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Focusing of evanescent vector waves
Optics Communications 283 (2010) 29–33
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Focusing of evanescent vector waves L.E. Helseth Department of Physics and Technology, University of Bergen, N-5007 Bergen, Norway
a r t i c l e
i n f o
Article history: Received 10 July 2009 Received in revised form 18 September 2009 Accepted 24 September 2009
a b s t r a c t We study focusing of two and three-dimensional evanescent vector waves, with a particular emphasis on identifying suitable intensity structures for applications in optical data storage. For two-dimensional evanescent waves large transverse spatial wave vectors result in purely circularly polarized evanescent states. We suggest that these may have applications in all-optical data storage through the inverse Faraday effect. On the other hand, for three-dimensional evanescent waves longitudinally polarized modes are observed to give the most tightly focused spot, and this may be utilized to confine light behind a solid immersion lens. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Focusing of electromagnetic waves has been studied for more than a century, and the diffraction behavior is well-understood in both the paraxial and non-paraxial regimes [1,2]. Conventional lens systems for propagation of homogeneous waves cannot focus light to a spot smaller than the classical diffraction limit [1]. To overcome this problem, immersion technologies using the fact that the wavelength in a medium may be smaller than that of air, have been proposed and tested. These lens, axicon or mirror systems utilize that the size of the focal spot scales with refractive index of the medium containing the focal plane [3–14]. Although the behavior of evanescent waves in the vicinity of scatterers has been studied extensively [15–18], much less is known about their contribution to focused fields. In fact, focusing of pure evanescent states was only recently suggested as a method of generating a small focal spot [19], followed by several studies analyzing the optical structures in the focal region [20–22]. An efficient lens for focusing of evanescent waves must exhibit two features. First, we note that the evanescent waves decay exponentially away from the aperture, such that if a strong signal is required at the focal plane one needs to find a scheme for amplifying these waves. Recent research has demonstrated that surface plasmons may facilitate such an enhancement [23,24], and several structured plasmonic lens systems have been presented to date [25–27]. Second, it is necessary to find the structure of the electric field right behind the aperture required to efficiently focus the evanescent waves at the focal plane. We will here only address this latter issue, and attempt to analyze it analytically in order to obtain insight into how one should
design the aperture. The underlying physics of evanescent wave focusing is not yet fully understood, in particular when it comes to the polarization behavior of such focused evanescent waves. This works aims at providing new insight into the focusing of both two and three-dimensional polarized evanescent waves. Based on our results, we suggest applications in optical data storage. 2. Focusing of two-dimensional waves Let us consider a two-dimensional quasimonochromatic electric field E in a linear, homogenous and isotropic medium. As opposed to Ref. [20], where the two-dimensional electric field could be treated as a scalar (transverse electric), we require here that it consists of two electric field components Ex and Ez (transverse magnetic). It should also be pointed out that the optical singularities were classified and the intensity of such polarized fields were analyzed in Refs. [21,22]. Here, we put an emphasis on finding suitable intensity structures for optical data storage. In the region of space where there are no sources, the Helmholtz wave equation is given by
0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.09.053
ð1Þ
where k ¼ x=v is the wavenumber, x is the angular frequency and v is the velocity of electromagnetic wave in the medium under consideration. In the angular spectrum representation, the field can be expressed as [2]
Eðx; zÞ ¼
1 2p
Z
1
Ek ðkx Þeikx xþikz z dkx ;
ð2Þ
1
where the angular spectrum is given by
Ek ðkx Þ ¼ E-mail address: [email protected]
r2 þ k2 Eðx; zÞ ¼ 0;
Z
1
1
0
0
Eðx0 ; 0Þeikx x dx ;
ð3Þ
30
L.E. Helseth / Optics Communications 283 (2010) 29–33
"
where
Ek ðkx Þ ¼
Ek;x ðkx Þ Ek;z ðkx Þ
ikz z0
Ek ðkx Þ ¼ e
:
The corresponding magnetic field is found using Maxwell’s equations. We also require that
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k2 k2x if k2 P k2x kz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : i k2 k2 if k2 < k2 x x 2
2
2
2 kx
k < corresponds to inhomogeneous or evanescent plane waves that propagate in the x direction and decay in the z-direction. In the paraxial approximation, we may set kx k, and the diffraction integral reduces to a form well known in the literature (see e.g., Ref. [2] and references therein). Here we are interested in an angular spectrum where kx k. The conditions of validity of such an approximation was discussed in detail in Ref. [20], and will therefore not be repeated here. In the case where only very large spatial frequencies are selected (e.g. by using a grating-like structure to ‘filter out’ homogeneous waves), we have kx k and the evanescent waves are the relevant states such that one can assume qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 kz ¼ i kx k ijkx j. According to Gauss’ law the electric field must fulfill $ E ¼ 0, which gives kx Ek;x ðkx Þ þ kz Ek;z ðkx Þ ¼ 0. That is, the two components of the angular spectrum are related as
Ek;z ðkx Þ ¼
kx Ek;x ðkx Þ: kz
ð4Þ
The ratio jEk;z ðkx Þ=Ek;x ðkx Þj is displayed in Fig. 1. For small kx the zcomponent of the electric field is small, but increases when jkx j ! k, and here the x-component approaches zero (such that jEk;z ðkx Þ=Ek;x ðkx Þj ! 1). It is therefore clear that when a nearly collimated circularly polarized wave is passed through lens system which strongly alters the spatial frequency kx , the ratio Ek;z ðkx Þ=Ek;x ðkx Þ is not conserved. Moreover, the amplitudes of the electric field components are dictated by kx , such that a circularly polarized wave cannot be maintained in the focal region. However, for evanescent waves the situation is different. When kx k we have kz ijkx j and the harmonic field components become circularly polarized such that Ek;z ðkx Þ iðkx =jkx jÞEk;x ðkx Þ. Let us now follow Ref. [20] and multiply the angular spectrum by a factor eikz z0 such that
4 3.5
|E k,z /Ek,x |
3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2 2.5 |k x /k|
3
3.5
4
Fig. 1. The ratio of the z and x-components of the angular spectrum, jEk;z ðkx Þ=Ek;x ðkx Þj, as a function of jkx =kj.
