1. Introduction

How small is a very small frequency shift in the optical domain? The answer to this depends heavily on who you ask. In the context of spectroscopy, two beams, each with a frequency of 100s of THz, separated by a few GHz (i.e. 1 part in 10⁵) are described as being of nearly equal frequency¹. As another example, in the laser cooling of cold atoms several atomic species require two laser systems with closely spaced frequencies, acting as a cooling laser and a re-pumping laser respectively, to ensure the atoms stay in the laser cooling cycle². This again requires frequency shifts in the GHz range.

Producing frequency shifts below 1 GHz is no small challenge, but such shifts are required for offset locking of diode laser systems to atomic features, again notably in the field of cold atoms. The conventional method uses acousto-optic modulators (AOMs) which can shift the frequency of a laser by tens or hundreds of MHz (1 part in 10⁷). If two AOMs are used, much smaller frequency differences of a few MHz or less can be achieved. However, this imposes stringent criteria on the drivers of the AOM systems if this shift is to go below the kHz regime (less than 1 part in 10¹¹). To obtain frequency shifts in the one hertz or less regime requires sophisticated phase locked synthesizers.

Optical conveyer belts formed from a standing wave pattern between two counter-propagating lasers can be used to transport atoms from a magneto-optical trap³. By treating the system in the temporal domain (as opposed to creating a phase shift in the spatial domain) the standing-wave pattern can be set in motion by creating a relative frequency shift of a few kHz or less between the two counter-propagating beams. Similar concepts can be applied in optical tweezing⁴, where microscopic particles can be trapped and translated in the transverse interference pattern formed between two focussed Gaussian beams⁵. Interference patterns employing Laguerre-Gaussian beams can rotate microscopic particles^6,7,8 and furthermore create three dimensional trapped structures. The linear or angular motion required for motion of the interference pattern can be created by a frequency shift between the interfering beams. Such frequency shifts need to be of a few Hz or less to induce motion of the interference pattern at appropriate velocities for optical trapping studies. This corresponds to a frequency shift 14 orders of magnitude smaller than the frequency of the laser itself.

In this paper we show a technique based on the angular Doppler-effect⁹ (more generally known as the rotational frequency shift¹⁰) that allows relative frequency shifts to be created from well below 1 Hz up to kHz. For the first time we use this technique to demonstrate the continuous scanning of a linear interference pattern formed between two Gaussian beams and the continuous rotation of an interference pattern formed between a Laguerre-Gaussian mode and a Gaussian TEM₀₀ mode. The beating of two overlapping Gaussian modes, similar to the result obtained by Simon et. al. from a dynamical manifestation of Berry’s phase shift¹¹, is used to determine the stability of the technique. Furthermore, our method also lends itself to absolute positional control of the interference pattern thus allowing placement of atoms and particles at predescribed locations, topics that are attracting considerable attention at the moment. This technique presents a far simpler and hence more reliable alternative to preexisting techniques such as: phase locked acousto-optic modulators which suffer from phase noise, the use of an electro-optic modulator where the requirements on the driver are very demanding and a linearly translated mirror with which it is very difficult to achieve good stability¹².

2. Theory – angular Doppler-effect

In a Michelson interferometer, when the mirror in one arm of the interferometer is moved the fringes observed at the output move normal to the fringes themselves. This is usually explained as the path length change induced by moving the mirror. However, it is equally valid to consider the fringe movement as being due to a frequency shift ∆ω, experienced by the beam as it is reflected off the moving mirror, compared to the beam in the other arm of the interferometer. The reflected light experiences a shift in frequency of ∆ω = 2ωv/c, where v is the relative velocity between the observer and the mirror, c is the speed of light and ω is the frequency of the light. This is a manifestation of the linear Doppler effect¹³.

In the angular equivalent of the linear Doppler effect, it is necessary to have a rotating E-field such as that present in circularly polarised light⁹. The frequency of circularly polarised light recorded by an observer is dependant on the rotational frequency of the incident light and in a rotating reference frame the observed frequency will be shifted by the angular frequency of that reference frame. When it is passed through a half-wave plate, circularly polarised light has its handedness reversed, analogous to the reversal of the k-vector of light on reflection at a mirror. If the half-wave plate is rotating then the observed frequency of the transmitted light ${\mathrm{\Omega }}_{\mathit{\text{rot}}}^{/}$ is shifted in frequency by twice the rotation frequency of the wave-plate either up, if the plate is rotating in the opposite sense as the incident E-field, or down if it is rotating in the same sense.

A simple calculation using the conservation of energy and angular momentum of a circularly polarised photon passing through a wave-plate shows how the frequency shift arises. Circularly polarised light has angular momentum of +ħ or - ħ per photon. Conservation of angular momentum of a photon passing through a rotating half-wave plate gives,

where Ω_rot and ${\mathrm{\Omega }}_{\mathit{\text{rot}}}^{/}$ are the rotation frequencies of the wave plate before and after the photon has passed through it, respectively, and I is the moment of inertia of the wave plate. Conservation of energy gives

where L₁ = IΩ_rot and L₂ = I${\Omega }_{\mathit{\text{rot}}}^{/}$ are the angular momenta of the wave plate and ω₁ and ω ₂ are the frequencies of the photon before and after the photon has passed through the wave plate. By comparing the conservation of momentum and energy we find the frequency shift due to the rotating wave plate to be

The above derivation has used the approximation ½(${{\mathrm{L}}_2}^2$ - ${{\mathrm{L}}_1}^2$) ≅L₁(L₂ - L₁), but the approximation sign has been dropped since it leads to an error of only 1 part in 10³⁰.

