The concepts of slow and fast light refer to the group velocity of a light wave being lower or higher than the speed of light in vacuum, $c$ [1]. For an optical beam, the group velocity is equal to the velocity of the moving intensity peaks along the beam axis [2]. Control over the group velocity can be achieved in several different ways by modifying the material dispersion and absorption or gain [3,4], spatially or spatiotemporally structuring the beam profile [5–7], or adjusting the angular dispersion of the beam’s plane wave components [8].

Self-healing non-diffracting optical beams in free space usually exhibit an on-axis group velocity that is slightly higher than $c$. However, beams with a group velocity that varies upon the beam propagation have also been demonstrated. Examples of such beams are transversely accelerating Airy, Mathieu, and Weber beams [9,10], as well as longitudinally accelerating Bessel beams [11]. Superluminal [12–14], subluminal [15–17], and even negative group velocities [18] have been observed for these beams in free space. Longitudinal acceleration, however, has been demonstrated only for femtosecond-pulsed laser beams with group velocities staying close to the speed of light in a vacuum [19,20]. In this work, we demonstrate experimentally that a two-frequency-component cw beam can be produced to achieve an essentially arbitrary longitudinal acceleration or deceleration including large positive and negative group velocities. Such beams are interesting in view of both fundamental and applied optics. They can find applications, e.g., in ultrafast optics, optical interferometry, and laser trapping and the manipulation of atoms, molecules, and nanoparticles.

Recently, a systematic theoretical description of structured pulsed and cw laser beams with varying group velocities in free space has been presented [21]. There it was shown that, for cw beams, the velocity of the intensity peaks can fluctuate or change deterministically both in space and time. In particular, by adjusting the angular spectra of two frequency components of a Bessel beam, one can achieve strong longitudinal acceleration of the intensity peaks along the beam propagation direction. Such a beam can be created by using a lens and an axicon, as shown in Fig. 1. The lens collimates one of the two incident beam components (orange) and slightly focuses the other one (red). Being transmitted by the axicon, the collimated component (of frequency $\omega _1$) forms a Bessel beam with a constant angular spectrum, while the focused component (of frequency $\omega _2$) forms a beam with a local (on-axis) angular spectrum that depends on the coordinate $z$. As a result, the group velocity of the combined Bessel beam becomes a function of $z$ given by [21]

 

Fig. 1. Formation of a longitudinally accelerating two-component Bessel beam. The frequency components are presented in red and orange colors. The acceleration is achieved in the intersection region of the two components within the length $L_\textrm {b}$ of the beam.

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For measuring the group velocity of the two-component beam, we introduce a method based on the dependence of the group velocity on the on-axis phase distributions of the beam components. The method is independent of the beam type. By definition, the on-axis group velocity of an optical beam is $v_\textrm {g}=d\omega /dk_z$, which gives $v_\textrm {g}/c=dk/dk_z$. For a two-component Bessel beam, one then obtains $\displaystyle \frac {v_\textrm {g}}{c}=\frac {(k_1-k_2)z}{(k_{1z}-k_{2 z})z} = \frac {\varphi _1(z)-\varphi _2(z)}{\varphi _1^\textrm {b}(z)-\varphi _2^\textrm {b}(z)}$, where $\varphi _i = \omega _i z/c$ is the phase of a plane wave (or a spherical wave) with frequency $\omega _i$ due to propagation along $z$ and $\varphi _i^\textrm {b} (z)$ is the phase profile of component $i$ of the beam along the beam axis. To measure the propagational phase profiles, we use a common-path optical interferometer made of a semitransparent metal film with a pinhole at the center [22]. We place the pinhole at a fixed coordinate $z$ behind the axicon and measure the longitudinal interference pattern for each frequency component of the beam. The pattern is the superposition of the on-axis beam field and a spherical wave produced by the pinhole. Hence, if for component $i$ at a given coordinate $z_\textrm {c}$ we measure $\varphi _i^\textrm {b}(z_\textrm {c}) = \varphi _i(z_\textrm {c})$ (constructive interference) and after a distance $z_{\pi i}$ along the beam axis we obtain $\varphi _i^\textrm {b}(z_\textrm {c} + z_{\pi i}) = \varphi _i(z_\textrm {c} + z_{\pi i}) -\pi$ (destructive interference), then the phase of the beam must satisfy $\varphi _i^\textrm {b}(z)=\varphi _i(z)-\pi z/z_{\pi i}.$ Here we have assumed that locally (close to $z_\textrm {c}$), the phase changes linearly with $z$. Therefore, the local group velocity can be expressed as

