L Peng

L Ran

N A Mortensen

The scattering of a cylindrical invisibility cloak: reduced parameters and optimization

Keywords: numbers: 41, 20, -q, 42, 25, Fx

We investigate the scattering of 2D cylindrical invisibility cloaks with simplified constitutive parameters with the assistance of scattering coefficients. We show that the scattering of the cloaks originates not only from the boundary conditions but also from the spatial variation of the component of permittivity/permeability. According to our formulation, we propose some restrictions to the invisibility cloak in order to minimize its scattering after the simplification has taken place. With our theoretical analysis, it is possible to design a simplified cloak by using some peculiar composites like photonic crystals (PCs) which mimic an effective refractive index landscape rather than offering effective constitutives, meanwhile canceling the scattering from the inner and outer boundaries.

Introduction

Keeping the form of Maxwell equations invariant, coordinate transformation (CT) provides us the possibility to control the electromagnetic field, by changing the distribution of permittivity and permeability, as well as to create an isolated space to hide any objects, i.e. invisibility cloaking [1,2]. In recent years, invisibility cloaking has been attracting significant attention [3,4,5,6,7,8,9,10,11,12], due to its novel electromagnetic (EM) characteristics and potential applications.

In general, the invisibility cloaks, realized through a CT approach, possess constitutive parameters being anisotropic and spatially varying. In particular, a 3D spherical invisibility cloak calls for constitutive parameters which are anisotropic and inhomogeneous, but with finite values, which seems realizable in practice [1]. However, in the 2D case, cylindrical invisibility cloaks commonly possess constitutives not only being anisotropic and spatially dispersive, but also requesting infinite components [4], which in practice leaves the realization of any ideal cloak impossible. To overcome this drawback, Schurig et al. suggested the simplification of the constitutive parameters, i.e. flexibly choosing the constitutives while keeping the spatial distribution of dispersion relations unchanged [4], which is based on the ray tracing concept. With such an approach, the first cylindrical cloak shell was successfully constructed.

Since we have to simplify the constitutives of a 2D cylindrical cloak in the practical design and fabrication, the scattering of the cloak itself is unavoidable, i.e. an invisibility cloak with reduced parameters is inherently visible [13]. In order to reduce the scattering of a cloak with reduced parameters, a variety of research work has been done. For instance, a second-order polynomial function was used in the transformation in order to avoid the reflection at the outer boundary [14], or to minimize the scattering of cloaks with reduced parameters by the optimized simplification [15,16,17]. Though people have achieved success in the invisibility cloak theory and experiments, a general description of the scattering by a cloak with reduced parameters is unrevealed.

In this paper, we analytically solve the scattering problem of a cylindrical invisibility cloak by applying the classical scattering theorem. With the aid of scattering coefficients, we show the full-wave solution of the scattering by cloaks with ideal as well as reduced parameters. According to our theoretical analysis, the scattering of a cloak with reduced parameters is contributed by both the inner and outer boundary, as well as the spatially varying internal impedance. We point out the way to minimize the scattering of a cloak in practice, in which case its scattering is dominated by the spatially distributed permittivity/permeability rather than the boundary conditions. In addition, the way to realize nearly zero scattering by an approximate cloaking shell is also proposed.

Scattering of cylindrical invisibility cloak

Consider the anisotropic and inhomogeneous invisibility cloaks obtained from a CT approach. If the mapping employed in the transformation is simple, i.e. only one coordinate variable is involved in the transformation like the ones proposed in Ref. [1] or Ref. [4], the cloak's ¯ and μ are diagonal and inhomogeneous. Though the analytical solution of the scattering problem of an ideal cloak can be found by applying the Maxwell equations, the scattered fields by those cloaks with reduced parameters have to be determined numerically [13]. However, the scattering by a multi-layered cloak shell can be found by applying the full wave expansion, satisfying the boundary conditions [18].

Suppose there is an N -layered cloak structure as shown in Fig. 1 (A). The inner and outer radius of this cloak shell are a and b, respectively. ρ j is the outer radius of the j-th shell, ρ 0 = a stands for the radius of the inner region, ρ N = b is the outer radius. Inside the j-th layer with thickness d j = ρ j -ρ j-1 , see Fig. 1, the permittivity and the permeability are constant and denoted by ¯ j and μj . If ¯ j and μj vary continuously in space, then a continuous cloak shell can be obtained if we let N → ∞ and max d j → 0 (j = 1, 2, 3...N ). For simplicity, the permittivity and permeability in each layer of the multi-layered cloak are assumed to be anisotropic but diagonal, e.g. ¯ j / 0 = diag r ρ,j , r φ,j , r z,j . Here, we consider the TM polarization case, i.e. the electric field is polarized along z direction, while the same procedure can be applied to a TE polarization case.

Basically, the field inside each layer can be easily expressed by a superposition of vector cylindrical waves [START_REF] Chew | Waves and fields in inhomogeneous media[END_REF]18]. We rewrite the field inside the j-th layer as

E z,j = ∞ m=-∞ [C j 1,m J ν j (k ρ,j ρ) + C j 2,m H (1) ν j (k ρ,j ρ)]e imφ , (1)
with ν j = m(k ρ,j /k φ,j ), k ρ,j = k 0 r z,j µ r φ,j , k φ,j = k 0 r z,j µ r ρ,j , and k 0 = ω √ 0 µ 0 . Here, C j 1,m and C j 2,m are the unknown expansion coefficients inside the j-th layer which need to be determined by matching the boundary conditions. In the outer space, C N +1 2,m are used to represent the total field in the outer free space, composed of the incident and the scattered fields. In Eq. ( 1), the first term represents the standing wave, while the second one is the scattered wave in each layer [START_REF] Chew | Waves and fields in inhomogeneous media[END_REF]. Boundary conditions are applied to determine the unknown coefficients. At the interface between the j-th and the j + 1-th layers, tangential electric and magnetic fields should be continuous, hence we find

C j+1 1,m C j+1 2,m = P -1 m,j+1 Qm,j C j 1,m C j 2,m , (2)
with

P m,j+1 =   J ν j+1 (k ρ,j+1 ρ j ) H (1) ν j+1 (k ρ,j+1 ρ j ) k ρ,j+1 µ r φ,j+1 J ν j+1 (k ρ,j+1 ρ j ) k ρ,j+1 µ r φ,j+1 H (1) ν j+1 (k ρ,j+1 ρ j )   , (3)
Qm,j =   J ν j (k ρ,j ρ j ) H (1) ν j (k ρ,j ρ j ) k ρ,j µ r φ,j J ν j (k ρ,j ρ j ) k ρ,j µ r φ,j H (1) ν j (k ρ,j ρ j )   . (4)
Similarly, we consider the boundary conditions at any other interface, and the two unknown coefficients in the free space can be evaluated by

P m,N +1 Q m,N +1 = N j=1 P -1 m,j+1 Qm,j C 1 1,m C 1 2,m . (5)
Equation ( 5) is the common relationship that a multi-layered cloak should satisfy, with which the scattered field of a cloak consisting of several layers can be conveniently worked out [18]. However, since the 2D invisibility cloak requires that k φ,j tends to zero near the inner interface, then ν j is singular which makes both T 1,m and T 2,m being ill-conditioned and the numerical evaluation of Eq. ( 5) fails.

However, since our purpose is to find out the scattered field, the scattering coefficients are of course very helpful and simplifying the formulation may also overcome the drawback of Eq. ( 5). The scattering coefficient for the m-th cylindrical wave in the j-th layer is defined as R j m = C j 2,m /C j 1,m . From Eq. ( 2), we easily arrive at

R j+1 m = A j m + B j m R j m C j m + D j mR j m , (6)
where A j m , B j m , C j m , and D j m are functions of ν, k ρ , k φ in both the j-th and j + 1-th layers. If we further assume that all the layers are very thin, and let N → ∞ and max d j → 0, i. e. the cloak structure gradually tends to be continuous, we can rewrite

R j+1 m , R j m , A j m , B j m , C j m and D j m as R m + ∆R m , R m , A m + ∆A m , B m + ∆B m , C m + ∆C m and D m + ∆D m , respectively. Hence we find ∆R m = A m + ∆A m + (B m -C m + ∆B m -∆C m )R m -(D m + ∆D m )R m 2 C m + ∆C m + (D m + ∆D m )R m . (7)
Balancing the first order differentials in Eq. ( 7), we get

R m (ρ) = πρ 2i A m + (B m -C m )R m (ρ) -D m R 2 m (ρ) , (8)
with

A m = -J ν (k ρ ρ) k ρ k ρ ρJ ν (k ρ ρ) + k ρ J ν (k ρ ρ) + k ρ ν ∂ ∂ν J ν (k ρ ρ) +J ν (k ρ ρ) k ρ k ρ ρJ ν (k ρ ρ) + k ρ µ φ µ φ J ν (k ρ ρ) + k ρ ν ∂ ∂ν J ν (k ρ ρ) , B m = -H (1) ν (k ρ ρ) k ρ k ρ ρJ ν (k ρ ρ) + k ρ J ν (k ρ ρ) + k ρ ν ∂ ∂ν J ν (k ρ ρ) +H (1) ν (k ρ ρ) k ρ k ρ ρJ ν (k ρ ρ) + k ρ µ φ µ φ J ν (k ρ ρ) + k ρ ν ∂ ∂ν J ν (k ρ ρ) , C m = J ν (k ρ ρ) k ρ k ρ ρH (1) ν (k ρ ρ) + k ρ H (1) ν (k ρ ρ) + k ρ ν ∂ ∂ν H (1) ν (k ρ ρ) -J ν (k ρ ρ) k ρ k ρ ρH (1) ν (k ρ ρ) + k ρ µ φ µ φ H (1) ν (k ρ ρ) + k ρ ν ∂ ∂ν H (1) ν (k ρ ρ) , D m = H (1) ν (k ρ ρ) k ρ k ρ ρH (1) ν (k ρ ρ) + k ρ H (1) ν (k ρ ρ) + k ρ ν ∂ ∂ν H (1) ν (k ρ ρ) -H (1) ν (k ρ ρ) k ρ k ρ ρH (1) ν (k ρ ρ) + k ρ µ φ µ φ H (1) ν (k ρ ρ) + k ρ ν ∂ ∂ν H (1) ν (k ρ ρ) ,
For more details of the technical work, please refer to the appendix. Containing the term involving R 2 m (ρ), Eq. ( 8) becomes the non-linear Riccati equation for continuous invisibility cloaks. It could be solved numerically with initial value at the boundaries.