kkxz
;
where we require that fk ðkx Þ is bandlimited (i.e. nonzero only in a range ka 6 kx 6 kb ) or converge sufficiently quickly such that the field integral (Eq. (2)) does not diverge. Then the electric field at any plane z > 0 is given by
Eðx; zÞ ¼
Here, k > kx represents the homogenous plane waves, whereas
fk ðkx Þ
#
1
1 2p
Z
1
"
1
# fk ðkx Þeikx xþikz ðzz0 Þ dkx :
kkxz
1
It is seen that when kx k, we have kz ijkx j and the harmonic field components become circularly polarized such that Ek;z ðkx Þ i ðkx =jkx jÞEk;x ðkx Þ. The electric field at any position behind the aperture is now given by
1 Eðx; zÞ 2p
Z
1
1
"
1 i jkkxx j
# fk ðkx Þeikx xjkx jðzz0 Þ dkx :
Note that the electric field is here circularly polarized independently of the exact polarization state of the field impinging on the aperture (as long as two electric field components Ex and Ez exist or are generated at the aperture), due to the fact that homogeneous waves carrying this information have been filtered out. It is therefore clear that a metamaterial superlens focusing both homogenous and evanescent waves (see Ref. [28]) cannot alone produce a focused, circularly polarized wave as seen here. However, by properly selecting the spatial frequencies one can also allow such a superlens to act as a device for focusing circularly polarized light. It is also clear that the focal line is located at z ¼ z0 , where the electric field is just a Fourier Transform of the modulation function fk ðkx Þ. As a simple example, assume that fk ðkx Þ ¼ E0 if ka 6 kx 6 kb and zero elsewhere. Here, ka k and E0 is a constant. The field can then be expressed as
Eðx; zÞ
E0 1 eikb xkb ðzz0 Þ eika xka ðzz0 Þ : 2p i ix ðz z0 Þ
The field at the focal line is then given by
kb ka E0 1 ikb þka x sin 2 x Eðx; z0 Þ e 2 : x p i Fig. 2a shows the normalized intensity ðI / jEj2 Þ at the aperture (dashed line) and the focal line z ¼ z0 (solid line) when kb z0 ¼ 2, whereas Fig. 2b shows the same distributions for kb z0 ¼ 5. In both cases we assume that kb ka . It is seen from Fig. 2 that for small kb z0 the intensity distribution at z0 is less confined than at the aperture. Note that the envelope of the field at z ¼ 0 is much broader than that at z ¼ z0 in Fig. 2b. This comes from the fact that as the intensity at the aperture ðz ¼ 0Þ is proportional to 1=ðx2 þ z2 Þ, the intensity at the focal line ðz ¼ z0 Þ is proportional 2 to sin ðkb x=2Þ=x2 . A possible criterium for focusing could be obtained by requiring that the aperture intensity envelope at x ¼ z0 , corresponding to the position half the maximum intensity, should be wider than the focused intensity distribution of width approximately given by p=kb . Thus, focusing would occur if we require kb z0 P p, which is consistent with what we see in Fig. 2. The intensity at the aperture will be stronger than that at the focus by a factor e2kb z0 P e2p . We also note that in order for only pure evanescent states to exist we must require kb k, which is fulfilled if we, as an example, set kb ¼ 10k ¼ 20p=k (k is the wavelength in the medium), such that the half-width of the intensity distribution at the focal line is narrower than p=kb ¼ k=20. Thus, subwavelength resolution is definitely within reach using this approach. The bad news is that in order to be able to focus light to such a small spot, the subwavelength aperture structure at z ¼ 0
31
L.E. Helseth / Optics Communications 283 (2010) 29–33
polarization state after passing through the system. This is related to the fact that electromagnetic waves are transversal, such that when the lens system changes the directions of the rays, it also alters the polarization state. The subwavelength grating structure studied above may utilize the two-dimensional geometry and circumvent this problem by producing highly confined, circularly polarized waves. The magnetization distribution at the focal line is then given by
1
(a)
Intensity (normalized)
0.5
2
Mðx; z0 Þ ¼ icE E / cjE0 j2
0 −20 1
−10
0
10
20
(b)
0.5
sin
kb ka 2
x
x2
ez ;
ð5Þ
where c is a magneto-optic constant [30] and ez is the unit vector along the optical axis. Thus, a magnetization distribution similar to that seen in Fig. 2 could be generated using this approach. The drawback is the fact that only lines, not dots, can be reproduced. However, only two-dimensional evanescent waves give rise to perfectly circularly polarized light in the focal region, and a threedimensional evanescent wave focusing system is therefore expected to produce a less perfect situation. 3. Focusing of three-dimensional waves
0 −20
−10
0
10
20
k x b Fig. 2. The intensity of a two-dimensional, circularly polarized evanescent wave. The solid line shows the intensity at the focal line z ¼ z0 , whereas the dashed line shows the intensity at the aperture z ¼ 0. Figure a) corresponds to kb z0 ¼ 2 whereas b) corresponds to kb z0 ¼ 5.
