The magnetic field in a realistic device is not uniform. For example, in a tokamak whose corresponding schematic plot is shown in the figure 8, the toroidal magnetic field in low--! plasma can be approximately written as B. [B0[1 (r/R) cos /]; where r andR are the minor and major radii, respectively, and /!is the poloidal angle. Thus, the variation of its magnetic field is sinusoidal in the poloidal direction and the magnetic field strength approximately decreases with the increase in its major radial position. When the directions of both the magnetic field minimum. On the other hand, when both directions are along the poloidal direction, one may find a magnetic profile with a sinusoidal nonuniformity. However, the Doppler shift effect of the alpha particles should be accessed while the velocity of slow ions is too small and the electron cyclotron
density. For that with a small velocity ratio and thus a smaller resonance ratio, it may still remain in the resonance for driving the RICI whose growth rate is proportional to the square root of the particle density. Note that the RICI is selective and only the alpha particles with a high perpendicular velocity (e.g. a small velocity ratio) are involved in the interaction with the wave [28–31].
Because the instability and resultant selective broadening can survive under the simplified nonuniformities, their consequences (e.g. on driving waves each occupying a space of a fraction of the minor radius as well as on significantly changing the perpendicular velocity and the pitch angle of energetic alpha particles) and implications (e.g. channeling the alpha particles energy and/or modifications in the particle orbits) for fusion plasmas deserve to be further studied.
Figure 8. A schematic plot of a tokamak device; the r and R are the minor and major radii, respectively, and /!represents the poloidal angle.
In figure 9, we show the growth rate and the real frequency versus )*(based on the local dispersion relation, equation (9), at x = 0; i.e. under a uniform magnetic field. It clearly shows that, for some unstable modes, the wave frequency can be lower than the harmonic ion cyclotron frequency [12] which apparently violates the positive requirement of the frequency mismatch. Although they are not the most unstable mode, as long as one of them can satisfy the absolute instability condition while the most unstable mode cannot, it can survive the convective damping due to the nonuniformity of the magnetic field. Thus, the localized mode can be excited and retains the characteristics of the survivable mode. Moreover, we have also run a simulation keeping only the mode of )*( = 17.25; which is close to the mode satisfying the absolute instability condition considered earlier. Figure 10 shows the instability can indeed exist with a
normalized growth rate and frequency mismatch at 3.7 × 10 3 and 2 × 10 3, respectively, which agree well with the theoretical predictions. Thus, even in a uniform plasma, the electrostatic cyclotron instabilities may not be constrained by the requirement of a positive frequency mismatch. While the relativistic cyclotron instability studied here is for energetic ions, the findings may still have important implications and thus applications for electrons. For relativistic electron cyclotron instabilities, it is well known that the frequency mismatch is required to be positive. The devices usually have a fixed magnetic field. Its operation frequency is determined by the magnetic field and the frequency mismatch. If it is possible to operate for both positive and negative mismatch, the tunability of the devices may be increased.
Figure 9. The normalized growth rate (solid) and the frequency mismatch between wave and resonant cyclotron harmonic (dashed) in a uniform magnetic field.
The RICI under sinusoidal nonuniform magnetic fields has been investigated via, respectively, simulation and analytic methods, adopting a parabolic magnetic field approximation. Other types of nonuniformity for fusion plasmas deserve further investigation. In the simulation, we use the particle-in-cell method to describe the alpha particles and consider the deuterium as a fluid to obtain results with high resolution. Localized wave eigenmodes are observed at the magnetic minimum when the external magnetic field has a sinusoidal nonuniformity. The spatial structure of the electrostatic wave looks like a twin-wavelet. In order to understand the physics and mechanism involved, an analytic theory based on
expansion around the absolute instability condition has been developed. The parabolic magnetic profile considered in theory is an approximation for the magnetic minimum employed in the simulation. Bounded eigenstates are found at the magnetic minimum as observed in the simulation. The structure, growth rate and frequency of the theoretical n = 1 eigenstate are in good agreement with the simulation twin-wavelet. On the other hand, the n = 0 eigenstate of a single packet wave mode from the theory is verified by another simulation; with more limited modes kept in the simulation, only the wave mode of a single packet can be excited, which corresponds to the n = 0 eigenstate. In contrast to the conventional requirement of a positive frequency mismatch, the instability can drive the unstable wave mode existing where the frequency mismatch is negative.
In addition to possible applications to fusion plasmas, this study may also be helpful for electron cyclotron instability and application, especially for high harmonic and for tunability. These results have been published [81-82].