To find out the scattering coefficients in the outer free space, we should find out the scattering coefficients on the inner interface first. For the ideal cloaks, the permittivity/permeability on the inner boundary is infinite. This prevents us from the evaluation of the scattering coefficients, hence we have to assume that the cloak's inner radius has a small perturbation δ, i. e. the inner radius is now a + δ but the rest of the parameters are kept unchanged. Notice that there is no scattered field inside the inner region (ρ < a + δ), then the scattering coefficients on the new inner boundary can be evaluated by applying Eq. ( 2)

R m (a+δ) = η a+δ J m [k i (a + δ)]J ν a+δ [k a+δ,ρ (a + δ)] -J m [k i (a + δ)]J ν a+δ [k a+δ,ρ (a + δ)] η a+δ J m [k i (a + δ)]H (1)
ν a+δ [k a+δ,ρ (a + δ)] -J m [k i (a + δ)]H (1)
ν a+δ [k a+δ,ρ (a + δ)] , (9)
with η a+δ = a+δ,z /µ a+δ,φ and k i being the wave number in the region ρ < a + δ. The initial scattering coefficients of any cloak structure can be obtained by letting δ → 0 in Eq. (9). For example, for ideal cloaks, k a+δ,ρ µ a+δ,φ → 0 and ν a+δ → 0 lead to R 0 (a + δ) → -J 0 (ka,ρa)

H (1) 0 (ka,ρa)
and R m (a + δ) → 0(m = 0).

To verify our theoretical formulation, here we assume there is a cylindrical invisibility cloak with a=0.024m, b=0.072m and its constitutive parameters ¯ r = μr = diag ρ-a ρ , ρ ρ-a , ( b b-a ) 2 ( ρ-a ρ ) . By using Eqs. ( 8) and ( 9), the scattering coefficients can be conveniently evaluated. Fig. 2 (A) shows the scattering coefficients for the lowest four cylindrical waves (m=0,1,2,3). From Fig. 2, we see that all the scattering coefficients (|R m |) are zero on the outer boundary. Since an ideal cloak perfectly matches the outer space, no scattered wave will be excited. Likewise, the inner boundary, |R m | is also zero, except the 0-th order case. Though non-zero R, no electric field can be excited inside the inner region [1], which can also be proved by applying Eq. ( 2). In Fig. 2 (A), R 0 varies fast near the inner boundary, hence the total scattered field is sensitive to any perturbation of the inner boundary, as reported in Ref. [START_REF] Ruan | [END_REF].

Cloaks with simplified parameters

In the previous section, we have obtained the scattering coefficients for an arbitrary invisibility cloak structure with cylindrical symmetry. However, the permittivity and permeability of ideal cloaks are not only anisotropic and inhomogeneous, but also have infinite components ( φ and µ φ ) on the inner boundary [4]. Hence, in order to realize the practical design and fabrication, we have to simplify the constitutive parameters, i. e. maintain the dispersion relation distribution, while releasing our choice of and µ to a more realistic set [4,21,14,15]. Unfortunately, this simplification generally affects the scattering coefficients, i.e. the mismatched inner and outer boundaries, as well as the variation of µ φ (for TM case), will cause non-zero scattering coefficients, which we see below in detail. In Ref. [14], to minimize the scattered field after the simplification, high order coordinate transformation was applied, for the purpose of impedance matching on the boundary ρ = b. While the inner boundary is usually assumed to be perfectly conducting [5,14,18], by doing this the inner region is isolated from the outer space. However, the 0-th order cylindrical wave can still see the inner region and detect the perfect conducting wall [13]. Although with the help of some numerical method like genetic algorithm (GA) we can optimize the scattered field by a multi-layered cloak [18], the physical interpretation of the scattering is missing. Returning to Eq. ( 9), the initial value of R is commonly determined by both the cloak and the object placed inside. However, we notice that R(a) can be independent of the permeability and keeps zero if k a,ρ = 0, which means that the scattering of a cloak will be irrelevant to the inner objects. Again, in Eq. ( 8), the alternative µ φ after the simplification may introduce the disturbance of scattering coefficients through the term containing µ φ /µ φ . To minimize the contribution from this term, the other terms should dominate, i.e. k ρ can not be zero.

If the coordinate transformation is ρ = f (ρ), φ = φ and z = z, all the above restrictions can be expressed as f (b) = 1, f (a) = 0, f (ρ) = 0, with natural conditions f (a) = 0 and f (b) = b. Any functions satisfying these restrictions can be applied to the cloak design. Here, a third-order polynomial function is adopted, i.e. ρ = Aρ 3 +Bρ 2 +Cρ+D with A = -a+b (b-a) 3 , B = 1 2(b-a) -3(a+b) 2 A, C = 1-3Ab 2 -2bB, and D = -(Aa 3 +Ba 2 +Ca). We should notice that the choice for higher-ordered polynomial function is not unique, which indicates that the optimization might be accomplished by using some numerical method like GA. Now we address some numerical calculations to verify our theoretical analysis. We suppose an infinitely long cylinder with radius r=24mm made of PEC is placed in free space. The scattering coefficients of this PEC cylinder under TM polarized wave illuminating are listed in the second column of Tab. 1, while the near field distribution is shown in Fig. 2 (B).

Next we assume that a cylindrical cloak with a=24mm and b=72mm is applied to hide the specified PEC cylinder. First, we apply a linear function to complete the CT, as previously stated. In practice, the constitutive parameters have to be simplified. Here we make µ φ =1 while we tune the other parameters correspondingly [4]. The near field distribution of the scattering by this simplified cloak is shown in Fig. 3 (A), while the scattering coefficients for the lowest four order cylindrical waves are listed in the third column in Tab. 1. We clearly find that the non-zero scattering coefficients make the cloak visible [13]. Comparing the second and the third columns in Tab. 1, we see that though with non-zero scattering coefficients, this simplified cloak has a larger size but less scattering than the PEC cylinder.

Second, a third-order polynomial function, which can be uniquely determined by the above analysis, is applied in the CT. Though the cloak still needs simplification, its scattering is only caused by the alternated µ φ , see Eq. ( 8), hence we would expect the reduction of all the scattering coefficients. We still make µ φ =1 after the simplification. The total field distribution is shown in Fig. 3 From Fig. 3 and Tab. 1, a simplified cloak structure related to a third-order polynomial CT has less scattering than the one related to a linear CT. However the scattering contributed from the µ φ term in Eq. ( 8) is still unavoidable. In fact, by applying high-order polynomial transformation functions and some numerical optimization, the cloak's parameters can be optimized to minimize the scattering involved by the µ φ term. Here, we point out one possible analytical solution.

We go to Eq. ( 8) and if the term containing µ φ /µ φ is zero, then the scattering of the cloak is dominated by the spatially dispersive refractive index. For this purpose, we have µ φ /µ φ = 0, or ρ = b 1-X ρ X , with X being determined by boundary conditions. Rigorously speaking, functions of this kind can not become the transformation function since ρ = 0 if ρ = 0. However, properly choosing the two unknown coefficients can still support an approximate cloak shell [22]. An approximate cloak shell coming from this kind of transformation function will possess constitutives as ¯ r = μr = diag X -1 , X, Xb 2-2X ρ 2C-2 , i.e. no singularity existing in both the permittivity and the permeability. Since this kind of invisibility cloak shell has been reported in Ref. [22], we do not place too much discussion here. The scattering coefficients are summarized in the last column in Tab. 1 and the electric field distribution is given in Fig. 3 (C). We see that the approximate cloak shell scatters little EM power, which in principle "shrinks" the non-zero scatter [22].

Conclusion

In conclusion, we have solved the scattering problem of 2D cloak structures with cylindrical symmetry, for both the ideal and simplified cases. We show that the scattering coefficients of a cloak are determined by both the inhomogeneous refractive index and the inhomogeneous µ φ ( φ ), as well as the inner and outer boundary conditions. In the practical fabrication, to avoid the infinities of an ideal cloak, the cloak's constitutive parameters have to be simplified, which commonly makes an otherwise invisible cloak visible.