For a quasimonochromatic, three-dimensional electric field E in regions without sources, the Helmholtz wave equation is given by
$2 þ k2 Eðx; y; xÞ ¼ 0:
In the angular spectrum representation, the field can be expressed as
2 Z 1 Z 1 1 Ek ðkx ; ky Þeiðkx xþky yþkz zÞ dkx dky ; 2p 1 1
Eðx; y; zÞ ¼ must be able to generate the electric field Eðx; 0Þ given above, which exhibits features much finer than k=20. Thus, considerable constraints are put on the manufacturing device. Typical wavelengths of visible light have a magnitude k 550 nm, giving a focal line of width 28 nm. Current electron beam lithography systems can reproduce features down to a few nanometers, such that the goal should be within reach. The amplitude and phase modulation could be generated using a planar grating-like structure as proposed in Ref. [19]. To this end, it should be noted that circular polarization is generated, if the grating is illuminated by a transverse magnetic wave or the grating somehow generates two electric field components, due to fact that only the large spatial frequencies remain in the angular spectrum, thus providing a polarization filter which generates the given polarization state. The exact structure to be employed will depend on the optical properties of the grating, and whether it does support plasmons or not, and will therefore not be considered here, where only the ideal situation has been analyzed. The focused evanescent circularly polarized waves considered above may have applications in all-optical data storage utilizing the inverse Faraday effect (IFE). The IFE was first predicted nearly 50 years ago [29], and has since been studied experimentally and theoretically [30–32]. The light-induced magnetization related to the IFE has been shown to follow M / iE E , and it is therefore obvious that the polarization properties of light play an important role [29,30]. Recent theoretical studies have shown that the IFE may be of importance in future all-optical data storage [33,34], but only relatively large magnetization imprints of dimensions >1 lm have so far been generated experimentally [32]. It is clear that one needs to go well below 300 nm if all-optical data storage is to challenge traditional optical data storage techniques, thus requiring new methods for focusing circularly polarized light. It is well known that a circularly polarized electromagnetic wave incident on a strongly focusing lens system will not maintain its
ð6Þ
ð7Þ
where
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k2 k2x k2y if k2 P k2x þ k2y kz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : i k2 þ k2 k2 if k2 < k2 þ k2 x y x y This is the angular spectrum representation, where Ek ðkx ; ky Þ is the angular spectrum,
Ek ðkx ; ky Þ ¼
Z
2
1
Z
1
1
1
2
2
Eðx; y; 0Þeiðkx xþky yÞ dxdy:
ð8Þ
When k > kx þ ky the plane waves are homogenous, whereas for 2 2 2 k < kx þ ky the plane waves are evanescent (inhomogeneous) and propagate in the transversal direction while decaying in the z direction. Let us now define Ek ðkx ; ky Þ ¼ f k ðkx ; ky Þ expðikz z0 Þ such that f k ðkx ; ky Þ is bandlimited and converges sufficiently quickly such that Eq. (7) does not diverge [20]. For a cylindrical system we may perform a change of coordinates such that kx ¼ q cos a; ky ¼ q sin a; x ¼ r cos b and y ¼ r sin b. Radially polarized homogeneous waves have recently studied in great detail due to their ability to generate tightly focused homogeneous waves [36,37]. Radially polarized homogeneous waves can be created by choosing the following angular spectrum:
2 kz
f k ðq; aÞ ¼
k 6 f0 ðqÞ4 kkz
cos a
3
7 sin a 5:
qk
The electric field vector is now given by
Eðr; zÞ ¼
1 2pk
Z 0
2
1
3 ikz J 1 ðqrÞ cos b 6 7 f0 ðqÞ4 ikz J 1 ðqrÞ sin b 5eikz ðzz0 Þ qdq: qJ 0 ðqrÞ
L.E. Helseth / Optics Communications 283 (2010) 29–33
It is seen that the electric fields presented in Refs. [35–37] can be recreated if we assume q < k; q ¼ k sin h; kz ¼ k cos h and h is the angle each optical ray forms with the z-axis. It is seen that when h ¼ 90 , both the x and y-components of the electric field vanish since q ¼ k here. Under such conditions a homogeneous wave originally of radial polarization is entirely polarized along the optical axis. Unfortunately, this interesting and highly desirable property does not transfer to the case of pure evanescent states, where the magnitude of the axial wave vector does not vanish, such that Maxwells equations do not support a polarization field vector aligned solely along the z-direction. In other words, introduction of evanescent waves alter the polarization component along the z-axis without altering azimuthal symmetry. For evanescent waves we have kz iq, such that the field distribution is given by
Eðr; zÞ
Z
1 2pk
0
2
1
3
J 1 ðqrÞ cos b 6 7 f0 ðqÞ4 J 1 ðqrÞ sin b 5eqðzz0 Þ q2 dq;
1 2p
Z
1
ð9Þ
0
Here, f ðqÞ is the angular spectrum, which must designed to obtain the optimal intensity distribution at z0 . As an example, we now assume that f ðqÞ ¼ f0 (where f0 is a constant), ka 6 q 6 kb ðka kÞ and zero elsewhere. The latter condition means that the subwavelength zone structure must be much smaller than the wavelength, since the spatial frequencies are much larger than k. The the electric field at z0 is now given by an Airy function 2
Eðr; 0Þ ¼
2
kb A J 1 ðkb rÞ ka f0 J 1 ðka rÞ ; 2p k b r 2p ka r
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0 −10
J 0 ðqrÞ
f ðqÞJ 0 ðqrÞeikz ðzz0 Þ qdq:
0.9
0.1