Figure 10. The growth of instability (a) and the measured frequency power spectrum (b) when only one mode )*(= 17.25 is kept in the simulation.
2 The interaction between surface plasmon polaritons and electromagnetic wave in subwavelength nano-structures
2.1 Introduction
Diffraction, as a general wave phenomenon which occurs whenever a traveling wave front encounters and propagates past an obstruction, was first referenced in the work of Leonardo da Vinci in the 1400s [45] and has being accurately described since Francesco Grimaldi in the 1600s [45]. Explanation based on the wave theory was not available until the 1800s [45]. The diffraction limit was the inspiration for Heisenberg’s quantum uncertainty principle [46,47]
that is a foundation of modern science; in fact, they can be deduced from each other [46-48]. The diffraction sets the smallest achievable line width or spot size [1-4], which is the ultimate manipulability and resolution [45,48] of numerous diagnostic and fabrication instruments. For a single aperture, the limitation can be given as +,/NA, where , is the wavelength in vacuum, NA = n sin- ŪŴġ ŵũŦġ ůŶŮŦųŪŤŢŭġ aperturen is the refractive index of the medium where the focused light locates, -*is the convergence angle of the light, and the constant + is 0.38, 0.5 and 0.61 for a ring, line, and circular aperture, respectively [45].
2.2 Background and current conditions
The diffraction limit concerns travelling light that can propagate freely in the free space, in contrast to the evanescent near-field [49,50] that is electrostatic [51-56] or magnetostatic [57,58], needs a preferred plane or surface for propagation and cannot propagate freely in the free space, such as occurs in a super lens [51-58]. For a dipole source, the near, intermediate (or mid) and far fields decrease away from the source as proportional to 1/r3, 1/r2and 1/r, respectively [76], where r is the distance; for a line source, the far field is proportional to 1/r1/2. The scaling laws of the far field conserve the field energy. Mathematically, the divergence of the electrostatic field and the Laplacian of the scalar potential are determined by the charge density according to Gauss’s law and Poisson’s equation, respectively, while the fields of propagating light concerned with the diffraction limit are decided by the spatial curl and temporal derivative equations (i.e., Faraday’s and modified Ampere’s equations) involving the scale of wavelength.
Thus, a new understanding and a different approach are needed to produce a beyond-the-limit focusing of travelling light.
Both the theory of the conventional diffraction limit of light and the Heisenberg’s quantum uncertainty principle consider wave function expanded into reciprocal space and a situation where the scale of the eigenfunction is half the wavelength or larger. For quantum mechanics, the surface integral involving the wave function is assumed to vanish when taken over a very large surface or an infinite potential well. For the latter, the spatial eigenvalue (i.e., the wave number) is k = m./a, where m is a natural number (i.e., non-zero) and a is the well width.
For the ground state, the well width is half a wavelength, while the wave function is the corresponding harmonic function.
2.3 Research purpose
The excitations of surface plasmon [59-61] on metallic surfaces and surface-plasmon-like modes [62-63] are claimed to enhance [64-65] the transmission of light and to beam/focus [66-68] it through subwavelength holes/slits [69]. In fact, the light and the surface plasma are coupled and hence are self-consistent within the slit. The wave function across the slit is close to a constant and drops sharply on the surface. This kind of function with k = 0 mode bounded within a sub-limit scale is not considered in the conventional theories and thus is not within their scope.
The innovative approach and physical mechanisms of the focusing aperture beyond the conventional diffraction limited line width (FAB) of half the wavelength [70] are demonstrated here with the FAB lens including a metallic film with a double-slit and a patterned exit structure, as shown in Fig. 11(a). The width of each slit is smaller than half the wavelength, and thus the limited line width. At the same time, the width of the central metal strip (the interslit spacing) is about a half-wavelength that gives rise to a half-wavelength dipole antenna which can effectively emit propagating fields. Besides the generation of sub-limit wave functions within the slits and at the central area resulted from a polarized field conversion and the surface current, the transmitted light of sub-limit scale will be shown to be bent toward the center and focused to achieve a lower diffraction limit.
2.4 Research method
Finite-Difference-Time-Domain (FDTD) simulation [71] is employed to verify the approach. A structured thin silver film with 20 and 2 grooves at the incident and exit sides, respectively, as shown in Fig. 11(b)is employed as our FAB lens. A simplified structure on silica substrate has been employed in an optical
experiment [72] that verifies this approach, in addition to another experimental confirmation in the microwave range.The refractive index of silver [73] used for ,
= 633 nm is 0.134 + i 3.99. The system has 1600 × 1000 cells of the Yee space lattice with a unit cell size of 5 nm. The top of the silver film is at the y = 800 cell.
