Since 2001 the intrinsic birefringence of fluorine has been accessible to experiment. It is known that its intrinsic anisotropy is entirely due to spatial dispersion, and that the index surface of fluorine and crystals with the same symmetry has seven optical axes, four of them intersecting this surface at pairs of conical points. I point out the fact that there is no internal conical refraction, but only simple refraction (and without walk-off), with these conical points. I also explain why the rays are not a priori normal to the index surface in the case of fluorine because of its spatial dispersion; and I discuss two particular cases of spatial dispersion where the Poynting vector remains orthogonal to the index surface.
© 2006 Optical Society of America
The semiconductor industry is considering an excimer laser line at wavelength 157 nm for optical lithography. The main material convenient for refractive optics systems at that wavelength is crystalline fluorine CaF2. Its crystal having cubic symmetry, it was thought that no intrinsic birefringence occurs: such crystals can be neither uniaxial nor biaxial. But this property is based upon the simplifying assumption that the material is local ; that approximation ends up loosing its validity when the wavelength to crystal unit cell ratio λ/a decreases below a certain point . For a plane monochromatic progressive wave of pulsation ω and wave vector noted q (instead of k, according to the recent quoted papers of Burnett and al.), the non-locality of a dielectric is easily described by the dependence of its electric permittivity tensor (ε) with q: this phenomenon is also called “spatial dispersion”, and adds to the usual dispersion described by the dependence of (ε) with ω. Any material shows some absorption, and for a dielectric it means that (ε) is not Hermitian, according to the Kramers-Kronig relations involving its usual dispersion. But when a medium is considered being transparent the effects of the anti-Hermitian part of (ε) are essentially negligible - and with fluorine in the optical domain this is the case over an exceptionally broad band going from thermal infrared to vacuum ultraviolet (9-0.12 μm approximately); so in this work we shall consider only Hermitian permittivity tensors and real wave vectors. In the visible domain, ∥q∥ is small enough compared to 1/a for (ε) (ω, q) to be nearly equal to (ε) (ω, 0) which is scalar for crystals having cubic symmetry, so that they seem isotropic. But that is not true in the vacuum ultraviolet, so a new form of anisotropy occurs: the intrinsic birefringence by spatial dispersion. That phenomenon has been predicted by Lorentz in 1878 , theoretically studied by Agranovitch and Ginzburg and experimentally observed in non-optical materials since 1960 , then recently rediscovered in fluorine where it complicates the design of lithography objectives .
In the ultraviolet domain, a good approximation of (ε) (ω, q) is given by its second order expansion for a ∥q∥ ≪1. For crystal of classes Td and Oh , (in Schoenflies notation, fluorine being of class ), the elements of (ε)(ω,q) - in an orthonormal basis whose unit vectors x̂, ŷ and ẑ are parallel to the edges (E) of the cubic conventional unit cell (C) - are given by 
where ε 0 denotes the electric permittivity of the vacuum, the Kronecker symbol, and indices j and l are equal to 1, 2, and 3 for axes x, y, and z respectively. Among the parameters n 0, α, β and γ, only the latter is responsible for a dielectric anisotropy: if γ = 0 , the refractive index n of the spatially dispersive medium has its isotropic value 
(where q 0 is the modulus of wave vectors in a vacuum for the given frequency) ; Eq. (2) results from the two Maxwell curl Eqs., which yield
and the three scalar components of (3) form a linear system whose determinant is null and gives n. α is not involved in Eq. (2) for γ = 0 because one verifies straightforwardly that the electric field E is parallel to the electric displacement D (a fact which will be useful later) hence E is transverse. For such crystals with γ ≠ 0 , the study of the index surface  shows that it presents eight conical points (twice as many as conventional biaxial crystals!) located on its four optical axes parallel to the diagonals (D) of (C). One should note that its other three optical axes, those parallel to (E), are not associated to conical points (like for conventional uniaxial crystals).
In an anisotropic crystal that is magnetically perfect (i.e. for which B = μ H where its magnetic permeability is a real constant μ, B and H being the magnetic induction and field), when wave vector q is parallel to an optical axis, any transverse direction is allowed for D, and (D,H,q) forms an orthogonal triplet. Generally there is a “walk-off angle” χ = (D,E), so that the direction of the Poynting vector R≡E×H makes with q the same angle χ which depends on the D-direction - because (E,H,R) forms another orthogonal triplet .
In a conventional uniaxial crystal, there is only one optical axis, and χ= 0 if q is parallel to it.
In a conventional biaxial crystal, there are two optical axes, and when q is parallel to one of them χ= 0 only for D normal to the plane of these axes but χ differs from zero for all other directions allowed for D, so that the set of refracted rays associated to one incident non polarized ray in that case forms a cone. That well-known phenomenon of internal conical refraction is geometrically understood by use of the classical property that the Poynting vector is orthogonal to the index surface at its point associated to the wave vector .
The fact that internal conical refraction occurs only if q is parallel to an optical axis is general. But is conical refraction possible in fluorine?
2. Absence of internal conical refraction in fluorine
In fluorine, when q is parallel to (E), the polarization properties of D and E are the same as in an uniaxial crystal with q parallel to its optical axis: χ= 0 for all the directions allowed for D, so there is no internal conical refraction, and there is no conical point associated to this optical axis on the index surface .