where f0 ðqÞ – 0 only if ka 6 q 6 kb . The fact that all polarization components remain is undesirable, as it gives rise to polarization aberrations which tend to make the spot at z ¼ z0 broader. We may term the evanescent wave quasi-radially polarized, due to the existence of the strong field component along the optical axis. The above polarization distribution is obviously not optimal in terms of resolution in optical data storage systems (if resolution is considered the most important factor). We must therefore ask ourselves for what polarization distribution can we expect to find the smallest intensity distribution at z ¼ z0 . The answer is, in analogy with the case of homogeneous waves, most likely a scalar-type of field, i.e. longitudinal polarization, where the light is linearly polarized along the optical axis. We will come back to a more practical account of this matter later, but for now we notice that in the angular spectrum representation one can write
Eðr; zÞ ¼
1
Intensity (normalized)
32
ð10Þ
while it at any other position can be evaluated numerically. A comparison of the (normalized) intensity of the electric field obtained using quasi-radial and longitudinal polarization is shown in Fig. 3 for kb =ka ¼ 5. It is seen that a scalar (longitudinal polarization) field gives rise to a more confined intensity distribution at z0 . The pure evanescent modes studied above may also be used to improve the resolution in conventional optical data storage techniques. To overcome the classical diffraction limit, much attention has been paid to hybrid immersion technologies [3–14]. In a recent study, a ridge waveguide placed at the flat surface of a solid immersion lens (SIL) was proposed and analyzed as method for creating a small focal spot [8]. However, such a system may not be optimal for collecting and focusing of evanescent waves. Here, we propose an entirely different approach based on the theory presented above, wherein the nanostructure at the surface of the SIL is used to shape the evanescent waves in a more optimal fashion. We propose combining a longitudinal polarized field generated using a SIL with a subwavelength zone structure in order to generate a more strongly confined evanescent wave, see Fig. 4. The coordinate
−5
0 k br
5
10
Fig. 3. The normalized intensity of a longitudinally (solid line)and quasi-radially (dashed line) polarized evanescent wave at z ¼ z0 . In both cases kb ¼ 5 (a.u.) whereas ka ¼ 1 (a.u.).
system is shown in Fig. 4, where the origin is placed at the focal point of the zone structure (which differs from the focal point of the waves focused by the SIL). Note that the incoming wave is radially polarized thus resulting in a strongly focused, z-polarized beam at the lower surface of the SIL. We will therefore assume that the electric field generated by the SIL at the focal plane is polarized entirely in the z-direction. In order to obtain the best possible resolution we must require that the zone structure placed at the focal plane of the SIL behaves as a continuous absorbtion and phase filter. That is, it will pass the electric field without changing its polarization, while at the same time attenuating its amplitude by a certain factor and modifying its phase by either 0° or 180°. In fact, we want our zone structure to utilize the evanescent waves and shape them into a well-defined focal spot, typically with an intensity distribution similar to that of Fig. 3. As stated above, pure evanescent waves can only be excited if q k. We also note that since the SIL focuses the electromagnetic wave to tight spot approximately given by 0:4k (when the numerical aperture is unity, see Ref. [10]), the zone structure must have much smaller dimensions than this. An important reason for using a SIL to obtain a tightly focused spot is simply to reduce the energy not participating in the focusing of evanescent waves. Moreover, the fact that the polarization of light is perpendicular to the zone structure ensures that circular symmetry is maintained (one may imagine the introduction of azimuthal asymmetry if the polarization was along the x or y directions). We may expect that kb k is fulfilled if we set kb ¼ 10k ¼ 20p=k (k is the wavelength in the medium). Furthermore, we know that J 1 ðkb rÞ is zero when qr 3:8, which gives a focal spot of diameter k=10. Typical wavelengths of visible light have a magnitude k 550 nm, which gives a focal spot diameter 55 nm. A disadvantage of using evanscent waves is the exponential decay of the intensity behind the aperture, found using Eq. (9). Fig. 5 shows an example of this, where the intensity drops by factor of 4 by changing kb ðz z0 Þ from 1 to 0. This corresponds to moving the data storage medium a distance k=125, which suggests that the system is extremely sensitive towards, e.g., vibrations, and must therefore be properly isolated. Moreover, thermally induced fluctuations start to play a role when approaching the nanometer scale, and such issues will ultimately decide whether subwavelength evanescent lenses can be a part of future optical data storage technologies.
L.E. Helseth / Optics Communications 283 (2010) 29–33
33
Fig. 4. Schematic drawing of a subwavelength structure located at the bottom surface of a SIL, thus generating evanescent waves focused a small distance behind the SIL.
1
increase in resolution over e.g. quasi-radial modes, and may have applications in extensions of conventional optical data storage.
Intensity (normalized)
0.9 0.8
References
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1
−0.5
0
0.5 1 k (z−z ) b
1.5
2
2.5
0
Fig. 5. The normalized intensity of a longitudinally polarized evanescent wave as a function of kb ðz z0 Þ. The vertical line denotes the position of the focal point.