The time, t, is normalized to the light period, and the time step is 0.005.
Figure 11(c) shows a snapshot of the magnetic field indicating the propagation of the focused light to the far zone and the near field along the surface.
As shown on Fig. 1(d), the Hzfield is almost in phase with the Exfield, except near the surface at y = 0.2 /m. Also, the profile fits better with the scaling of a far field than that of a mid or near field. Thus, the focused fields are dominated by the propagating field.
For better understanding of the approach and the physics involved, let us consider the dynamics of the focusing. The plane wave propagating downward is transmitted through the double-slit being enhanced by surface plasma and split into
Fig. 11 The schematic structure of the aperture, the approach and the simulation result. (a) Schematic diagram of the approach, aperture structure and the paths of the light transmitted, bent and focused, including the generator of sub-limit wave function at the central area. (b) The schematic structure of the aperture on a silver film used in the FDTD simulation. The depth, the width and the distance in between of the periodical grooves at the incident side are 80 nm, 200 nm and 200 nm, respectively; the slit width is 80 nm; both the width and depth of grooves at the exit side is 80 nm; the distance between the slit and the exit groove is 160 nm;
the film thickness is 280 nm; the thickness and the width of the central strip are 200 nm and 320 nm, respectively. In order to clarify the possible near-field involvement, the central metal width of 320 nm and the exit width of 480 nm are larger than half the wavelength, 316.5 green dots) for the fields being the far, mid and near fields, respectively, according to a dipole source model [32].
two beams which are bent toward the symmetry (y-) axis. In addition, due to the surface currents the metal strip between the slits acts like a half-wavelength dipole antenna (a generator). The interference of the diffracted wave functions from the bent beams and that of the generator result in the subwavelength focal spot, as indicated by the time-averaged Ex field energy contour plot shown in Fig. 12(a).
The Ey field, the polarized surface charge and the x component of time-averaged Poynting vector of the diffracted light from one slit, shown on Fig. 12(b), are cancelled with those from the other. The cancellation converts the energy of the polarized surface charge and the Eyfield to increase the focused Hzfield [Fig. 12(c), at the time (and the phase) defined as t = tf] and hence explains the behavior of the Exfield (Fig. 12a) at the central region so as to generate a sub-limit wave function there. The overall line width of the focused field can be further squeezed by the diffracted field, which is focused at a time half of the period earlier than the focused field, connecting with the transmitting field, which is focused at a time half of the period later, as shown in Fig. 12(c-d). In order to form this field connection,
there is a requirement on the width of the central metal strip and the field diffraction has to be slowed down by the surface plasma including the effect of the grooves on the structure. The snapshot of the focused Hzfield (Fig. 12c) is taken at the moment that there is a peak of focused positive magnetic field outside the
Fig. 2 The contour plots of the fields. (a) The time-averaged contours of the Exenergy. (b) The time-averaged contours of the Poynting vector in the x direction. (c) The snapshot of the focused Hzfield at t = tf. (d) The enlarged snapshot of the Hzfield at t = tf+ 0.12. All are normalized to the incident light.
surface and near the peak position of the time-averaged Poynting vector in the original y propagation direction shown later in Fig. 14(a). After being focused at t
= tf, the light diffracts to forward angles, and the Hzfield propagates out while it is still squeezed as shown in Fig. 12(d). Then, the focused light propagates out to the far zone, as also evidenced from the movies of the electric and magnetic fields.
The line width of the focused light can be defined by the full-width-half-maximum (FWHM) or the width of the spot of the Hzenergy averaged-along-x as twice the position uncertainty that is defined as x = (<x2>-<x>2)1/2, where <f> ='Hz2
f dx / Hz2
dx. Figure 13(a)shows that the FWHM of the time-averaged Hzfield energy agrees well with that for the snapshot of Hzfield energies. While the peak intensity of the focused light remains to be higher than that of the incident light, the FWHMs at the normalized distance kr up to 4.17 is smaller than the diffraction limited line width of half the wavelength. The x profiles of the peak focused light at three different times/phases are shown in Fig. 13(b). At the time t = tf, FWHM is 0.286 while the width is 0.217 when averaged over the focused line; it is !"#$!% when averaged over the profile. The FWHM becomes 0.394% %at the time of 0.12 period later and 0.498 %at the time of t = tf+ 0.29 while the peak of the focused light has reached at a distance of 0.664 %away from the surface. All the widths discussed above are smaller than the diffraction limited line width of half the wavelength. Obviously, the diffraction limit has been locally overcome by a result at the intermediate zone in which there is such a small single-line width occurring with regard to the focused light.
