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Generation of nonparaxial accelerating fields through mirrors. I: Two dimensions

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Abstract

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. Recent constructions of nonparaxial accelerating beams cannot span more than a semicircle. Here, we present a ray based analysis for nonparaxial accelerating fields and pulses in two dimensions. We also develop a simple geometric procedure for finding mirror shapes that convert collimated fields or fields emanating from a point source into accelerating fields tracing circular caustics that extend well beyond a semicircle.

© 2014 Optical Society of America

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Supplementary Material (5)

Media 1: MOV (951 KB)     
Media 2: MOV (316 KB)     
Media 3: MOV (519 KB)     
Media 4: MOV (5240 KB)     
Media 5: MOV (679 KB)     

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Figures (7)

Fig. 1
Fig. 1 (a Media 1)) In the paraxial regime, ray families (black lines) that form a parabolic caustic (orange line) preserve their structure under a “paraxial rotation” (a linear shear x, zxpz, z) followed by a displacement. (b Media 2) In the nonparaxial regime, ray families forming a circular caustic preserve their structure under rotations around the circle’s center.
Fig. 2
Fig. 2 (a, Media 3) Construction for drawing mirror shapes that focus a collimated beam onto a caustic of a given shape. (b) Illustration of the mirror shape (black curve) that reflects rays (blue lines) to form a circular caustic (orange). Media 4 shows the propagation of a wavefront or the crest of a pulse.
Fig. 3
Fig. 3 Apodization of the collimated incident field’s intensity needed to achieve intensity uniformity along the caustic, for several values of T/R.
Fig. 4
Fig. 4 Parallel incident rays (yellow), and after one (green) and two (orange) reflections, for (a) the same size mirror as in Fig. 2 (T = 6.2R), and (b) a significantly larger segment of the same mirror. In (b) the ray highlighted in blue, whose first incidence is at x = X(ϕc) = −27.64R, retroreflects at the second reflection. This ray is the boundary between those that cross the caustic a second time and those that don’t: rays incident at x < X(ϕc) cross the caustic but diverge away from each other, so that their disruption to the caustic pattern is not too significant. The segment corresponding to the box in (b) is expanded in (c).
Fig. 5
Fig. 5 Intensity around the caustic for (a) the field reflected by the mirror, and (b) the total field (reflected plus incident), for a mirror with kR = 80, T = 6.2R and an incident field leading to an angular spectrum after reflection with amplitude |A(ϕ)| = exp[−(1 − cosϕ)10/1.710].
Fig. 6
Fig. 6 Intensity around the caustic for a Gaussian pulse reflected by a mirror with k0R = 80, k0w = 0.2, T = 6.2R at several times with a time separation of R/c. Media 5 shows the continuous evolution of the pulse.
Fig. 7
Fig. 7 Mirror shapes that focus collimated beams into fields with elliptic caustics of different orientations. The density of the incident rays reflects the intensity apodization needed to achieve a field resembling a Mathieu field.

Equations (13)

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[ X ( ϕ ) , Z ( ϕ ) ] = [ R ( R ϕ + T ) sin ϕ 1 + cos ϕ , ( R ϕ + T ) cos ϕ R sin ϕ 1 + cos ϕ ] ,
| U inc [ X ( ϕ ) ] | 2 | d ϕ d X | = 1 + cos ϕ R ( ϕ + sin ϕ ) + T ,
U ref ( r ) = k 2 π | A ( ϕ ) | exp { i k [ R ϕ + r u ( ϕ ) ] } d ϕ ,
| A ( ϕ ) | = | U inc [ X ( ϕ ) ] | R ( ϕ + sin ϕ ) + T 1 + cos ϕ ,
2 R ( π ϕ c ) = T + 2 R cos ϕ c tan ϕ c 2 .
U ref ( r ; t ) = | A ( ϕ ) | P [ t r u ( ϕ ) / c R ϕ ] d ϕ ,
P ( t ) = exp ( c 2 t 2 2 w 2 + i k 0 c t ) ,
U ref ( r , φ , t ) [ a Ai ( h ) + b Ai ( h ) ] exp [ Ω ( ϕ 1 ) + Ω ( ϕ 2 ) 2 ] ,
Ω ( ϕ ) = [ r cos ( ϕ φ ) + R ϕ c t i k 0 w 2 ] 2 2 w 2 k 0 2 w 2 2 ,
h = { 3 [ Ω ( ϕ 1 ) Ω ( ϕ 2 ) ] 4 i } 2 / 3 ,
a = i π [ 2 i k 0 h Ω ( ϕ 1 ) A ( ϕ 1 ) + 2 i k 0 h Ω ( ϕ 2 ) A ( ϕ 2 ) ] ,
b = i π [ 2 i k 0 h Ω ( ϕ 1 ) A ( ϕ 1 ) 2 i k 0 h Ω ( ϕ 2 ) A ( ϕ 2 ) ] ,
ϕ 1 , 2 = φ + π 2 ± arccos ( R r ) .
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