We point out the way to avoid the scattering from both the outer and the inner boundaries, i.e. to have the outer boundary matched to the free space and make the refractive index tend to zero to minimize the scattering from the inner boundary, e.g. high-order polynomial CT can be applied to the cloak design. However, though numerical results show the reduction of scattering, the third-order polynomial based CT can not guarantee the zero scattering due to the changing of µ φ after the simplification. Some numerical method like GA can be adopted to further optimize the scattering of a simplified cloak. Our theoretical work shown in this paper clearly exhibits the role of the inhomogeneous refractive index. With minimizing the scattering involved by the µ φ term and the two boundaries, dielectric metamaterials and photonic crystals, which prefer effective spatial dispersion rather than effective constitutive parameters, could be conveniently applied to construct an invisibility cloak [23,24,25], in which case the topology optimization may stimulate the practical design and fabrication [START_REF] Sigmund | AIP Conf. Proc[END_REF]. Our theoretical work in this paper would improve the practical fabrication and application of invisibility cloaks in future.

In deriving Eq. (A.3), we use the Wronskian of Hankel functions [START_REF] Abramowitz | Handbook of mathematical functions[END_REF] H (1) v (x)J v (x) -J v (x)H Balancing the first order differentials in Eq. (A.5), we can get Eq. ( 8).

Figure 1 .

 1 Figure 1. Configuration (A) and Cross section (B) of a multi-layered cylindrical cloak.

Figure 2 .

 2 Figure 2. (Color online) (A) The absolute scattering coefficients for an ideal cloak with linear transformation function. (B) The total near field distribution with a bare perfect electric conducting (PEC) cylinder scattering. (C) The near field distribution with a PEC cylinder covered by an ideal cloak. In the calculation, operation frequency is 7GHz.

  (B), while the lowest four scattering coefficients are listed in the fourth column in Tab. 1. We see that though the scattering coefficients are non-zero again, this cloak scatters less EM power than the one shown in

Figure 3 .

 3 Figure 3. (Color online) Total electric field distribution for a PEC cylinder covered by (A) a simplified cloak coming from a linear CT; (B) a simplified cloak coming from a third-order polynomial CT; (C) an approximate cloak shell. (D) The magnitude of the scattered electric field on the virtue contour C (ρ=0.084m). Due to the symmetry, here we only show the distribution in 0 < φ < π. In all of the calculation, f =7GHz.

Fig. 3 (

 3 Fig. 3 (A).From Fig.3and Tab. 1, a simplified cloak structure related to a third-order polynomial CT has less scattering than the one related to a linear CT. However the scattering contributed from the µ φ term in Eq. (8) is still unavoidable. In fact, by applying high-order polynomial transformation functions and some numerical optimization, the cloak's parameters can be optimized to minimize the scattering involved by the µ φ term. Here, we point out one possible analytical solution.We go to Eq. (8) and if the term containing µ φ /µ φ is zero, then the scattering of the cloak is dominated by the spatially dispersive refractive index. For this purpose, we have µ φ /µ φ = 0, or ρ = b 1-X ρ X , with X being determined by boundary conditions. Rigorously speaking, functions of this kind can not become the transformation function since ρ = 0 if ρ = 0. However, properly choosing the two unknown coefficients can still support an approximate cloak shell[22]. An approximate cloak shell coming from this kind of transformation function will possess constitutives as ¯ r = μr = diag X -1 , X, Xb 2-2X ρ 2C-2 , i.e. no singularity existing in both the permittivity and the permeability. Since this kind of invisibility cloak shell has been reported in Ref.[22], we do not place too much discussion here. The scattering coefficients are summarized in the last column in Tab. 1 and the electric field distribution is given in Fig.3 (C). We see that the approximate cloak shell scatters little EM power, which in principle "shrinks" the non-zero scatter[22].

  Eq. (A.2), we get∆R m (C m +∆C m +∆D m R m ) = ∆A m +(∆B m -∆C m )R m -∆D m R m 2 .(A.5)

Table 1 .

 1 The absolute scattering coefficients for the lowest four cylindrical waves are shown. In the calculation, f =7GHz. |R c |: the bare PEC cylinder case; |R l |: PEC cylinder covered by a simplified cloak from linear CT; |R 3ord |: PEC cylinder covered by a simplified cloak from a third-order polynomial CT; |R app |: PEC cylinder covered by an approximate cloak shell.

	m |R c |	|R l |	|R 3ord | |R app |
	0	0.9036 0.801 0.354	0.1187
	1	0.3004 0.197 0.135	0.0068
	2	0.9934 0.082 0.031	0.0066
	3	0.7418 0.245 0.060	0.0120

Acknowledgments

This work was financially supported by the Danish National Advanced Technology Foundation (No. 004-2007-1), the Sino-Danish Scientific and Technological Cooperation, Danish Agency for Science, Technology and Innovation (No. 09-075623), the National Science Foundation of China (NSFC) (No. 61071063), 863 Project (No. 2009AA01Z227) and NCET-07-0750.

Appendix A. The derivation of Eq. (8)

We rewrite Eq. ( 2) as

and after some algebra operation we find

H (1) ν j+1 (k ρ,j+1 ρ j )J ν j (k ρ,j ρ j ),

From Eq. (A.1), we can derive Eq. ( 6). Furthermore, assuming that all the cylindrical shells are thin, i.e. max d j → 0, we can rewrite

The ∆ sign appears in those terms involving ∆ρ = ρ j+1 -ρ j . Thus Eq. ( 6) has a new form as Eq. ( 7) which is rewritten here


D Dobrovolskas
email: darius.dobrovolskas@ff.stud.vu.lt

J Mickevi Ius

E Kuokštis

G Tamulaitis

M Shur

M Shatalov

J Yang

R Gaska

Confocal spectroscopy of InGaN LED structures

Photoluminescence of InGaN structures for green light emitting diodes with multiple quantum wells as an active medium was studied with spatial and spectral resolution using confocal microscopy. Bright spots of ~200 nm in diameter were observed. Emission from these bright areas was up to 8 times more intense than from the rest of the sample surface and the band peak position in these areas was blueshifted in respect to the band position in the background surface of lower photoluminescence intensity. The data on emission properties in bright and dark areas and the dependence of these properties on the excitation power density are interpreted by assuming inhomogeneous distribution of defects acting as nonradiative recombination centers.

Introduction

Localization and inhomogeneous spatial distribution of nonequilibrium carriers (excitons) in InGaN in blue and white light emitting diodes (LEDs) and blue laser diodes play a crucial role in the light emission of InGaN epilayers and heterostructures [1] but peculiarities of these processes are still not fully understood.

Cathodoluminescence (CL) [2,3,4,5], µ-PL [6,7,8], confocal spectroscopy [9,10,11], and scanning near field optical microscopy (SNOM) [16,12,13,14] all reveal inhomogeneous photoluminescence intensity distribution in InGaN epilayers and QWs. Many publications [15,16,17,18] point out to the Confidential: not for distribution. Submitted to IOP Publishing for peer review 19 January 2011 well width fluctuations as the main origin of inhomogeneous spatial PL distribution in InGaN QWs, and dislocations are claimed to have a significant effect on the local emission efficiency [14,19].

Localization of excitons on a nanometer scale might be also caused by the localization of the hole wave functions around indium in randomly formed In-Ga-In chains, as was theoretically predicted for cubic InGaN [20].

In InGaN quantum wells (QWs), optical transitions and carrier dynamics are affected by the quantum-confined Stark effect (QCSE) due to the built-in electric field, which can be screened by nonequilibrium carriers [11,21], and by filling-in of the localized states that often have a similar effect on the InGaN QW characteristics.

The present paper reports on the study of spatial distribution of photoluminescence (PL) in LED structures with InGaN multiple quantum wells (MQWs) as an active region. PL spectroscopy in the confocal mode was employed and revealed unusual features of the inhomogeneous PL spatial distribution. For our sample, the strongly pronounced PL features cannot be interpreted as caused by localization and field-induced effects.

Experimental

The green LED structure under study was grown on the c-plane sapphire by metal-organic chemical vapor deposition. Trimethyl indium (TMI), trimethyl gallium (TMGa) and ammonia (NH 3 ) were used as In, Ga and N sources, respectively. The structure consisted of a 0.3 µm thick AlN buffer layer, 3 µm thick Si-doped n-GaN layer, InGaN compliance layer, 5-period InGaN/GaN MQW active layer, 20 nm thick p-AlGaN electron block, and 0.2 µm thick p-type contact layer. The active layer contained 2 nm wide In 0.27 Ga 0.73 N wells and 5 nm wide GaN barriers.

The PL intensity mapping and study of spatial variations in PL spectra on submicrometer scale have been performed using WITec confocal microscope Alpha 300 S. Objective with numerical aperture NA = 0.9 was used ensuring in-plane spatial resolution of ~200 nm. For spectral resolution, the microscope was coupled by an optical fiber with a Cromex spectrometer followed by a thermoelectrically cooled CCD camera. A CW He-Cd laser emitting at 442 nm was employed for the excitation. All the experiments have been carried out at room temperature.

Excitation power density dependence of the relative quantum efficiency of spatially-integrated PL in a wide dynamic range have been carried out under quasi-steady-state excitation by the third harmonic of Q-switched YAG:Nd laser radiation (3.49 eV).

Results

The mapping images of spectrally integrated PL intensity are presented in figure 1(a). The figure reveals a strongly inhomogeneous spatial distribution of emission intensity in the MQW plane: regions of stronger emission are evident on the field of lower emission intensity. The bright spots in the mapping image typically have a round shape of ~200 nm in diameter (close to the spatial resolution of the confocal microscope used). The PL intensity in the bright spots exceeds the background intensity by a factor of up to 8.

