REPORTS intersecting the plane o= k v with charged particle in the same time. Thus, the possess a backward-pointing overall cone the dispersion surface o= o (k)(Fig. IB). radiation wavefront for each mode lies on a (o >T/2), according to the analysis of Fig The CR behavior in real space, however, "group cone"(3, 6) with its apex on the 1C. This is therefore the reversed-cone re must be obtained from the group velocity u= moving particle and a half-apex angle a, the gime. Such a reversed cone is forbidden by dwak; that is, the gradient vector to a disper- angle between v-H and v. A superposition causality (1) in a uniform passive medium. on surface. which can be shown to be of all group cones gives the overall CR pat- 3)In the - 0. 2csvs-04c regime, all energy transport vector for each mode(25). term, and the maximum angle am of all such the radiation modes in the lowest emission The emission angle is given by the direction as is the half-apex angle for the overall cone. band reside in the region t/a <k. < 2/a of u. Moreover, the radiation pattern can be Propagating Bloch modes can only exist on with u, <0, which implies that the energy deduced from u via the group velocity con- the rear side of this overall cone, whereas the flows opposite to v in this band. The overall tour(Fig. IC). We plot both the charge ve- radiation fields are evanescent on the forward cone is now pointing forward and, as de city v and the group velocities u of all the side, and across the overall cone the radiated scribed below, the radiation becomes colli- CR modes in a velocity space(26). The field amplitude experiences a drop. In the mated in a backward direction. CR here is magnitude of u is proportional to the distance special case of a uniform material, only the strongly similar to the predicted behavior in a traveled by the wavefront of the associated G=0 modes are excited and our approach negative-index material. However, because photon mode, and the magnitude of v is pro- yields the characteristics of conventional CR. k. u>0 in this regime (where k' is the portional to the distance traveled by the For an electron traveling near a grating sur- Bloch-reduced k in the first Brillouin zone), face, one employs the dispersion relations of the photonic crystal may be regarded as an air along with diffraction to obtain the Smith- effective positive-index medium (22). W A Purcell radiation call this the backward- flux regime In the present photonic crystal, we solved 4) In the v 2 v regime, the solution to Eq 0.2 Eq. 1 using photon bands calculated by I starts from zero frequency, and the group- planewave expansion(27). We focused on velocity g0.1 06 higher-order modes with larger G or in higher now travels faster than all of the excited? bands can be analyzed similarly. Figure 2 modes, and constructive interference is 0.15 06010.2 0.40. 0.8 We see that for v<< ve, the radiation normal dispersive medium. Hence, this is the g (c around≈G…v. For larger v, the resonances The CR showed less interesting transition Bo merge together to form emission bands out- al behavior for intermediate charge velocities g 0.75 side which CR is inhibited(Fig. 2A). As v between these regimes. For example, when on increases k and u within each emission band 0.4c<y<y. the cr showed a mixture are strongly influenced by the photonic band of forward and backward emissions similar to structure. We can identify four regimes of the that depicted in Fig. IC. harge velocity(Fig. 2, B and C)with four We confirmed our analysis by performing o qualitatively different CR behaviors in a pho- finite-difference time-domain(FDTD) simu- tonic crystal: lations of radiation for a moving charge in g D)vs-oIc corresponds to the Smith- this photonic crystal (28). We reproduce in o 0.5 1.0 1.5 Purcell regime, in which the CR arises pri- Fig. 3A the velocity diagrams from Fig. 2C E k,(2/a) marily through constructive interference be- and present the resulting radiation-field sim tween consecutive unit cells and in the first ulations in Fig. 3B. To quantitatively demon- O emission band corresponds to the first res- strate the backward radiation, we also plot in 0.4 onance near k. 2la, where a is the Fig. 3C the simulated flux through a fixed 0.15 lattice period. The group velocity contour line perpendicular to v as a function of w. At 0.2 Because y<< y the overall wave front shows a smith-Purcell behavior of near-iso- 0 will be roughly circular and centered on the tropic wavefront. As the velocity increases to slowly moving particle 0. 15c. the overall radiation cone indeed 2)The regime v s 0. 15c is unique to the reversed as predicted. Further increasing of v photonic crystal: The first and higher Smith- to v=0.3c steers the radiation to the back Purcell resonances merge into one band, in ward direction, and if v is increased to v which the slow photon modes near the pho- 0.6c> v the familiar CR with a sharp for- -0.4-0.20 tonic band edge are in coexistence with the ward-pointing cone as in a uniform medium uz(c) fast modes. As can be seen from the group is recovered. The angular distribution of ra- velocity contours, in going from v= 0.lc to diation is directly visible (Fig. 3B).When Fig. 2. Calculated CR modes wit v=0.15c. the modes in the forward direction v=0.1c or 0.15c. the radiation is distributed begin to travel slower than the charged par- over a wide range of emission angles without (A)The CR emission band structure ticle(and can even have negative group ve- producing a cone of intensity maxima. For in the first photonic band as a function of locities), eventually producing a contour at = 0.3c or 0.6c, however, the Cr becomes different CR behaviors. (B) The solution k for a v=0. 15c that winds around without enclos- collimated, and a definite emission angle in few vs indicated by the numbers in italics. (c) ing v Because in this band there are some fast both the forward and the backward direction The group velocity contours for the represe modes in the forward direction whose u, ex- for most of the radiation energy can be ob- ative values of v ceeds v, the radiation pattern here should served. In particular, the crystal-induced dis- www.sciencemagorgSciEnceVol29917JanUary2003space by intersecting the plane 
k
v with the dispersion surface 
n(k) (Fig. 1B). The CR behavior in real space, however, must be obtained from the group velocity u