The experimental setup used to create a two-component Bessel beam and measure its local group velocity is shown in Fig. 2. Two orthogonally polarized laser beams of wavelengths $\lambda _1 = 632.8$ nm (Uniphase 1135P) and $\lambda _2=657.5$ nm (CrystaLaser DL655-025-SO) and bandwidths $\Delta \lambda < 0.01$ pm are combined by a polarizing beam splitter. The combined beam is expanded by a $40\times$ microscope objective and collimated by a lens with a 40-cm focal distance. The collimated beam (having a 5-cm diameter) is split into the two arms of a Michelson interferometer by another polarizing beam splitter so that each laser beam propagates into one of the arms only. Both arms contain a quarter-wave plate (QWP) and a mirror M that rotate the polarization of the reflected beam by $90^{\circ}$ for the beam to pass through the beam splitter to the detector. One of the arms contains a lens that focuses the beam such that it becomes collimated by another lens in front of the axicon. The wavelength of this laser is 657.5 nm. A linear polarizer (P) oriented at $45^{\circ}$ is placed before the lens. The intensities of the laser beams can be adjusted by rotating the polarizer. The axicon forms a two-component Bessel beam corresponding to Fig. 1 (with $f = 7$ cm). The propagation angle of the collimated component is $\alpha _1 = 10^{\circ}$. Because the beam is formed by two independent laser beams with a finite bandwidth, the group velocity fluctuates. However, its relative deviation from the mean value,

 

Fig. 2. Experimental setup: PBS, polarizing beam splitter; $\mathrm {MO}_1$ and $\mathrm {MO}_2$, $40\times$ and $20\times$ microscope objectives, respectively; L, lens; QWP, quarter-wave plate; M, mirror; P, linear polarizer; A, axicon. At the output, $\mathrm {MO}_2$ and the camera can be translated along the direction shown by the arrows. The wavelength of Laser 1 is 632.8 nm and that of Laser 2 is 657.5 nm. The inset shows the profiles of the beam as a whole (top) and its components produced by Laser 1 (middle) and Laser 2 (bottom).

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Each measurement starts with adjusting the imaging system ($\mathrm {MO_2}$ and C) to observe a sharp image of the aperture. Then, the system is moved away from the aperture in small steps and the transverse profile of the beam is recorded together with its longitudinal coordinate for each beam component. The inset of Fig. 2 shows the profiles of the beam as a whole and its components at $z = 12$ mm as an example. The intensity is averaged over $\mathrm {3\times 3}$ central pixels – to reduce possible errors due to small shifts of the beam with respect to the pixels – and is plotted as a function of $z$. An example of such a measurement is shown by orange and blue dots in Fig. 3. The distance between the first minimum and the next maximum of the interference pattern, i.e., the distance $z_{\pi i}$ in Eq. (3), is measured for each beam component. In Fig. 3, the orange and blue lines show fitting curves obtained by numerically superimposing a spherical wave with an expected Bessel beam:

 

Fig. 3. Measured longitudinal intensity profiles of the beam components with $\lambda _1=632.8$ nm (orange dots) and $\lambda _2=657.5$ nm (blue dots). The solid lines show fitting of the measured data with theoretical curves.

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To minimize the error, each measurement is repeated ten times and the mean value $\overline {z_{\pi i}}$ and standard deviation $\sigma _{\pi i}$ of $z_{\pi i}$ are obtained and used to evaluate the group velocity. In the present case, corresponding to Fig. 3, we obtain $\overline {z_{\pi 1}}=7.3$ $\mathrm{\mu} \mathrm {m}$, $\sigma _{\pi 1}=0.48$ $\mathrm{\mu} \mathrm {m}$, $\overline {z_{\pi 2}}=25.4$ $\mathrm{\mu} \mathrm {m}$, and $\sigma _{\pi 2}=0.84$ $\mathrm{\mu} \mathrm {m}$. The lower and upper limits of the measured group velocity $(v_\textrm {g}^{\pm })$ corresponding to the measured standard deviation are calculated by using Eq. (3) as