The spectral PL distribution in the MQW plane was studied by comparing the emission spectra in small spots (50×50 nm, 5×5 pixels) picked up from the regions of low and high emission intensity (dark and bright areas, as referred to further in this paper). Several of these PL spectra are presented for bright and dark areas in figure 1(b) and (c), respectively. The spots, from which the corresponding spectra were acquired, are marked and labeled in the mapping images [figure 1 The spatially-integrated dark area band peak is redshifted by 10 nm in respect to the band position in the bright areas. The comparison of the PL bands recorded from different spots shows that this redshift and broadening of the spatially-integrated PL band in the dark areas is caused by a lower intensity of PL emitted from spots corresponding to shorter wavelengths, while the emission intensities in the long wavelength region are similar in the dark and bright areas (see figure 2).

To gain more data on possible mechanisms of the inhomogeneous PL intensity distribution, we studied the PL properties of our sample at different nonequilibrium carrier densities. Figure 3 presents the PL spectra measured in a dark area at different excitation power densities. The spectrum is basically dominated by the single band, which is peaked at ~500 nm and shows a considerable band shift with increasing excitation. In addition to the prevailing band, another PL band [peaked at considerably longer wavelengths (~560 nm)] was observed in the dark areas, but its intensity saturated with increasing excitation. This PL band probably involves defect-related states. No long-wavelength band was observed in the bright areas.

The excitation power density dependences of the peak position of the main PL band for the bright and dark areas are compared in figure 4(a). The band peak in the dark areas gradually blueshifts by ~25 nm when the excitation power density is increased from 4×10 -3 to 2.4 MW/cm 2 . A striking feature is that the band peak in the bright areas at low excitation power densities is at a shorter wavelength than that in the dark areas. With increasing excitation intensity, the PL band in bright areas first blueshifts slower than that in dark areas but the shift rate becomes similar in both areas at high excitation intensities.

The excitation power density dependences of the spectrally integrated PL intensity in bright and dark areas are presented in figure 4(b). The dependence is linear at low excitation, but becomes superlinear at elevated excitation power densities. To study the PL efficiency dependence on excitation power density in a wider dynamic range, experiments have been performed under quasisteady-state pulsed (pulse duration 4.4 ns) [figure 4(c)]. Note that the photon energy in quasi-steadystate pulsed excitation (3.49 eV) was high enough to excite the 5 nm wide barriers, while in CW excitation (photon energy 2.81 eV) only the 2 nm-wide quantum wells were excited. Thus, at the same excitation power density, the expected carrier density under the pulsed excitation is larger than that under selective excitation approximately by a factor of 3. Moreover, the small excess energy of the photoexcited carriers might also diminish the spillover effect, which probably is one of the mechanisms causing the efficiency droop effect [22]. Comparison of figure 4(b) and figure 4(c) shows that the droop is observed at a higher carrier density than that achieved in our spatially-resolved experiments.

Discussion

The PL band shift with increasing excitation intensity in InGaN QWs is usually explained by two effects: gradual population of the lowest localized states and screening of the built-in electric field.

Both effects result in a blue shift of the PL band and their contributions are difficult to distinguish.

We calculated the energy levels in quantum well with a built-in electric field by simultaneously solving the stationary Schrödinger equation and the Poisson equation describing the asymmetric space-charge distribution in the well. Using material parameters taken from Ref. [23] and approach suggested in Ref. [24] results in the built-in field at GaN/In 0.27 Ga 0.73 N of 3.4 MV/cm. Such a field shifts the energy of the lowest optical transition by ~200 meV. The built-in field of 1.5 MV/cm, which is a typical low-bound value estimated for similar GaN/InGaN interfaces [25], results in a shift of ~60 meV. Carrier densities sufficient to cause a significant blue shift of the lowest optical transition due to screening of the built-in field are lower in wider QWs. In the 2 nm-wide well under study, a blue shift of 10% from the entire QCSE shift occurs at densities of ~2×10 12 cm -2 . The estimation of the carrier density in our PL experiments is obscured by uncertainty in the carrier lifetime, which locally depends on the carrier localization depth and on the carrier density. Assuming the lifetime to be 1 ns, the excitation power densities in our experiments (from 4 kW/cm 2 to 2.4 MW/cm 2 ) correspond to the carrier densities ranging from 2×10 10 cm -2 to 10 13 cm -2 . Thus, the faster blue shift of the PL band at the highest excitation power densities in our experiments should be influenced by the screening effect but the lowest excitation intensities used hardly influence screening. This conclusion is supported by the excitation power density dependence of the PL intensity [see figure 4(b)], which is linear at a low excitation but becomes superlinear at the excitation power densities exceeding approximately 0.1 MW/cm 2 . There is no return of the linear dependence, which is expected after the built-in field is completely screened. This might be an indication that the built-in field is not completely screened even at the highest excitation power densities used in our experiments. On the other hand, the superlinear growth of PL intensity at increasing excitation power density might also be expected due to the saturation of nonradiative recombination centers.

Localization and inhomogeneous spatial carrier distribution due to the fluctuations in indium content and/or well width definitely plays an important role in the samples under study. It is usually assumed that the deeper carrier (exciton) localization (corresponding to a lower energy of the photon emitted via radiative recombination in the localized states) results in a higher PL intensity due to decreased probabilities of carrier hopping and reaching nonradiative recombination centers. Thus, the inhomogeneous PL intensity distribution could be explained by inhomogeneous distribution of deep 6 localized states. Contrary to this assumption, the PL peak position in the bright areas of the sample under study is higher in energy than in the dark areas.

The difference in In x Ga 1-x N content with x ≈ 0.27 corresponding to the PL band shift observed (~10 nm) would be rather small (of the order of x = 0.01) and hardly could cause the observed increase in PL intensity by a factor of 8 due to possibly higher structural quality in the areas with lower In content.

The bright spots could have been interpreted as regions with narrower well width, where the blue shift of the PL band is stronger and the PL intensity is higher due to stronger overlap of the electron and hole wavefunctions. Though qualitatively consistent with experimental results, this effect is expected to be considerably weaker than observed. At low carrier density, the square of the wavefunction overlap for 1.75 nm wide well exceeds that for 2.25 nm well by a factor of two. This difference will be even lower due to stronger screening in the thicker regions, where a higher carrier density should be expected. Such difference is insufficient to explain the observed intensity difference of up to the factor of 8.

The lower PL intensity in areas where emission occurs at longer wavelength might have been explained by an inhomogeneous spatial distribution of the built-in field: a higher field causes a larger PL band redshift due to QCSE and, hence, a smaller overlap of the electron and hole wavefunctions and, consequently, a smaller rate of radiative recombination and a lower emission intensity. However, according to our calculations discussed above, this effect cannot explain such a big difference in the observed PL intensity in the bright and dark areas. The difference of ~25 nm observed for PL band peak positions in bright and dark areas at low excitation power density [see figure 4(a)] corresponds to variation of the built-in field by ~30%. The built-in field of 3.4 MV/cm causes a decrease of the squared electron and hole wavefunction overlap in a 2 nm QW down to 25% of the value for a rectangle QW (with zero built-in field). Variation of the field by 30% results in 1.5 times change, in the PL intensity due to the overlap change, which is much smaller than the observed factor of 8.

We suggest that the inhomogeneous PL intensity distribution observed in the structure under study can be explained by an inhomogeneous distribution of nonradiative recombination centers, as previously reported in [26]. The probability of nonradiative recombination is higher for the carriers (excitons) occupying the states with smaller localization energy. These excitons, which are responsible for the short-wavelength side of the PL band, have a higher probability to reach nonradiative recombination centers by hopping. Thus, the nonradiative recombination suppresses emission mainly on the high energy side of the PL band resulting in a redshift. The higher is the density of nonradiative recombination centers, the stronger is the redshift. Thus, the PL band in the dark areas is at longer wavelength than that in the bright ones. This interpretation is strongly supported by the data presented in figure 2. As seen, the intensity is equal both in bright and dark areas at the long-wavelength tail of the PL band, but the intensity in bright areas is progressively higher at the shorter wavelengths.

As the carrier density increases, more localized states get occupied. This effect results in a blue shift, which compensates the red shift caused by the nonradiative recombination centers. The blue shift due to the state filling is stronger in the spectral region corresponding to the tail of the density of the localized states and becomes less pronounced in the spectral region corresponding to a higher density of states. The results presented in figure 4(a) are consistent with this interpretation: the PL band blue shift at increasing excitation power density is more pronounced in the dark areas. As discussed above, the overall blue shift, which is observed in the bright areas, should be influenced by screening of the built-in field.

The conclusion that the inhomogeneous PL spatial distribution is strongly affected by an inhomogeneous distribution of the nonradiative recombination centers is indirectly supported by observation that the additional long-wavelength emission band is present in the dark but not in the bright areas. As shown in Ref. [27] for GaN, the densities of nonradiative recombination centers and the centers of radiative defect-related recombination (causing yellow luminescence) are correlated.

Provided that a similar conclusion is valid also for InGaN, the presence of the long-wavelength PL band in dark areas is an indication that these areas contain also a higher density of nonradiative recombination centers. The unusual PL characteristics and their dynamics under increasing excitation power density are consistent with the assumption that this sample has inhomogeneous distribution of nonradiative recombination centers. This feature can be favorable for higher emission efficiency. In the areas with a lower density of nonradiative recombination centers, the carrier density is higher and screens the built-in field more effectively.