4. Conclusion In conclusion, we have studied the structure of evanescent waves in focal regions, and suggested possible applications of these waves in optical data storage. For two-dimensional waves the field structures are restricted by geometry and may therefore produce purely circular polarization. We have suggested that such polarization states may be efficiently used in all-optical data storage, although it should be pointed out the geometry restricts the resolution to one dimension. Furthermore, we anticipate that circularly polarized evanescent waves may find use in manipulation and detection of chiral molecules, where such radiation is beneficial. Three-dimensional waves evanescent do, on the other hand, give rise to rich polarization structures in the focal region, whereof only two have been studied here. However, we show that scalar-type waves give, in addition to a Airy-like beam signature, significant
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Focusing of evanescent vector waves L.E. Helseth Department of Physics and Technology, University of Bergen, N-5007 Bergen, Norway
a r t i c l e
i n f o
Article history: Received 10 July 2009 Received in revised form 18 September 2009 Accepted 24 September 2009
a b s t r a c t We study focusing of two and three-dimensional evanescent vector waves, with a particular emphasis on identifying suitable intensity structures for applications in optical data storage. For two-dimensional evanescent waves large transverse spatial wave vectors result in purely circularly polarized evanescent states. We suggest that these may have applications in all-optical data storage through the inverse Faraday effect. On the other hand, for three-dimensional evanescent waves longitudinally polarized modes are observed to give the most tightly focused spot, and this may be utilized to confine light behind a solid immersion lens. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Focusing of electromagnetic waves has been studied for more than a century, and the diffraction behavior is well-understood in both the paraxial and non-paraxial regimes [1,2]. Conventional lens systems for propagation of homogeneous waves cannot focus light to a spot smaller than the classical diffraction limit [1]. To overcome this problem, immersion technologies using the fact that the wavelength in a medium may be smaller than that of air, have been proposed and tested. These lens, axicon or mirror systems utilize that the size of the focal spot scales with refractive index of the medium containing the focal plane [3–14]. Although the behavior of evanescent waves in the vicinity of scatterers has been studied extensively [15–18], much less is known about their contribution to focused fields. In fact, focusing of pure evanescent states was only recently suggested as a method of generating a small focal spot [19], followed by several studies analyzing the optical structures in the focal region [20–22]. An efficient lens for focusing of evanescent waves must exhibit two features. First, we note that the evanescent waves decay exponentially away from the aperture, such that if a strong signal is required at the focal plane one needs to find a scheme for amplifying these waves. Recent research has demonstrated that surface plasmons may facilitate such an enhancement [23,24], and several structured plasmonic lens systems have been presented to date [25–27]. Second, it is necessary to find the structure of the electric field right behind the aperture required to efficiently focus the evanescent waves at the focal plane. We will here only address this latter issue, and attempt to analyze it analytically in order to obtain insight into how one should
design the aperture. The underlying physics of evanescent wave focusing is not yet fully understood, in particular when it comes to the polarization behavior of such focused evanescent waves. This works aims at providing new insight into the focusing of both two and three-dimensional polarized evanescent waves. Based on our results, we suggest applications in optical data storage. 2. Focusing of two-dimensional waves Let us consider a two-dimensional quasimonochromatic electric field E in a linear, homogenous and isotropic medium. As opposed to Ref. [20], where the two-dimensional electric field could be treated as a scalar (transverse electric), we require here that it consists of two electric field components Ex and Ez (transverse magnetic). It should also be pointed out that the optical singularities were classified and the intensity of such polarized fields were analyzed in Refs. [21,22]. Here, we put an emphasis on finding suitable intensity structures for optical data storage. In the region of space where there are no sources, the Helmholtz wave equation is given by
0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.09.053
ð1Þ
where k ¼ x=v is the wavenumber, x is the angular frequency and v is the velocity of electromagnetic wave in the medium under consideration. In the angular spectrum representation, the field can be expressed as [2]
Eðx; zÞ ¼
1 2p
Z
1
Ek ðkx Þeikx xþikz z dkx ;
ð2Þ
1
where the angular spectrum is given by
Ek ðkx Þ ¼ E-mail address: [email protected]
r2 þ k2 Eðx; zÞ ¼ 0;
Z
1
1
0
0
Eðx0 ; 0Þeikx x dx ;
ð3Þ
30
L.E. Helseth / Optics Communications 283 (2010) 29–33
"
where
Ek ðkx Þ ¼
Ek;x ðkx Þ Ek;z ðkx Þ
ikz z0
Ek ðkx Þ ¼ e
:
The corresponding magnetic field is found using Maxwell’s equations. We also require that
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k2 k2x if k2 P k2x kz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : i k2 k2 if k2 < k2 x x 2
2
2
2 kx
k < corresponds to inhomogeneous or evanescent plane waves that propagate in the x direction and decay in the z-direction. In the paraxial approximation, we may set kx k, and the diffraction integral reduces to a form well known in the literature (see e.g., Ref. [2] and references therein). Here we are interested in an angular spectrum where kx k. The conditions of validity of such an approximation was discussed in detail in Ref. [20], and will therefore not be repeated here. In the case where only very large spatial frequencies are selected (e.g. by using a grating-like structure to ‘filter out’ homogeneous waves), we have kx k and the evanescent waves are the relevant states such that one can assume qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 kz ¼ i kx k ijkx j. According to Gauss’ law the electric field must fulfill $ E ¼ 0, which gives kx Ek;x ðkx Þ þ kz Ek;z ðkx Þ ¼ 0. That is, the two components of the angular spectrum are related as
Ek;z ðkx Þ ¼
kx Ek;x ðkx Þ: kz
ð4Þ
The ratio jEk;z ðkx Þ=Ek;x ðkx Þj is displayed in Fig. 1. For small kx the zcomponent of the electric field is small, but increases when jkx j ! k, and here the x-component approaches zero (such that jEk;z ðkx Þ=Ek;x ðkx Þj ! 1). It is therefore clear that when a nearly collimated circularly polarized wave is passed through lens system which strongly alters the spatial frequency kx , the ratio Ek;z ðkx Þ=Ek;x ðkx Þ is not conserved. Moreover, the amplitudes of the electric field components are dictated by kx , such that a circularly polarized wave cannot be maintained in the focal region. However, for evanescent waves the situation is different. When kx k we have kz ijkx j and the harmonic field components become circularly polarized such that Ek;z ðkx Þ iðkx =jkx jÞEk;x ðkx Þ. Let us now follow Ref. [20] and multiply the angular spectrum by a factor eikz z0 such that
4 3.5
|E k,z /Ek,x |
3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2 2.5 |k x /k|
3
3.5
4
Fig. 1. The ratio of the z and x-components of the angular spectrum, jEk;z ðkx Þ=Ek;x ðkx Þj, as a function of jkx =kj.
