term is a decreasing function of v(T) and γ (due to the fact that γ signifies the coupling strength between a bare photon and virtual nonpolar phonons, and v(T) is an increasing function of γ [4]). The decreasing of γ means that the coupling strength between a photon and phonons will be enforced and so more nonpolaritons will arise. This effect will lead to a result that both the density of energy uV(ω) and the fluctuation of u(ω) increase simultaneously, so ρ˜ðωÞ will increase with decreasing of γ. Besides, it also can be found that the relative fluctuation ΔuðωÞ = juðωÞ−uV ðωÞj = uV ðωÞ almost remains unchanged with decreasing of γ. (b) For fixed γ, RS(T) is an increasing function of temperature T. The basic physical mechanism about this phenomenon can be explained as follows: as the temperature T is increased, the velocity of a nonpolariton increases correspondingly, ˜ and so RV(T), RðTÞ and the absolute value of RC(T) will become larger simultaneously. Due to the fact that RC(T) is negative, the sum ˜ of RC(T) and RðTÞ will approach zero with increasing of T, and consequently RS(T) will increase and finally approach one. (c) For fixed T,
Fig. 3. For three values of γ of an actual SKNB and a normal spherical blackbody, variation of RS(T) with temperature T, where RS(T) is the relative total energy emitted per unit of time and area. The solid line corresponds to the normal blackbody.
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RS(T) is a decreasing function of γ. The reason is that with increasing of parameter γ, the nonpolar phonon weight in a nonpolariton is decreased but the velocity of a nonpolariton increases, so that the energy of the nonpolariton system and RV(T) becomes smaller, and R˜ ðT Þ becomes larger. At the same time, considering that RC(T) is a constant with a fixed temperature and RC ðT Þ + R˜ ðT Þ≤0, so RS(T) will become smaller with increasing of parameter γ. In addition, we can find that RS(T) of a KNB is larger than RS(T) of a normal blackbody, and the difference of RS(T) between a KNB and a normal blackbody becomes larger with decreasing of parameter γ, whereas the differences of RS(T) among all these blackbodies approach zero with increasing of the temperature T. The reason is that more and more virtual nonpolar phonons will evaporate when the temperature T approaches the transition temperature TC, and so the behavior of a nonpolariton is more and more like that of a bare photon. At T ≥ TC, the KNB behaves just as a normal blackbody. In conclusion, by using semiclassical techniques, we have obtained explicit expressions for FSCs of a KNB to the radiation laws, these corrections can be denoted as functions of the temperature, size and shape of the cavity. Moreover, we have investigated a SKNB in detail and discussed the suitable parameters which may be suitable to detect these corrections. This work may lay the foundation for experimental verification of a KNB and have a widely prospect of application.
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