Spectrally-integrated PL intensity dependence on carrier density [see figure 4(b)] is the same in bright and dark areas but a higher excitation power density is necessary to reach the same carrier density in the dark areas due to a higher density of nonradiative recombination centers.

The origin of inhomogeneous distribution of nonradiative recombination centers is unclear. It is worth noting, however, that the structure under study contains 5 QWs. The spatial resolution in the Z direction (perpendicular to the layers) is 0.75 µm, i.e. includes all the QWs. Hence, the bright areas vertically coincide within ~200 nm in all five QWs. Otherwise, the image would be washed out, and no distinct bright areas of ~200 nm in diameter would be observed. Such a vertical alignment implies possible involvement of dislocations propagating vertically though the QWs. At dislocation density of 10 9 cm -2 , the 2×2 µm sample area depicted in figure 1(a) should contain 40 dislocations on average. It can be expected that these dislocations concentrate in the dark areas and are absent in the bright areas.

To prove the model of inhomogeneous distribution of nonradiative recombination centers and to reveal the origin of the centers, results on microstructural analysis could be helpful and will be published elsewhere.

Conclusions

Confocal microscopy reveals spots of a more intense photoluminescence (~200 nm in diameter) in green LED structures with In 0.27 Ga 0.73 N MQWs as an active region. The spectral features in the dark and bright areas and dynamics of the photoluminescence characteristics under increasing excitation power density point out to an inhomogeneous distribution of nonradiative recombination centers affecting the carrier density distribution and screening of the built-in field in the QWs leading to nonuniform emission intensity.

from US ARMY RDECOM SBIR Phase I contract (W911NF-07-C-0072) monitored by Dr. Marc Ulrich (ARO). Authors would like to thank Vytautas Liuolia for his contribution to calculations.

  (a)]. A considerable scattering in intensities and peak wavelengths of the PL bands acquired from different spots in the bright and dark areas is quite evident. The resulting FWHM of the spatially-integrated PL band (measured at high excitation intensities when a single band prevails) is 38 nm in bright and 51 nm in dark areas.

Figure 1 .

 1 Figure 1. Typical PL intensity mapping (2×2 µm) image of InGaN-based LED structure (a) and

Figure 2 .

 2 Figure 2. Spatially-integrated PL spectrum in bright areas (blue curve) and spectral distribution of

Figure 3 .

 3 Figure 3. Excitation power density dependence of PL spectrum in dark area. The spectra are

Figure 4 .

 4 Figure 4. PL band peak position (a) and spectrally integrated PL intensity (b) in dark and bright areas

14

 14

Acknowledgments

This work was supported by the National Science Foundation (NSF) Smart Lighting Engineering Research Center (# EEC-0812056) and by the National Science Foundation (NSF) I/UCRC "Connection One". The work at Vilnius University was partially supported by the Lithuanian Science and Study Foundation (Grant No. T-74/09). Authors would also like to acknowledge partial support


E M Kanters

F Van Veldhuizen

M Manders

Haverlag

A Sobota
email: a.sobota@tue.nl

J H M Kanters

E M Van Veldhuizen

F Manders

M Haverlag

Statistical time lags in AC discharges

Keywords:

HAL is

Introduction

The breakdown process in gasses is a phenomenon that features gradual changes as the transition from the gaseous to the plasma state occurs. For a successful breakdown event it is necessary to supply the energy to the gas gap high enough to initiate electron avalanches; this is usually done by placing the gas gap in an electrode system, thus creating an electric field that gives energy to the free electrons in the gas. In high pressures and moderate to large electrode gaps, the breakdown process features a growth of thin, ionized channels. The development of the ionized channels is followed by the bridging of the electrode gap, building up the channel conductivity and finally a sharp drop of the potential difference between the electrodes. This point in time is known as the breakdown moment. The potential difference across the gas gap associated with the breakdown moment is called the breakdown voltage.

The duration of the breakdown process, starting from the first electron multiplication that will lead to breakdown until the breakdown moment, is called formative time lag. Ideally, the breakdown process in the gas starts at the exact same moment at which a sufficiently high electric field is brought to the gas gap. However, this Confidential: not for distribution. Submitted to IOP Publishing for peer review 5 January 2011 would require an abundance of free electrons in the gas gap, to ensure that at least one of them would start an electron avalanche that would lead to breakdown. In reality, there is a delay between the onset of the electric field and the start of the formation process, due to the absence of free electrons in the area of adequate electric field strength. This delay is called the statistical time lag. The statistical time lag comprises of two main parts: the time spent on waiting for a free electron to appear in the electrode gap and the time it takes to start an avalanche process that will lead to breakdown.

Formative time lags have been examined for pulsed breakdown processes in low pressure and for a number of gasses [START_REF] Fisher | Formative time lags of spark breakdown[END_REF][START_REF] Moss | Avalanche process in an idealized lamp: I. Measurements of formative breakdown time[END_REF], as a function of the overvoltage and experiment repetition frequency or the afterglow period. It was found that the decrease in the overvoltage used to drive the breakdown process causes an increase in the formative time lag. This is due to the specific form of the Townsend's first ionization coefficient α as a function of the reduced electric field E/p [START_REF] Yu P Raizer | Gas discharge physics[END_REF] α/p = A exp(-Bp/E) (

A and B are coefficients specific for the gas type. The ionization coefficient is a steep function of the reduced electric field and the amplification of the number of electrons in an avalanche exponentially rises with α [START_REF] Yu P Raizer | Gas discharge physics[END_REF]. Already a very small increase of the reduced electric field above the threshold value is thus sufficient for a large reduction of the formative time lag. The experimental studies of statistical time lags have been conducted in a broader set of conditions than the considerations on the formative lag, but still by and large in low gas pressures [START_REF] Meek | Electrical breakdown of gases[END_REF][START_REF] V Lj Marković | Surface recombination of atoms in a nitrogen afterglow[END_REF][START_REF] Radović | On the complex structure of the breakdown timedelay distribution in a neon-filled diode at 4 mbar pressure[END_REF][START_REF] V Lj Marković | Breakdown probability and influence on breakdown delay[END_REF][START_REF] V Kurdle | Breakdown delay times and memory effects in helium at low pressure[END_REF][START_REF] V Lj Marković | Breakdown probability of neon under the influence of field electron emission and surface charges on cathode surface[END_REF]. It has been observed that the increase in the overvoltage applied across the gas gap causes a decrease in the statistical time lag. This means that the number of electrons required to start a successful breakdown process decreases with the increase of the reduced electric field in the gas gap. It also shows that one electron is not always sufficient to start a successful breakdown process, otherwise the statistical time lag would be independent of the overvoltage. This result was anticipated by Wijsman [START_REF] Wijsman | Breakdown probability of a low pressure gas discharge[END_REF] in 1949. He developed an analytical formulation of the probability P (n, d) that one electron causes an avalanche of n electrons at distance d from the electrode

P (n, d) ≈ (1/n) exp(-n/n) n = exp d 0 α(x)dx (2)
This probability is exponentially decreasing with the number of electrons in the avalanche n. He also gave a probability P 0 that the succession of avalanches goes indefinitely, i.e. that the avalanche will lead to breakdown

P 0 = 0 ,q < 1 1 -1/q , q > 1 q = γ(n -1) (3)
γ is the secondary electron emission coefficient for ions in the breakdown process. If one wishes to determine the probability of breakdown as a function of the applied voltage, one should know q as a function of the voltage. In general, q increases rapidly with the reduced electric field in the region above the breakdown value, again due to the specific form of the Townsend's first ionization coefficient as presented in equation 1. As a result, the statistical time lag indeed decreases very quickly with just a small increase in the electric field in the gas gap.

It has also been shown from the experiments varying the afterglow time, that the statistical time lag depends on the amount of pre-ionization. The more charged particles remain from the previous discharge, the shorter the statistical time lag. Related to this, the presence of UV irradiation or radioactive materials were observed to have a decreasing effect on the statistical time lag with the increase of the gas gap length because of the increased probability of the appearance of an electron to initiate the discharge [START_REF] Meek | Electrical breakdown of gases[END_REF].