kkxz
;
where we require that fk ðkx Þ is bandlimited (i.e. nonzero only in a range ka 6 kx 6 kb ) or converge sufficiently quickly such that the field integral (Eq. (2)) does not diverge. Then the electric field at any plane z > 0 is given by
Eðx; zÞ ¼
Here, k > kx represents the homogenous plane waves, whereas
fk ðkx Þ
#
1
1 2p
Z
1
"
1
# fk ðkx Þeikx xþikz ðzz0 Þ dkx :
kkxz
1
It is seen that when kx k, we have kz ijkx j and the harmonic field components become circularly polarized such that Ek;z ðkx Þ i ðkx =jkx jÞEk;x ðkx Þ. The electric field at any position behind the aperture is now given by
1 Eðx; zÞ 2p
Z
1
1
"
1 i jkkxx j
# fk ðkx Þeikx xjkx jðzz0 Þ dkx :
Note that the electric field is here circularly polarized independently of the exact polarization state of the field impinging on the aperture (as long as two electric field components Ex and Ez exist or are generated at the aperture), due to the fact that homogeneous waves carrying this information have been filtered out. It is therefore clear that a metamaterial superlens focusing both homogenous and evanescent waves (see Ref. [28]) cannot alone produce a focused, circularly polarized wave as seen here. However, by properly selecting the spatial frequencies one can also allow such a superlens to act as a device for focusing circularly polarized light. It is also clear that the focal line is located at z ¼ z0 , where the electric field is just a Fourier Transform of the modulation function fk ðkx Þ. As a simple example, assume that fk ðkx Þ ¼ E0 if ka 6 kx 6 kb and zero elsewhere. Here, ka k and E0 is a constant. The field can then be expressed as
Eðx; zÞ
E0 1 eikb xkb ðzz0 Þ eika xka ðzz0 Þ : 2p i ix ðz z0 Þ
The field at the focal line is then given by
kb ka E0 1 ikb þka x sin 2 x Eðx; z0 Þ e 2 : x p i Fig. 2a shows the normalized intensity ðI / jEj2 Þ at the aperture (dashed line) and the focal line z ¼ z0 (solid line) when kb z0 ¼ 2, whereas Fig. 2b shows the same distributions for kb z0 ¼ 5. In both cases we assume that kb ka . It is seen from Fig. 2 that for small kb z0 the intensity distribution at z0 is less confined than at the aperture. Note that the envelope of the field at z ¼ 0 is much broader than that at z ¼ z0 in Fig. 2b. This comes from the fact that as the intensity at the aperture ðz ¼ 0Þ is proportional to 1=ðx2 þ z2 Þ, the intensity at the focal line ðz ¼ z0 Þ is proportional 2 to sin ðkb x=2Þ=x2 . A possible criterium for focusing could be obtained by requiring that the aperture intensity envelope at x ¼ z0 , corresponding to the position half the maximum intensity, should be wider than the focused intensity distribution of width approximately given by p=kb . Thus, focusing would occur if we require kb z0 P p, which is consistent with what we see in Fig. 2. The intensity at the aperture will be stronger than that at the focus by a factor e2kb z0 P e2p . We also note that in order for only pure evanescent states to exist we must require kb k, which is fulfilled if we, as an example, set kb ¼ 10k ¼ 20p=k (k is the wavelength in the medium), such that the half-width of the intensity distribution at the focal line is narrower than p=kb ¼ k=20. Thus, subwavelength resolution is definitely within reach using this approach. The bad news is that in order to be able to focus light to such a small spot, the subwavelength aperture structure at z ¼ 0
31
L.E. Helseth / Optics Communications 283 (2010) 29–33
polarization state after passing through the system. This is related to the fact that electromagnetic waves are transversal, such that when the lens system changes the directions of the rays, it also alters the polarization state. The subwavelength grating structure studied above may utilize the two-dimensional geometry and circumvent this problem by producing highly confined, circularly polarized waves. The magnetization distribution at the focal line is then given by
1
(a)
Intensity (normalized)
0.5
2
Mðx; z0 Þ ¼ icE E / cjE0 j2
0 −20 1
−10
0
10
20
(b)
0.5
sin
kb ka 2
x
x2
ez ;
ð5Þ
where c is a magneto-optic constant [30] and ez is the unit vector along the optical axis. Thus, a magnetization distribution similar to that seen in Fig. 2 could be generated using this approach. The drawback is the fact that only lines, not dots, can be reproduced. However, only two-dimensional evanescent waves give rise to perfectly circularly polarized light in the focal region, and a threedimensional evanescent wave focusing system is therefore expected to produce a less perfect situation. 3. Focusing of three-dimensional waves
0 −20
−10
0
10
20
k x b Fig. 2. The intensity of a two-dimensional, circularly polarized evanescent wave. The solid line shows the intensity at the focal line z ¼ z0 , whereas the dashed line shows the intensity at the aperture z ¼ 0. Figure a) corresponds to kb z0 ¼ 2 whereas b) corresponds to kb z0 ¼ 5.