/k; that is, the gradient vector to a disper￾sion surface, which can be shown to be the energy transport vector for each mode (25). The emission angle is given by the direction of u. Moreover, the radiation pattern can be deduced from u via the group velocity con￾tour (Fig. 1C). We plot both the charge ve￾locity v and the group velocities u of all the CR modes in a velocity space (26). The magnitude of u is proportional to the distance traveled by the wavefront of the associated photon mode, and the magnitude of v is pro￾portional to the distance traveled by the charged particle in the same time. Thus, the radiation wavefront for each mode lies on a “group cone” (3, 6) with its apex on the moving particle and a half-apex angle , the angle between v u and v. A superposition of all group cones gives the overall CR pat￾tern, and the maximum angle m of all such ’s is the half-apex angle for the overall cone. Propagating Bloch modes can only exist on the rear side of this overall cone, whereas the radiation fields are evanescent on the forward side, and across the overall cone the radiated field amplitude experiences a drop. In the special case of a uniform material, only the G
0 modes are excited and our approach yields the characteristics of conventional CR. For an electron traveling near a grating sur￾face, one employs the dispersion relations of air along with diffraction to obtain the Smith￾Purcell radiation. In the present photonic crystal, we solved Eq. 1 using photon bands calculated by planewave expansion (27). We focused on the solutions with the lowest frequencies; higher-order modes with larger G or in higher bands can be analyzed similarly. Figure 2 shows the results for the CR emission fre￾quency , wave vector k, and group velocity u. We see that for v vc , the radiation coalesces into Smith-Purcell resonances around   G
v. For larger v, the resonances merge together to form emission bands out￾side which CR is inhibited (Fig. 2A). As v increases, k and u within each emission band are strongly influenced by the photonic band structure. We can identify four regimes of the charge velocity (Fig. 2, B and C) with four qualitatively different CR behaviors in a pho￾tonic crystal: 1) v
0.1c corresponds to the Smith￾Purcell regime, in which the CR arises pri￾marily through constructive interference be￾tween consecutive unit cells and in the first emission band corresponds to the first res￾onance near kz
2 /a, where a is the lattice period. The group velocity contour is approximately circular, with radius vc. Because v vc, the overall wavefront will be roughly circular and centered on the slowly moving particle. 2) The regime v  0.15c is unique to the photonic crystal: The first and higher Smith￾Purcell resonances merge into one band, in which the slow photon modes near the pho￾tonic band edge are in coexistence with the fast modes. As can be seen from the group velocity contours, in going from v
0.1c to v
0.15c, the modes in the forward direction begin to travel slower than the charged par￾ticle (and can even have negative group ve￾locities), eventually producing a contour at v
0.15c that winds around without enclos￾ing v. Because in this band there are some fast modes in the forward direction whose uz ex￾ceeds v, the radiation pattern here should possess a backward-pointing overall cone (m  /2), according to the analysis of Fig. 1C. This is therefore the reversed-cone re￾gime. Such a reversed cone is forbidden by causality (1) in a uniform passive medium. 3) In the 0.2c
v
0.4c regime, all the radiation modes in the lowest emission band reside in the region /a kz 2 /a with uz 0, which implies that the energy flows opposite to v in this band. The overall cone is now pointing forward and, as de￾scribed below, the radiation becomes colli￾mated in a backward direction. CR here is strongly similar to the predicted behavior in a negative-index material. However, because k
u  0 in this regime (where k is the Bloch-reduced k in the first Brillouin zone), the photonic crystal may be regarded as an effective positive-index medium (22). We call this the backward-flux regime. 4) In the v vc regime, the solution to Eq. 1 starts from zero frequency, and the group￾velocity contour becomes an open-ended curve with positive uz . The charged particle now travels faster than all of the excited modes, and constructive interference is achieved throughout the whole photonic crys￾tal. This behavior is identical to CR in a normal dispersive medium. Hence, this is the normal regime. The CR showed less interesting transition￾al behavior for intermediate charge velocities between these regimes. For example, when 0.4c v vc , the CR showed a mixture of forward and backward emissions similar to that depicted in Fig. 1C. We confirmed our analysis by performing finite-difference time-domain (FDTD) simu￾lations of radiation for a moving charge in this photonic crystal (28). We reproduce in Fig. 3A the velocity diagrams from Fig. 2C and present the resulting radiation-field sim￾ulations in Fig. 3B. To quantitatively demon￾strate the backward radiation, we also plot in Fig. 3C the simulated flux through a fixed line perpendicular to v as a function of . At the low velocity, v
0.1c, the radiation shows a Smith-Purcell behavior of near-iso￾tropic wavefront. As the velocity increases to v
0.15c, the overall radiation cone indeed reversed as predicted. Further increasing of v to v
0.3c steers the radiation to the back￾ward direction, and if v is increased to v
0.6c  vc the familiar CR with a sharp for￾ward-pointing cone as in a uniform medium is recovered. The angular distribution of ra￾diation is directly visible (Fig. 3B). When v
0.1c or 0.15c, the radiation is distributed over a wide range of emission angles without producing a cone of intensity maxima. For v
0.3c or 0.6c, however, the CR becomes collimated, and a definite emission angle in both the forward and the backward direction for most of the radiation energy can be ob￾served. In particular, the crystal-induced dis￾Fig. 2. Calculated CR modes with the lowest frequencies for the photonic crystal of Fig. 1. (A) The CR emission band structure (red region) in the first photonic band as a function of v. Colored vertical lines mark representative vs for different CR behaviors. (B) The solution k for a few vs indicated by the numbers in italics. (C) The group velocity contours for the represen￾tative values of v. R EPORTS www.sciencemag.org SCIENCE VOL 299 17 JANUARY 2003 369 on June 8, 2007 www.sciencemag.org Downloaded from