We find that at a distance of 21.6 mm from the axicon tip, the group velocity has a mean value of $5.6c$ with lower and upper limits of $3.8c$ and $11.7c$. We have measured the group velocity at 15 points along the $z$ axis between $z = 11.6$ mm and $z = 26.1$ mm. The results are shown by the blue dots together with the corresponding error bars in Fig. 4. The blue line shows the theoretically predicted group velocity [Eqs. (1) and (2)] for the two-component beam we have created. The group velocity is seen to gradually increase from approximately $1.3c$ to infinity when $z$ increases toward 23.8 mm, which is a discontinuity point of the curve. At this point, the denominator in Eq. (2) crosses zero and the group velocity changes sign, becoming negative infinite. Then it continues to increase, staying negative. The intensity peaks in this range move toward the light source with high acceleration. The most extraordinary mean values of the group velocity that we have measured are $7c$ and $-7c$. We note that when the magnitude of the group velocity increases, the measurement becomes more uncertain. Therefore, the points close to the coordinate of the curve discontinuity have been excluded from the measurement results. It was also impossible to measure the group velocity for $z > 26.1$ mm, because these distances are already outside the range of the beam intersection (see Fig. 1).

 

Fig. 4. Measured group velocity of the longitudinally accelerating Bessel beam (see the blue dots and the estimated error bars). The blue line shows the theoretically predicted group velocity of the beam. The curve corresponds to the theoretical curve shown in Fig. 6  of Ref. [21] for the current parameter values.

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To obtain a Bessel beam with zero group velocity (i.e., with a standing longitudinal interference pattern), Laser 2 is switched off and the polarization plane of Laser 1 ($\lambda _1=632.8$ nm) is rotated by $45^{\circ}$. The beam propagates into both arms of the Michelson interferometer, forming the Bessel beam components at the output of the axicon. The group velocity measurement based on the common-path interferometer is not needed in this case, because the longitudinal standing-wave pattern can be observed directly with the camera. The period of the pattern, $\Lambda _z$, is calculated as

 

Fig. 5. Measured period $\Lambda _z$ of the longitudinal standing-wave pattern as a function of distance $z$ from the tip of the axicon (blue dots). The solid line corresponds to the theoretically predicted dependence of $\Lambda _z$ on $z$.

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By swapping the lasers in the setup and rotating their polarization planes by $90^{\circ}$, we change the wavelengths of the collimated and focused beams incident onto the axicon respectively to 632.8 nm and 657.5 nm. This results in a longitudinally decelerating Bessel beam at the output of the axicon. The group velocity measurement procedure is the same as for the accelerating Bessel beam. Figure 6 shows intensity profiles obtained with the common-path interferometer for the two components of the beam at $z=12.5$ mm (see the blue/orange dots and the corresponding fitting curves for $\lambda = 632.8/657.5$ nm). The distance between the first minimum and the next maximum of the pattern at $\lambda = 657.5$ nm is $z_{\pi 1}=12.5$ $\mathrm{\mu} \mathrm {m}$ and at $\lambda = 632.8$ nm it is $z_{\pi 2}=19.6$ $\mathrm{\mu} \mathrm {m}$. The average values and standard deviations are $\overline {z_{\pi 1}}=12.5$ $\mathrm{\mu} \mathrm {m}$, $\sigma _{\pi 1}=0.71$ $\mathrm{\mu} \mathrm {m}$, $\overline {z_{\pi 2}}=21.3$ $\mathrm{\mu} \mathrm {m}$, and $\sigma _{\pi 2}=1.06$ $\mathrm{\mu} \mathrm {m}$. Therefore, the group velocity is $v_\textrm {g}=0.78c$, as follows from Eq. (3), and the lower and upper limits for it are $0.75c$ and $0.82c$, as follows from Eq. (6). Similar measurements were done at eight other distances from the axicon. The results are shown in Fig. 7. They are again in good agreement with the theoretical results introduced in the figure by the blue solid line.

 

Fig. 6. Measured longitudinal intensity profiles of the beam components with $\lambda _1=657.5$ nm (orange dots) and $\lambda _2=632.8$ nm (blue dots). The solid lines show fitting of the measured data with theoretical curves.

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Fig. 7. Measured group velocity of the longitudinally decelerating Bessel beam (see the blue dots and the estimated error bars). The blue line shows the theoretically predicted group velocity of the beam.

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To summarize, we have created a longitudinally accelerating two-component cw Bessel beam and achieved subluminal, superluminal, negative, and zero group velocities in free space with a single experimental setup. We have also proposed and demonstrated a method to measure the group velocity of a two-component optical beam. The method is interferometric and makes use of a simple common-path interferometer. The obtained measurement results are in good agreement with theoretical predictions. The methods and findings of this research can be applied to other types of optical beams and are of interest to scientists working in fields such as optical interferometry, ultrafast optics, nonlinear optics, and optical tweezers.