For a quasimonochromatic, three-dimensional electric field E in regions without sources, the Helmholtz wave equation is given by
$2 þ k2 Eðx; y; xÞ ¼ 0:
In the angular spectrum representation, the field can be expressed as
2 Z 1 Z 1 1 Ek ðkx ; ky Þeiðkx xþky yþkz zÞ dkx dky ; 2p 1 1
Eðx; y; zÞ ¼ must be able to generate the electric field Eðx; 0Þ given above, which exhibits features much finer than k=20. Thus, considerable constraints are put on the manufacturing device. Typical wavelengths of visible light have a magnitude k 550 nm, giving a focal line of width 28 nm. Current electron beam lithography systems can reproduce features down to a few nanometers, such that the goal should be within reach. The amplitude and phase modulation could be generated using a planar grating-like structure as proposed in Ref. [19]. To this end, it should be noted that circular polarization is generated, if the grating is illuminated by a transverse magnetic wave or the grating somehow generates two electric field components, due to fact that only the large spatial frequencies remain in the angular spectrum, thus providing a polarization filter which generates the given polarization state. The exact structure to be employed will depend on the optical properties of the grating, and whether it does support plasmons or not, and will therefore not be considered here, where only the ideal situation has been analyzed. The focused evanescent circularly polarized waves considered above may have applications in all-optical data storage utilizing the inverse Faraday effect (IFE). The IFE was first predicted nearly 50 years ago [29], and has since been studied experimentally and theoretically [30–32]. The light-induced magnetization related to the IFE has been shown to follow M / iE E , and it is therefore obvious that the polarization properties of light play an important role [29,30]. Recent theoretical studies have shown that the IFE may be of importance in future all-optical data storage [33,34], but only relatively large magnetization imprints of dimensions >1 lm have so far been generated experimentally [32]. It is clear that one needs to go well below 300 nm if all-optical data storage is to challenge traditional optical data storage techniques, thus requiring new methods for focusing circularly polarized light. It is well known that a circularly polarized electromagnetic wave incident on a strongly focusing lens system will not maintain its
ð6Þ
ð7Þ
where
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k2 k2x k2y if k2 P k2x þ k2y kz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : i k2 þ k2 k2 if k2 < k2 þ k2 x y x y This is the angular spectrum representation, where Ek ðkx ; ky Þ is the angular spectrum,
Ek ðkx ; ky Þ ¼
Z
2
1
Z
1
1
1
2
2
Eðx; y; 0Þeiðkx xþky yÞ dxdy:
ð8Þ
When k > kx þ ky the plane waves are homogenous, whereas for 2 2 2 k < kx þ ky the plane waves are evanescent (inhomogeneous) and propagate in the transversal direction while decaying in the z direction. Let us now define Ek ðkx ; ky Þ ¼ f k ðkx ; ky Þ expðikz z0 Þ such that f k ðkx ; ky Þ is bandlimited and converges sufficiently quickly such that Eq. (7) does not diverge [20]. For a cylindrical system we may perform a change of coordinates such that kx ¼ q cos a; ky ¼ q sin a; x ¼ r cos b and y ¼ r sin b. Radially polarized homogeneous waves have recently studied in great detail due to their ability to generate tightly focused homogeneous waves [36,37]. Radially polarized homogeneous waves can be created by choosing the following angular spectrum:
2 kz
f k ðq; aÞ ¼
k 6 f0 ðqÞ4 kkz
cos a
3
7 sin a 5:
qk
The electric field vector is now given by
Eðr; zÞ ¼
1 2pk
Z 0
2
1
3 ikz J 1 ðqrÞ cos b 6 7 f0 ðqÞ4 ikz J 1 ðqrÞ sin b 5eikz ðzz0 Þ qdq: qJ 0 ðqrÞ
L.E. Helseth / Optics Communications 283 (2010) 29–33
It is seen that the electric fields presented in Refs. [35–37] can be recreated if we assume q < k; q ¼ k sin h; kz ¼ k cos h and h is the angle each optical ray forms with the z-axis. It is seen that when h ¼ 90 , both the x and y-components of the electric field vanish since q ¼ k here. Under such conditions a homogeneous wave originally of radial polarization is entirely polarized along the optical axis. Unfortunately, this interesting and highly desirable property does not transfer to the case of pure evanescent states, where the magnitude of the axial wave vector does not vanish, such that Maxwells equations do not support a polarization field vector aligned solely along the z-direction. In other words, introduction of evanescent waves alter the polarization component along the z-axis without altering azimuthal symmetry. For evanescent waves we have kz iq, such that the field distribution is given by
Eðr; zÞ
Z
1 2pk
0
2
1
3
J 1 ðqrÞ cos b 6 7 f0 ðqÞ4 J 1 ðqrÞ sin b 5eqðzz0 Þ q2 dq;
1 2p
Z
1
ð9Þ
0
Here, f ðqÞ is the angular spectrum, which must designed to obtain the optimal intensity distribution at z0 . As an example, we now assume that f ðqÞ ¼ f0 (where f0 is a constant), ka 6 q 6 kb ðka kÞ and zero elsewhere. The latter condition means that the subwavelength zone structure must be much smaller than the wavelength, since the spatial frequencies are much larger than k. The the electric field at z0 is now given by an Airy function 2
Eðr; 0Þ ¼
2
kb A J 1 ðkb rÞ ka f0 J 1 ðka rÞ ; 2p k b r 2p ka r
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0 −10
J 0 ðqrÞ
f ðqÞJ 0 ðqrÞeikz ðzz0 Þ qdq:
0.9
0.1
where f0 ðqÞ – 0 only if ka 6 q 6 kb . The fact that all polarization components remain is undesirable, as it gives rise to polarization aberrations which tend to make the spot at z ¼ z0 broader. We may term the evanescent wave quasi-radially polarized, due to the existence of the strong field component along the optical axis. The above polarization distribution is obviously not optimal in terms of resolution in optical data storage systems (if resolution is considered the most important factor). We must therefore ask ourselves for what polarization distribution can we expect to find the smallest intensity distribution at z ¼ z0 . The answer is, in analogy with the case of homogeneous waves, most likely a scalar-type of field, i.e. longitudinal polarization, where the light is linearly polarized along the optical axis. We will come back to a more practical account of this matter later, but for now we notice that in the angular spectrum representation one can write
