Precise amplitude, trajectory, and beam-width control of accelerating and abruptly autofocusing beams

We show that it is possible to independently control both the trajectory and the maximum amplitude along the trajectory of a paraxial accelerating beam. This is accomplished by carefully engineering both the amplitude and the phase of the beam on the input plane. Furthermore, we show that the width of an accelerating beam is related only on the curvature of the trajectory. Therefore, we are able to produce beams with predefined beam widths and amplitudes. These results are useful in applications where precise beam control is important. In addition we consider radially symmetric abruptly autofocusing beams. We identify the important parameters that affect the focal characteristics. Consequently, we can design autofocusing beams with optimized parameters (such as sharper focus and higher intensity contrast). In all our calculations the resulting formulas are presented in an elegant and practical form in direct connection with the geometric properties of the trajectory. Finally we discuss methods that can be utilized to experimentally realize such optical waves.

Figure 1

Accelerating beams following a parabolic trajectory [Eq. (10) with $\alpha =2$ and $\beta =0.01]$. In the first column we can see the amplitude dynamics and the theoretical prediction for the trajectory (dashed curve). In the second column the maximum amplitude as a function of the propagation distance (solid curve) along with the theoretical prediction (shown in circles) is depicted. In the third and fourth columns cross sections of the beam intensity at different propagation distances are presented with the theoretical predictions shown in circles. In the three rows the theoretical maximum values of the field amplitudes are $U\left(z\right)=1,U\left(z\right)=exp[-{(z-160)}^2/{100}^2]$, and $U\left(z\right)=1+0.5{sin}^2[(z-160)/80]$, respectively. The horizontal dashed lines in the first column correspond to the cross sections shown in the third and fourth columns.

Figure 2

Accelerating beams following a power-law trajectory [Eq. (10) with $\alpha =3/2$ and $\beta =1/5]$. In the first column we can see the amplitude dynamics and the theoretical prediction for the trajectory (dashed curve). In the second column the maximum intensity as a function of the propagation distance (solid curve) along with the theoretical prediction (shown in circles) is depicted. In the third and fourth columns cross sections of beam intensity at different propagation distances are presented with the theoretical predictions shown in circles. In the three rows the theoretical maximum values of the field amplitudes are $U\left(z\right)=1,U\left(z\right)=0.5+0.5exp[-{(z-160)}^2/{110}^2]$ and $U\left(z\right)=0.1z+6{sin}^2(z/80)$, respectively. The horizontal dashed lines in the first column correspond to the cross sections shown in the third and fourth columns.

Figure 3

Accelerating beams following a cubic trajectory [Eq. (10) with $\alpha =3$ and $\beta =1/32000]$. In the first column we can see the amplitude dynamics and the theoretical prediction for the trajectory (dashed curve). In the second column the maximum intensity as a function of the propagation distance (solid curve) along with the theoretical prediction (shown in circles) is depicted. In the third and fourth rows cross sections of beam intensity at different propagation distances are presented with the theoretical predictions shown in circles. In the three rows the predicted maximum values of the field amplitudes are $U\left(z\right)=1,U\left(z\right)=z+15sin(z/20)$, and $U\left(z\right)=1+tanh\left[0.015\right(z-160/2\left)\right]$, respectively. The horizontal dashed lines in the first column correspond to the cross sections shown in the third and fourth columns.

Figure 4

An abruptly autofocusing beam following a parabolic trajectory ($\alpha =2$) with $\beta =1,r_0=20,c=0.06,w_1=1$. In (a) we see the spectrum and the amplitude profile on the input plane. In (b) the three-dimensional wave dynamics are depicted along with the theoretical prediction for the trajectory (black-white dashed curve). In (c) the intensity contrast is presented as a function of the propagation distance along with the theoretical prediction (shown with circles). In the last two rows we depict the intensity profile of the horizontal cross sections shown in (b) with the theoretical predictions shown with circles. Specifically, in the third row the cross sections are taken before the focus, and the analytic prediction is obtained from Eq. (15). In the last row (f) is computed exactly at the focus, and (g) is computed after the focus whereas the theoretical estimates are given by Eq. (23).

Figure 5

The same as in Fig. 4 for a cubic trajectory ($\alpha =3$) with $\beta =1,r_0=10,c=0.05,w_1=1$.

Figure 6

Intensity contrast as a function of the propagation distance. In the insets the initial conditions are shown. The parameters for the trajectory are the same as in Fig. 5. In (a) the amplitude on the input plane is the same as in Fig. 5, but the aperture is reduced to $r_a=45$ and $w_2=1$. In (b) the amplitude is constant $c=0$ whereas $r_a=45$ and $w_2=5$.