For a far (near) field [32], the electric field is in phase (out of phase) with the magnetic field so that the Poynting vector (i.e., the energy flow) is not (is) zero.
Fig. 13 The profiles and the width of the focused light. (a) The FWHM vs. normalized r profiles of the snapshot of the Hzfield energy (blue squares) and the time-averaged Hzfield energy (red curve), where r is the y distance from the metal surface. (b) The profiles of the focused Hzfield energy at t = tf(red curve), tf+ 0.12 (blue dashes) and tf+ 0.29 (green dots).
The focused light beyond the diffraction limited line width of half the wavelength is located at the intermediate zone [74]. To characterize it further, the time-averaged Poynting vector in the original y propagation direction shown in Fig.
14(a)also indicates that the focused fields are propagating and hence are capable of travelling to the far zone [74]. As a travelling light is an indicator of the capability of our approach for moving the focal point away from the surface, this makes the FAB lens superior to evanescent near-field solutions [49-58] for many critical applications.
The involvement of the near field on the line width of the focused light is quantitatively investigated in-depth. At the middle location of the central metal strip surface, the magnetic field is out of phase with the electric field and in phase
Fig. 14 The Poynting vector contours and the field profiles. (a) The time-averaged contours of the Poynting vector (i.e., the energy flow) in the y direction. (b) The temporal profiles of the magnetic field Hz(red curve), the electric field Ex(blue dashes) and the current Jx(green dots) at x = 0 of the central metal surface. (c) The x profiles of peak Ex(blue dashes) and ZHz(red curve) fields at t = tf
+ 0.12; where |Z| = 0.832 and the phase is 3 cells or 0.0237 wavelength. (d) The r profiles of the Ex
field at t = tf(light blue short dashes), t = tf+ 0.13 (blue dashes), and t = tf+ 0.29 (purple short dashes and dashes) when the FWHM of the Hzenergy is still smaller than half the wavelength, as well as the ZHzfield (red curve; the estimated far Exfield; where |Z| = 0.840), the near Exfield (dark green dots;
the difference of the overall and far fields), and the near Ex field for the case of an almost perfect electric conductor (light green dots and short dashes).
with the surface current -Jx, as shown in Fig. 14(b). Thus, it is dominated by the near field. But, at the time of t = tf + 0.12, the focused light has moved out. Its magnetic field near the surface is close to zero on average and is a small positive number at the middle. The focused Ex field has a similar contour as the Hzfield.
The ratio and locations of their peaks determines the impedance Z. The estimated far Ex field is ZHzthat agrees well with the measured Ex as shown in Fig. 14(c).
This is one more indication [76] for the focused light to be dominated by the radiative field, as also evidenced by the propagation of the Ex field at different times shown in Fig. 14(d). At the time 0.01 later, the magnetic field at the middle of the central metal strip has dropped to a negative value very close to zero. As shown in Fig. 14(d), the estimated far Ex field along kr based on the analytical theory is in good agreement with the focused field obtained from the simulation.
The near Exfield including the intermediate field resultsfrom their difference and is decreasing away from the surface as expected with the length of half the field energy being kr = 0.476 or less than 0.1; that is consistent with the length scale of the near zone measured by NSOM [76-76]. The near field is smaller than the far field at kr > 1. Although the mid field may not be completely ignored because of the small phase difference (see Fig. 11(d), Fig. 14(c)); the effect of the near field is negligible at the intermediate zone of 2 < kr < 4, where the line width of the focused light is smaller than the diffraction limited value. Since the near field is expected to be caused by the plasma effect on the metal surface, we also calculate the near field for the case of an almost perfect electric conductor whose plasma
The near Exfield including the intermediate field resultsfrom their difference and is decreasing away from the surface as expected with the length of half the field energy being kr = 0.476 or less than 0.1; that is consistent with the length scale of the near zone measured by NSOM [76-76]. The near field is smaller than the far field at kr > 1. Although the mid field may not be completely ignored because of the small phase difference (see Fig. 11(d), Fig. 14(c)); the effect of the near field is negligible at the intermediate zone of 2 < kr < 4, where the line width of the focused light is smaller than the diffraction limited value. Since the near field is expected to be caused by the plasma effect on the metal surface, we also calculate the near field for the case of an almost perfect electric conductor whose plasma