Eðr; zÞ ¼
1
Intensity (normalized)
32
ð10Þ
while it at any other position can be evaluated numerically. A comparison of the (normalized) intensity of the electric field obtained using quasi-radial and longitudinal polarization is shown in Fig. 3 for kb =ka ¼ 5. It is seen that a scalar (longitudinal polarization) field gives rise to a more confined intensity distribution at z0 . The pure evanescent modes studied above may also be used to improve the resolution in conventional optical data storage techniques. To overcome the classical diffraction limit, much attention has been paid to hybrid immersion technologies [3–14]. In a recent study, a ridge waveguide placed at the flat surface of a solid immersion lens (SIL) was proposed and analyzed as method for creating a small focal spot [8]. However, such a system may not be optimal for collecting and focusing of evanescent waves. Here, we propose an entirely different approach based on the theory presented above, wherein the nanostructure at the surface of the SIL is used to shape the evanescent waves in a more optimal fashion. We propose combining a longitudinal polarized field generated using a SIL with a subwavelength zone structure in order to generate a more strongly confined evanescent wave, see Fig. 4. The coordinate
−5
0 k br
5
10
Fig. 3. The normalized intensity of a longitudinally (solid line)and quasi-radially (dashed line) polarized evanescent wave at z ¼ z0 . In both cases kb ¼ 5 (a.u.) whereas ka ¼ 1 (a.u.).
system is shown in Fig. 4, where the origin is placed at the focal point of the zone structure (which differs from the focal point of the waves focused by the SIL). Note that the incoming wave is radially polarized thus resulting in a strongly focused, z-polarized beam at the lower surface of the SIL. We will therefore assume that the electric field generated by the SIL at the focal plane is polarized entirely in the z-direction. In order to obtain the best possible resolution we must require that the zone structure placed at the focal plane of the SIL behaves as a continuous absorbtion and phase filter. That is, it will pass the electric field without changing its polarization, while at the same time attenuating its amplitude by a certain factor and modifying its phase by either 0° or 180°. In fact, we want our zone structure to utilize the evanescent waves and shape them into a well-defined focal spot, typically with an intensity distribution similar to that of Fig. 3. As stated above, pure evanescent waves can only be excited if q k. We also note that since the SIL focuses the electromagnetic wave to tight spot approximately given by 0:4k (when the numerical aperture is unity, see Ref. [10]), the zone structure must have much smaller dimensions than this. An important reason for using a SIL to obtain a tightly focused spot is simply to reduce the energy not participating in the focusing of evanescent waves. Moreover, the fact that the polarization of light is perpendicular to the zone structure ensures that circular symmetry is maintained (one may imagine the introduction of azimuthal asymmetry if the polarization was along the x or y directions). We may expect that kb k is fulfilled if we set kb ¼ 10k ¼ 20p=k (k is the wavelength in the medium). Furthermore, we know that J 1 ðkb rÞ is zero when qr 3:8, which gives a focal spot of diameter k=10. Typical wavelengths of visible light have a magnitude k 550 nm, which gives a focal spot diameter 55 nm. A disadvantage of using evanscent waves is the exponential decay of the intensity behind the aperture, found using Eq. (9). Fig. 5 shows an example of this, where the intensity drops by factor of 4 by changing kb ðz z0 Þ from 1 to 0. This corresponds to moving the data storage medium a distance k=125, which suggests that the system is extremely sensitive towards, e.g., vibrations, and must therefore be properly isolated. Moreover, thermally induced fluctuations start to play a role when approaching the nanometer scale, and such issues will ultimately decide whether subwavelength evanescent lenses can be a part of future optical data storage technologies.
L.E. Helseth / Optics Communications 283 (2010) 29–33
33
Fig. 4. Schematic drawing of a subwavelength structure located at the bottom surface of a SIL, thus generating evanescent waves focused a small distance behind the SIL.
1
increase in resolution over e.g. quasi-radial modes, and may have applications in extensions of conventional optical data storage.
Intensity (normalized)
0.9 0.8
References
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1
−0.5
0
0.5 1 k (z−z ) b
1.5
2
2.5
0
Fig. 5. The normalized intensity of a longitudinally polarized evanescent wave as a function of kb ðz z0 Þ. The vertical line denotes the position of the focal point.
4. Conclusion In conclusion, we have studied the structure of evanescent waves in focal regions, and suggested possible applications of these waves in optical data storage. For two-dimensional waves the field structures are restricted by geometry and may therefore produce purely circular polarization. We have suggested that such polarization states may be efficiently used in all-optical data storage, although it should be pointed out the geometry restricts the resolution to one dimension. Furthermore, we anticipate that circularly polarized evanescent waves may find use in manipulation and detection of chiral molecules, where such radiation is beneficial. Three-dimensional waves evanescent do, on the other hand, give rise to rich polarization structures in the focal region, whereof only two have been studied here. However, we show that scalar-type waves give, in addition to a Airy-like beam signature, significant
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