Zuber and von Laue developed a theory in 1925 [START_REF]Bemerkung zu k. zubers messung der verzögerungszeiten bei der funkenentladung[END_REF][START_REF] Zuber | über die verzögerungszeit bei der funkenentladung[END_REF] that deals with the structure of the total time lag. The theory involves several parameters important for the characterization of the time lag, such as the rate at which the electrons are produced in the gap by an external ionizing agent β, the probability of an electron appearing in a region of the gas gap where it can start an avalanche p 1 and the probability that such an electron will indeed lead to a spark p 2 . The number of sparks that occur in the time interval t + dt is

dn = -βp 1 p 2 n t dt (4)
n t is the number of sparks with time lags greater than t. Integration over the whole time interval [0, t] gives

ln(n t /n 0 ) = - t 0 βp 1 p 2 dt ( 5
)
n 0 is the total number of time lags observed. Assuming that parameters β, p 1 and p 2 do not change with time, the equation above can be expressed as

n t = n 0 exp(-βp 1 p 2 t) = n 0 exp(-t/σ) ( 6 )
According to the statistical theory, σ is the mean time lag, although it can be easily approximated to the mean statistical time lag, providing that the formative lag in the experiment is small enough compared to the total time lag. The work of Zuber and von Laue has been proven to be valid in a number of experiments, but its limitation lies in the fact that the parameters β, p 1 and p 2 are presumed to be constant in time. When voltage is varied in time, these coefficients vary with it. It has been experimentally determined by Cobine and Easton [START_REF] Easton | Time lag of impulse breakdown at high pressures[END_REF] that the equation 6 can best be modified in the following empirically obtained form

n t = n 0 exp(-bt c ) ( 7 )
The parameter c can be more or less than unity, depending on the trend of the voltage across the electrodes. For example, if the voltage is decreasing, c will be less than unity. This has been explained in the following way: the longer the time lag, the lower the voltage across the gap when a free electron appears in the volume. This causes a lower probability that the electron will lead to breakdown. As a result, an electron arriving late is less likely to cause breakdown than the electron arriving early in the experiment. Therefore, a greater number of the time lags fall in the long time range than would be the case at a constant voltage. If the voltage across the gas gap would happen to increase during the experiment, the result would be the opposite, c would be greater than one. We took a particular interest in the statistical time lags in high pressure argon and xenon discharges for their importance in the lighting industry. While the formative lag is of the order of microseconds or less in high intensity discharge (HID) systems [START_REF] Moss | Avalanche process in an idealized lamp: I. Measurements of formative breakdown time[END_REF][START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], the statistical lag can be orders of magnitude larger and have a large spread of values. This is of particular interest for lamp driver design.

In this paper we show the measured values of the statistical lag as a function of the gas type and pressure, electrode gap, frequency and voltage amplitude slope for high frequency AC breakdown in argon and xenon. We also test the validity of the theory presented by Zuber and von Laue [START_REF]Bemerkung zu k. zubers messung der verzögerungszeiten bei der funkenentladung[END_REF][START_REF] Zuber | über die verzögerungszeit bei der funkenentladung[END_REF] and later adapted by Cobine and Easton [START_REF] Easton | Time lag of impulse breakdown at high pressures[END_REF], on the structure of the time lags during breakdown in gasses. We show that their theories, although developed for quite different voltage forms, are still general enough to hold for high-frequency AC breakdown, provided that one additional parameter is introduced. Setup details were given in [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] and will be shortly presented in this section. We used a pin-pin electrode geometry enclosed in 70 W Philips YAG (Yttrium Aluminium Garnet) burners as designed for metal-halide HID lamps. The electrodes were rodshaped and 0.6 mm in diameter, forming an electrode gap of 4 or 7 mm. The volume of the lamp burners we used was 3.6 × 10 -7 m 3 . They were filled with 0.3 or 0.7 bar of argon or xenon, without the addition of salts or mercury.

Details of the experiments and the analysis

Experimental parameters

The experiments were done using a high-frequency sine voltage signal of linearly rising amplitude on the active electrode, while the second electrode was grounded. The AC signal was generated in a way which allowed us to change the frequency between 100 kHz and 1 MHz and the slope of the voltage amplitude between 10 and 1280 V/ms. We performed breakdown voltage measurements for each breakdown process, using a high voltage probe and an oscilloscope. Breakdown voltages were measured for different frequencies and voltage amplitude slopes.

We have made and measured several lamps for each type (lamp type in this case denotes a single combination of gas type, gas pressure and electrode gap) to avoid our measurements being influenced by possible defects in electrodes or burners or errors in lamp filling or sealing procedure. The list of lamp types used in the experiments is given in table 1. Ten breakdown voltage measurements were performed for each lamp in table 1 and for each data point. One data point is a specific combination of the voltage frequency and the slope of the voltage amplitude. The average values and the standard deviations were calculated for each lamp, and subsequently for each lamp type. The data points in the graphs throughout this paper represent the average values and the standard deviations for the corresponding lamp type.

The experiment repetition frequency was held constant at 0.1 Hz. This choice provided sufficient time between two breakdown processes for them not to influence each other. Our experimental systems contained only noble gasses and the discharges were not allowed to burn after breakdown, which ensured that no gas heating would take place. A current limiter was used to ensure that the power to the lamps is cut off after breakdown [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. The dielectric surface of the lamp burner in which we have enclosed our experimental system was, however, very likely to have been charged during one breakdown process and still remained charged during the next one, due to the very long decay time of charge on dielectric surfaces. However, we found this to be an acceptable starting condition, for two reasons. First, the discharges in our geometry do not develop on the burner surfaces, as has been shown previously [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. The charges on the surfaces are, thus, not actively participating in the discharge development. Second, the starting conditions were the same for each breakdown event, or similar to the point that we could not distinguish one from another. We know this because during the measurements we did not observe a systematic change in the breakdown voltage. Large changes in the amount of free electrons in the gas volume where the electric field was high enough to sustain an avalanche process would have changed the outcome of the breakdown voltage measurements. Even if the changes of the surface charge were gradual, the breakdown voltages would gradually change over time. Nothing similar was observed.

The breakdown processes were at times aided by a flux of UV irradiation at 254 nm. Between 10 13 and 10 14 photons entered the lamp volume every second, and about 5% of those hit one of the electrodes. With the estimated efficiency of 10 -3 for photoeffect on the cathode [START_REF] Yu P Raizer | Gas discharge physics[END_REF], the UV irradiation ensured the emergence of between 5 × 10 8 and 5 × 10 9 electrons per second in the lamp volume. Consequently, the preionization levels were high enough to minimize the statistical time lag, the assumption which will be later supported in the results section of the paper. Another function of UV irradiation in these experiments was to clean the charge off the surfaces of the lamp burner. This effect has been discussed already in 1981 [START_REF] Kindel | Wall charge behaviour in external electrode gas discharges[END_REF]. The lack of charge on the burner surfaces in the case of UV-assisted breakdown is an essential difference between the UV-aided and non-aided breakdown events in the case where the discharges develop on the insulating surfaces. However, as the discharges in our experiments did not develop on the dielectric surfaces [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], this effect is not considered important. The procedure of determining the statistical time lag in AC breakdown required the comparison of breakdown voltage of non-assisted and the UV-aided breakdown processes. The idea is illustrated in figure 1. The experiments we performed were done by using sine voltage of linearly rising amplitude. In the first part of the voltage signal, denoted as 1 in the figure, the voltage was too low to start an electron avalanche in the lamp. At the transition from part 1 to part 2, the voltage crossed the threshold at which the reduced electric field in areas of the lamp became high enough for an electron avalanche to start. At this point it became possible for the discharge to start forming near the electrode tips and subsequently lead to breakdown. However, the preionization levels in non-aided breakdown events were very low, which caused an appreciable statistical time lag to occur, while the voltage amplitude was still rising in a linear way. This portion of the breakdown process is denoted as 2 in the figure. Finally, at the transition from part 2 to part 3, an avalanche started that later lead to breakdown. The final part of the process is therefore the formative time lag.

Determination of statistical time lags

The influence of UV irradiation on the breakdown process was already discussed at length in [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. It was concluded that there were two effects expected to have a notable influence on the breakdown process. One, the statistical time lag in UV-assisted measurements was expected to be negligible. Therefore, the part 2 in the voltage form shown in figure 1 does not exist for the UV-aided breakdown. The formation of the avalanches that will lead to breakdown take place after the voltage amplitude crosses the threshold value for sustaining the breakdown process. This is why the UV-aided breakdown takes place at lower voltages, which leads us to the second UV-related effect with notable influence on the breakdown process. The formative time lag was expected to be prolonged because the breakdown process took place at lower voltage. We will show that even the prolonged formative time lag in UV-aided breakdown processes as measured and reported in [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] is still orders of magnitude lower than the statistical time lag encountered in the non-aided case, and can be neglected in the calculations.

As stated before, by comparing the breakdown voltages of UV-aided and non-aided breakdown processes for the same frequency and voltage amplitude slope, we can derive the statistical time lag in the non-aided breakdown events. Let T 0 refer to the time interval denoted as part 1 in the figure 1, when the electric field in the gap is too low to sustain a breakdown process. T f is the formative time lag, and refers to the part 3 of the breakdown process as represented graphically in figure 1. Let the formative times in UV-assisted case be denoted as T f,UV . The statistical time lag T s is presumably present only in non-assisted breakdown. The structure of the total time used for one breakdown event for non-assisted (T tot ) and UV-assisted (T tot,U V ) events can be represented in the following way

T tot = T 0 + T s + T f (8) T tot,U V = T 0 + T f,UV (9)
The difference in the total time can give the statistical time lag, presuming that the difference in the formative times can be neglected when compared to the statistical lag,

T s T f,UV -T f T tot -T tot,U V = T s + (T f -T f,UV ) ≈ T s ( 10
)
The total time can be expressed as the ratio between the breakdown voltage V bd and the slope of the voltage amplitude R

T tot = V bd /R (11)
The statistical time lag can be expressed as a function of slope and voltage difference Figure 2 shows the structure of the measurements used to calculate statistical lag in AC-driven breakdown processes. We compared the breakdown voltage measurements for UV-assisted and non-assisted breakdown processes and calculated the time lags from the difference in the breakdown voltages measured at the same frequency and the same voltage amplitude slope. For these calculations to be valid it is necessary to assume that the UV-assisted breakdown is virtually free of statistical lag and that the UV irradiation does not affect the discharge evolution in any other way.