Now I investigate whether internal conical refraction exists when q is parallel to (D). In that case relation (1) becomes
Then (ε) has the same structure as the permittivity tensor of an isotropic spatially dispersive medium, satisfying relation (2) where β is replaced by . Hence its index is
(the three numbers in square brackets are proportional to the relevant D components on the crystallographic axes). For our purpose, the important consequence is that E is parallel to D here (like in the case leading to Eq. (2)) - in other terms, one easily sees that ⌊11̅0⌋ and ⌊112̅⌋ are two eigenvectors, not only of the projection (ε -1)⊥ of (ε -1) - the inverse of the permittivity (used by ) - orthogonal to q, but also of the whole tensor (ε -1) and of (ε) given by Eq. (4). Therefore χ= 0, hence R is again parallel to q for all the directions allowed for D, and there is no internal conical refraction.
Nevertheless, on the index surface there are conical points associated to the optical axes parallel to (D) ! This puzzling situation is allowed because the derivation of the classical property of orthogonality of R to the index surface no longer holds for crystal showing spatial dispersion.
3. Discussion of the orthogonality of the Poynting vector to the index surface
3.1 General case of a linear spatially dispersive dielectric
To investigate that problem, I shall generalize a classical derivation (, footnote 1 p. 414), considering a plane monochromatic progressive wave propagating in a linear dielectric. The two Maxwell curl Eqs. yield, for the complex representations (denoted by underlining) of the fields:
where (ε) depends on ω and q because of conventional as well as spatial dispersions. ω being fixed, an infinitesimal variation dq of q leads to variations of the fields which are related by the differential forms of Eqs. (6) and (7):
let us stress that the term (dε) is entirely due to spatial dispersion. Hence, the scalar product of both sides of Eq. (9) by E̲* , subtracted from the scalar product of (8) by H̲* , gives
The right-hand side can be simplified, because
where 〈R〉 is the mean value of the Poynting vector. Up to now, the wave vector q could in fact be complex (case of an absorbing medium, or of an evanescent wave, etc.); but we want to consider that a point of the index surface suffices to determine completely q, so we restrict ourselves to the case where q is real, hence we consider that the medium is transparent. This leads to a new simplification, because with q real Eq. (6) gives
so relation (10) and (11) lead to
where (ε)+ denotes the Hermitian conjugate of (ε), and (ε)ah its anti-Hermitian part. We know that for a transparent medium, (ε) is not automatically real (it is complex when the spatial dispersion involves gyrotropy), but it is always Hermitian, hence (ε)ah is null  and finally
When the medium is local, (dε) is null and for a given q all the directions of the infinitesimal vectors dq span the tangent plane of the index surface at its point associated to q, so 〈R〉 is perpendicular to that surface at this point (let us note that this result holds also if the magnetic permeability is a Hermitian matrix independent of q).
But with a fluorine-type crystal, the absence of internal conical refraction associated to the conical points of their index surface shows that 〈R〉 is not always perpendicular to that surface; this is possible because the medium is spatially dispersive then (dε) is not null.
To finish, let us stress the fact that in some particular cases of non local media however, (dε) is not null but E̲*.(dε)E̲ is null for each q, hence 〈R〉 is always perpendicular to their index surface; I shall point out two such cases.
3.2 Particular case of a purely gyrotropic medium
where i 2 = -1, and is the classical tensor, totally antisymmetric, such that
= + 1 for j = 1, l = 2, m = 3, and any even permutation
-1 for any odd permutation
0 if any two indices are equal;
moreover, the medium being transparent, and p are real. But it is well-known that, in such a case ( pp. 433–434), E and D are always parallel (their two eigenvibrations being left and right-handed circularly polarized), the index surface is composed of two spheres centered at the origin; then R and q are always parallel, and R is normal to the index surface, although (dε) is not null! This is possible because here
and E̲×E̲* is parallel to q (E̲ and E̲* being transverse like D), dq is orthogonal to q because of the form of the index surface, hence
3.3 Particular case of a spatially dispersive non gyrotropic medium
The second case is a spatially dispersive medium, non gyrotropic, but transparent and isotropic: it obeys relation (1) with γ = 0 ; moreover, its index surface is a sphere of radius ni given by relation (2), and E̲ is orthogonal to q. In this case, we find
which is null from the same reasons as the precedent case - E̲.q , E̲*.q and q.dq are null.
The major conclusion of this paper is that, when an incident ray gives q parallel to any of the seven optical axes of the fluorine, there is neither double nor internal conical refraction, only a simple refraction and without walk-off - the refractive index of the crystal being given by relation (5). This may in fact simplify the correction of the birefringence of fluorine in photolithography objectives, because one way used to correct it is to combine fluorine lenses whose geometrical axes are parallel to (D), the transverse crystal axes being rotated by odd multiples of 60° from a lens to the following one ; in such a corrected objective, no internal conical refraction will occur for the light ray propagating on the geometric axis of the objective.
The author wants to thank Pierre Chavel for his comments. The author is also with Lycée Blaise Pascal, 36 av. Carnot, F-63037 Clermont-Ferrand Cedex, France
References and links
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