T s = V bd -V bd,U V R . ( 12
)

Results and discussion

Breakdown voltage measurements

The effects the UV irradiation has on a growing discharge in our experimental conditions have already been discussed in [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] and in the previous two sections. Our conclusions were that the main effect is the significant lowering of the statistical time lag. Figure 2 shows that the statistical time lag is not only lowered, but virtually gone in the case of UV-aided events, by demonstrating the difference in behaviour of the breakdown voltage as a function of the voltage amplitude slope. When the breakdown process was aided by UV irradiation, the breakdown voltage did not change with the slope, which was not the case with the non-aided breakdown events.

The link between the observed phenomenon and the assumption about the lack of the statistical time lag in UV-aided processes follows from figure 1. There, a portion of the voltage signal at the active electrode in the non-aided case is inefficiently used to wait for an electron avalanche that will subsequently develop into a full discharge. This part is designated as region 2 in the figure and causes statistical time lag. During this time the voltage amplitude keeps rising, due to its positive slope. The faster the slope, the more the voltage amplitude rises in the same amount of time. In an event where a breakdown process would not suffer from statistical time lag and where the formative time lag would be negligible, the measured breakdown voltage would be the threshold voltage amplitude needed for breakdown and the breakdown voltage would not change with the amplitude slope.

The contribution of the formative time lag to the increase of the breakdown voltage would be important in the case where the formative time lag would be large or in the case of fast voltage amplitude slopes. Formative time lags in our experiments were, however, very short. As shown in [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF], formative times rise with the increase in the voltage frequency. The longest formative times relevant to the calculations presented in this paper were in argon at 0.7 bar and 7 mm electrode gap at 800 kHz, xenon at 0.3 bar and 7 mm electrode gap at the same frequency and xenon at 0.7 bar and 7 mm electrode gap at 400 kHz. The formative times for these discharges were 0.041 ms, 0.121 ms and 0.037 ms, respectively, for UV-aided breakdown. We take the values for the UVassisted events as a reference because these formative times are expected to be longer than the corresponding formative time lags in non-assisted breakdown processes because they occur at lower voltage [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. At the highest voltage amplitude slopes at which the statistical time lags were calculated, the increase in voltage due to the formative time lags was 26 V, 39 V and 47 V, respectively. Such small increases in voltage fall within the error bars of the breakdown voltage measurements. In addition, all formative time lags were at least one to two orders of magnitude lower than the corresponding statistical time lags, which will be presented in the next section. The values were measured as the time delay from the first light emitted from the electrode tip to the breakdown moment. Even if the formation time took twice as long, the formative time lags would still be small enough to be neglected in comparison with the statistical time lags.

In conclusion, our measurements have shown that the UV-aided processes were virtually free of statistical time lag, that the UV irradiation did not affect the discharge development in a way that would disallow us to compare the UV-aided and non-aided breakdown events [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF] and that the formative time lags were negligible compared to the corresponding statistical time lags [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF]. Consequently, the assumptions on which rests the validity of the procedure for determining statistical time lag as described in section 2.2 are satisfied for AC-driven breakdown events in our experimental conditions.

Statistical time lags

Figures 3 to 8 show the statistical time lags as a function of voltage amplitude slope, calculated at 100, 400 and 800 kHz. The bars in the graphs represent the standard deviation calculated for a function of multiple (two in our case) independent variables where each variable has a known variance.

The term overvoltage is used quite often in this paper. It is associated with the excess voltage applied to the electrode, with respect to the threshold breakdown voltage at given conditions. If the threshold breakdown voltage is given by V b , and the voltage at which the measurements are performed is V b + ∆V , than the ∆V is the overvoltage applied to the electrode.

It is usual [START_REF] Fisher | Formative time lags of spark breakdown[END_REF][START_REF] Moss | Avalanche process in an idealized lamp: I. Measurements of formative breakdown time[END_REF][START_REF] Meek | Electrical breakdown of gases[END_REF][START_REF] V Lj Marković | Surface recombination of atoms in a nitrogen afterglow[END_REF][START_REF] Radović | On the complex structure of the breakdown timedelay distribution in a neon-filled diode at 4 mbar pressure[END_REF][START_REF] V Lj Marković | Breakdown probability and influence on breakdown delay[END_REF][START_REF] V Kurdle | Breakdown delay times and memory effects in helium at low pressure[END_REF][START_REF] V Lj Marković | Breakdown probability of neon under the influence of field electron emission and surface charges on cathode surface[END_REF] to show time lags as a function of overvoltage, not of the slope of the voltage amplitude. It is also easier to relate the statistical time lag to voltage or electric field than to the voltage amplitude slope, because it is intuitively clear that the higher the electric field, the higher the probability that one electron will cause avalanches that will lead to breakdown [START_REF] Wijsman | Breakdown probability of a low pressure gas discharge[END_REF]. We chose to present our results as a function of the voltage amplitude slope because the voltage amplitude slope is a parameter which we were able to control very precisely in our experiments. In addition, even though the slope can be related to the overvoltage in a simple way, the interpretation of the results is not straightforward. Because of the reasons given, all the determined statistical time lags are presented as a function of the voltage amplitude slope, and the extension to the scope of overvoltage is presented in the next section.

Figures 3 to 8 show that the statistical time lag decreases as a steep function of the voltage amplitude slope. This is caused by different levels of overvoltage achieved at different voltage amplitude slopes during statistical time lags. The faster the slope, the more the voltage amplitude grows in a given time increment. The resulting overvoltage is higher for the faster rising voltage. Higher overvoltage causes a higher Townsend's first ionization coefficient and results in more efficient electron avalanches. The probability of a successful breakdown greatly increases with the number of free electrons available in the electrode gap. This is a crucial factor in the behaviour of the statistical time lags, as we have discussed in the introductory section.

Statistical time lag in all the given figures is always bigger in breakdown events at higher gas pressure and in xenon compared to the breakdown events in argon. The time lags do not appear to be a function of frequency. This is a result one would expect given that there is no transition between the drift-dominated to the diffusion-dominated mode in our experimental conditions, as we have shown in [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF][START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: II. Simulations[END_REF]. With further increase in frequency we would expect the statistical time lag to be lowered, as the electron loss processes become lower and diffusion-dominated.

The statistical time lags reaching 120 ms for the slowest slope (Ar at 0.7 bar and 4 mm electrode gap) are quite large compared to the values obtained for pulsed discharges [START_REF] Meek | Electrical breakdown of gases[END_REF]. However, given our slow voltage amplitude slope and the fact that we performed the measurements using high frequency AC voltage, we expected lower electric fields in the electrode gap and consequently larger statistical time lags.

A more interesting observation is that the statistical lag in shorter electrode gaps is larger than the corresponding lag in the larger gaps. As was explained in [START_REF] Meek | Electrical breakdown of gases[END_REF], increasing the gap length allows many more electrons to be produced in the gap by means of cosmic radiation or similar events responsible for the occurrence of first free electrons; this, in turn, reduces time lags. The effects are very clear for the difference in electrode gaps used in our experiments -reducing the gap from 7 mm to 4 mm effectively reduces the volume where the initial electrons have to be created by 50%. Time lags in general are usually presented as a function of overvoltage, which is understandable given the fact that they can be related to the Townsend's first ionization coefficient, whose value depends greatly on the imposed electric field. Mean statistical time lag can be expressed [START_REF] Meek | Electrical breakdown of gases[END_REF][START_REF] V Lj Marković | Surface recombination of atoms in a nitrogen afterglow[END_REF] as Y is the electron yield, and P is the probability that one electron will cause avalanches that will lead to breakdown. P was given in introductory section in equation 3. This equation holds low pressure and in cases where the formative time lag is smaller than the time lag. the condition was fulfilled in our experiments, we were dealing with the discharges in high pressures, which makes deviations from the theoretical predictions possible. It is not simple to express the statistical time lags as a function of overvoltage in our experiments for the simple reason that the overvoltage was changing during the statistical lag. One should recall that we were doing experiments with voltage amplitude rising in time and a sine voltage signal. Nevertheless, time lags can be expressed as a function of the final overvoltage reached at the end of the breakdown process, as presented in figure 9.

Statistical time lags as a function of overvoltage

Ts = 1 Y P . (13)
The figure shows statistical time lags for argon discharges at 100 kHz. The overvoltage was calculated in the following way

OV = V bd -V bd,U V V bd,U V . ( 14
)
Large error bars in x direction are a consequence of large statistical spread of breakdown voltages in non-aided breakdown events, as shown in figure 2.

The data in figure 9 can be fitted to a double exponential function of the form

Ts = ae -be -c/OV . ( 15
)
The same is true for the data obtained at two other frequencies and for breakdown in xenon. The large range of values of the coefficients a, b and c that can be used in the fitting procedure makes it inappropriate to ascribe a high degree of certainty to any theory trying to explain these measurements. A good fit can be made with c ranging from 1 to 10 20 . Coefficients a and b cannot be chosen in such a wide range of values, their values are typically limited within one or two orders of magnitude. Still, we would like to comment on the possible origin of the double-exponent dependency on the overvoltage. Equation 13shows that in principle, the statistical time lag is a function of parameters. Combining equations and 3 that the mean statistical time lag is proportional to

Ts Y γ(exp(αd) 1) . ( 16
)
The electron yield Y is difficult to since there are numerous theories explaining breakdown in high pressures [START_REF] Meek | Electrical breakdown of gases[END_REF] and we are working in an AC regime, where electron production and loss happens on a larger time scale than in typical pulsed discharge [START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: I. Experiment[END_REF][START_REF] Sobota | AC breakdown in near-atmospheric pressure noble gasses: II. Simulations[END_REF]. However, if we assume an exponential evolution as a function of α, as is so often seen in the literature [START_REF] Meek | Electrical breakdown of gases[END_REF], the mean statistical lag becomes proportional to

Ts ∝ 1 γ exp(αd)(exp(αd) -1) . ( 17
)
Assuming that exp(αd) >> 1 leaves us with the exponential function in the denominator. With the form of Townsend's first ionization coefficient as given in equation 1, equation 17 can be rewritten as following, for a single value of the electric field Ts ∝ γ(e -2Apde -Bp/E ).

The electric field is presumably a linear function of the applied voltage. Therefore, it is a linear function of the overvoltage, and we can assume E = E 0 + E 1 , where E 0 is the threshold electric field needed for breakdown and E 1 is the excess electric field that is present in the experiments. The coefficients A and B depend on the gas type. The reasoning described above can be considered valid for discharges at a constant overvoltage. In our experiments the voltage amplitude slope was positive, and we were working with AC voltage signal. The overvoltage was not constant. However, one should look more closely to the parameter α, which is a steep function of the electric field [START_REF] Yu P Raizer | Gas discharge physics[END_REF].

Consequently, α at voltage extremes is a steep function of time in our experiments. Figure 10 shows its evolution as a function of time for the voltage amplitude slopes used in our experiments. The calculations for this figure were done with parameters A = 12 cm -1 Torr -1 and B = 180 Vcm -1 Torr -1 [START_REF] Yu P Raizer | Gas discharge physics[END_REF], for 0.7 bar argon discharge in a homogenous electrode gap of 4 mm. We used a simple expression for the electric field in a homogenous electrode gap, thus underestimating the electric field in the experiments. Our purpose was, however, only to demonstrate the behaviour of α with time. Time zero on the x-axis designates the moment where the voltage passed the threshold value needed for breakdown -effectively, the x-axis shows the statistical time lag in the experiments.

The behaviour of α with time and electric field suggests that a small change in the experimental conditions or timing causes a big change in α, and consequently in the values of P and Y in expression 13. If the biggest contribution is also the one governing the final value of P and Y , we can say that these parameters and the mean statistical time lag are local functions of electric field, both in time and in space. We expect this to be the reason why a theory valid only for homogenous electric fields constant in time can also be applied to our case. The theory on the structure of time lags by von Laue and Zuber was published in 1925 [START_REF]Bemerkung zu k. zubers messung der verzögerungszeiten bei der funkenentladung[END_REF][START_REF] Zuber | über die verzögerungszeit bei der funkenentladung[END_REF] and corrected in 1943 by Cobine and Easton [START_REF] Easton | Time lag of impulse breakdown at high pressures[END_REF]. As we have described in the introductory section, the theory was conceived to describe breakdown in DC voltage conditions. However, it was developed in a way that did not explicitly make use of the DC voltage form. The theory of von Laue and Zuber uses the spread of time lags in the experiments to determine the mean time lag σ, as shown in equation 6. According to their theory, sigma is given by the production rate of electrons by an external ionizing agent β, the probability that an electron will appear in the part of the gas gap where it can lead to breakdown p 1 and the probability that the electron in question will indeed lead to breakdown p 2 . These probabilities should not be a function of time.

The Cobine-Easton expression

σ = 1/βp 1 p 2 . ( 19
)
As we were working with AC voltage, these three probabilities were not sufficient to describe our system. One more parameter was always important, and that was the phase in the voltage cycle in which an electron would appear in the electrode gap. The electric fields that governed the breakdown processes in our experiments were the ones at the voltage cycle extremes, as we were working with voltage amplitudes that were not high above the threshold values. Therefore, if an electron would appear in the electrode gap at a moment when the voltage phase was far from an extreme, chances of this electron to cause an electron avalanche would be very small. This additional parameter, the probability that the electron would appear in the electrode gap at a favourable voltage phase, can be denoted as p AC . The aforementioned equation can now be rewritten as following σ = 1/βp 1 p 2 p AC .

(20)

This holds only for constant electric fields, or in our case, for the electric fields with amplitudes that are constant in time.

Cobine and Easton have shown that for voltage that is not constant in time, a slight modification should be made, as in equation 7. This modification comes from the fact that for breakdown events in which voltage with slope different than zero is applied, the probabilities p 1 and p 2 are a function of voltage and therefore a function of time. Our addition p AC is also a function of the voltage amplitude.

The new form the von Laue -Zuber expression 6 takes is shown in equation 7. It contains the parameter b in the exponent, which has to have the dimension of [time] -c . However, we do not know how the probabilities p 1 , p 2 and p AC develop with time, and we cannot calculate the mean statistical time lags as a consequence.

From our experiments can be deduced that the theory of Cobine and Easton can in principle be applied to AC-driven breakdown processes as well as DC. Figure 11 shows a log(ln)-log plot of the number of breakdown events with the time lag greater than t, as a function of t. According to equation 7, this plot should give a linear curve for each combination of experimental parameters (frequency, slope, lamp type). ln(n 0 /n t ) = bt c log(ln(n 0 /n t )) = log(b) + c log(t) (21)

Figure 11 shows that the data obtained in the experiments indeed forms sets that can be approximated by linear curves. One should notice that the shapes of the plots greatly depend on the amount of the data points that we have available, because the ordinate axis basically shows the ratio of the number of the events that fulfill certain condition and the total number of events. When observing various plots of this type, we have noticed that the linearity indeed improves with the number of data points.

Conclusions

In this paper, we have presented the statistical time lag measurements for AC-driven breakdown events, using a slow voltage amplitude slope. Voltage frequency used in the experiments ranged from 100 to 800 kHz and the electrode gap was either 4 or 7 mm.

The statistical time lags varied between roughly 1 and 150 ms, depending on the gas type, pressure, electrode gap and the voltage amplitude slope. The statistical time lag was greater in xenon than in argon, at higher pressure and smaller electrode gap, which was explained by a smaller number of free electrons available at a given moment in the gas volume.

The statistical time lags were presented as a function of the voltage amplitude slope and as a function of overvoltage. A double-exponential function fit was made to the statistical time lags as a function of overvoltage. We suggest this to be the result of specific forms of the mean statistical time lag as a function of Townsend's first ionization coefficient.

Finally, we have shown that the theory presented in 1943 by Cobine and Easton [START_REF] Easton | Time lag of impulse breakdown at high pressures[END_REF] for breakdown events driven by long square pulses is general enough to show the expected results in the case of AC-driven discharges as well. Further research is needed to examine the extent of validity of this theory in the AC case.

Figure 1 .

 1 Figure 1. Schematics of the time lag structure. The figure shows the voltage on the active electrode, depicts the concept of breakdown moment and breakdown voltage.The voltage signal is divided into three sections, further explained in the text. The length of the three sections is not in scale in the figure, as section 1 takes a much bigger part of the voltage signal in reality. In addition, section 2 is larger compared to section 3 than shown in the figure. Threshold voltage mentioned in the text is the voltage amplitude at the transition from part 1 to part 2.

Figure 2 .

 2 Figure 2. Breakdown voltage measured as a function of the voltage amplitude slope, in UV-assisted and non-assisted breakdown events. The data shown belongs to the discharges in argon, at 0.7 bar and at 400 kHz. The inset shows the same data, only on a logarithmic x-axis. The same trends as shown in this graph were obtained at other gas pressures, frequencies and also for xenon.

Figure 3 .

 3 Figure 3. Statistical time lags for argon at 100 kHz, presented as a function of voltage amplitude slope.

Figure 4 .

 4 Figure 4. Statistical time lags for xenon at 100 kHz, presented as a function of voltage amplitude slope. The measurements of breakdown properties in 0.7 bar xenon could not be performed due to the limitations of the equipment.

Figure 5 .

 5 Figure 5. Statistical time lags for argon at 400 kHz, presented as a function of voltage amplitude slope.

Figure 6 .

 6 Figure 6. Statistical time lags for xenon at 400 kHz, presented as a function of voltage amplitude slope.

Figure 7 .

 7 Figure 7. Statistical time lags for argon at 800 kHz, presented as a function of voltage amplitude slope.

Figure 8 .

 8 Figure 8. Statistical time lags for xenon at 800 kHz, presented as a function of voltage amplitude slope. The measurements of breakdown properties in 0.7 bar xenon and with 4 mm electrode gap could not be performed due to the limitations of the equipment.

Figure 9 .

 9 Figure 9. Statistical time lags for argon discharges at 100 kHz, presented as a function of overvoltage at the end of the breakdown process.

Figure 10 .

 10 Figure 10. Townsend's first ionization coefficient α calculated as a function of time for different voltage amplitude slope. The data belonging to argon discharges at 0.7 bar, 4 mm electrode gap at 100 kHz were used for calculations. See the text for more details.

Figure 11 .

 11 Figure 11. The data on statistical time lags belonging to argon at 0.3 bar and 7 mm electrode gap, at 400 kHz. The data is presented in accordance with the Cobine-Easton relation. Further details can be found in the text.

Table 1 .

 1 Lamp types used in the experiments. A lamp type is one specific combination of gas type, pressure and electrode gap.

			Electrode number of
		Pressure	gap	available
	Gas	[bar]	[mm]	lamps
	Ar	0.3	4	6
			7	5
		0.7	4	3
			7	5
	Xe	0.3	4	3
			7	2
		0.7	4	3
			7	3

Acknowledgments

This work was supported by Philips Lighting.