3. Application – Moving Interference Patterns

When creating a frequency difference between two co-propagating laser beams the most practical scheme is that of a Mach-Zehnder interferometer such as that seen in Fig. 1. In this scheme a linearly polarised (at 45° to the horizontal) beam is split by a polarising beamsplitter. Each beam is then incident on a quarter-wave plate that makes them circularly polarised with opposite handedness (+ħ and -ħ) to each other. One beam is then passed through a rotating half-wave plate. The handedness of the two beams is again the same, so that they interfere when they are recombined at a 50:50 (non-polarising) beam splitter.

 

Fig. 1. Adapted Mach-Zehnder interferometer for creating two co-propagating laser beams with a frequency shift between them of 2Ω_rot. PBS, polarising beam splitter; M, mirror; λ/2, half wave plate; Rλ/2, rotating λ/2; BS, 50:50 beam splitter; +ħ , right hand circularly polarised light; - ħ , left hand circularly polarised light; Ω_rot, rotation frequency of half-wave plate.

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When the input to the interferometer in Fig. 1 is a Gaussian beam and the half-wave plate is stationary the output lies on or between complete constructive or destructive interference. As soon as the plate starts to rotate the output beats at twice the frequency of the half-wave plate rotation frequency. Fig. 2 shows a beat signal as typically observed at the output of the interferometer. A fast Fourier-transform (FFT) analysis of this beat signal reveals a single peak at 14 Hz, exactly double the rotation rate of the half-wave plate, as expected. The peak frequency of the beat signal is stable within less than 1 Hz of the central frequency for over 1 hour and the bandwidth of the beat signal is less than 3 Hz. The laser used for this and the other results in this paper was a CW diode-pumped Nd:YLF laser operating at 1064nm with an approximate bandwidth of 10 GHz. The frequency stability of the laser is not crucial in this work since it is the relative frequency shift between the two interfering beams that gives the evolution in the interference pattern. However, it is necessary that the laser does not suffer from longitudinal mode hops since this would lead to discontinuous hops in the pattern. The narrow bandwidth is necessary to ensure complete constructive or destructive interference even in the presence of small inequalities in the path lengths of the two interfering beams.

 

Fig. 2. The beat signal produced by interfering two Gaussian beams separated in frequency using the angular Doppler effect.

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If the two interfering Gaussian beams are not perfectly overlapping then instead of a beat signal we get a set of linear fringes that scan from left to right or vice-versa, depending on whether ∆ω is positive or negative. The appearance of new fringes at the edge of the interference pattern occurs at the same frequency as the beat signal (2Ω), so that the fringes scan more slowly the larger the number of fringes in the pattern. For a beam waist d the number of fringes N produced by interfering two beams of wavelength λ at an angle a is N = d sin α/λ. Hence, the velocity V at which a particle trapped in a fringe of the interference pattern travels as the pattern is scanned across is

To achieve rotation of trapped particles, rather than translation, one of the Gaussian modes can be replaced by a Laguerre-Gaussian (LG) mode. LG beams have two indices: p, which describes the number (p + 1) of radial nodes and l, which describes the number of 2π cycles of phase round the circumference of the annular shaped mode. When an LG mode with a radial index p = 0, but with azimuthal index l ≥ 1, is interfered with the Gaussian beam the azimuthal phase variation is converted to an azimuthal intensity variation. At a focus this results in a pattern consisting of l spots but outside of a focus the pattern consists of l spiral arms due to the mismatch in the Gouy phase elements of the two modes⁶. If instead two LG beams of opposite helicity are interfered there is no beam parameter mismatch and the resultant interference pattern of |l ₁ - l ₂| spots propagates invariant in space (with radial scaling equal to the divergence of the interfering LG beams⁷. If the two modes are separated in frequency the interference pattern will rotate around the axis of propagation.

For the case of two interfering LG beams with opposite helicities, regardless of its polarisation state which should be the same for both beams, the corresponding electric field can be written as

Where ∆ω = ±2Ω is the frequency shift introduced due to the angular Doppler effect, and Ω. is the rotation rate of the half-wave plate. Then the intensity of the resulting field is

But in this case, since |E ₀₁|² = |E ₀₂|² = I ₀ (p, z) we obtain

And finally, this can be written as

Where ψ/(φ, t)=(|l|φ + Ω_rot t). According to the last expression, the rotation rate of the interference pattern is

It is worth emphasising that the frequency difference between the two arms of the interferometer is 2Ω_rot. On the other hand, in the case of the interference between a plane wave and an LG beam, we can take expression (5) for the LG beam and for the plane wave we have

The corresponding intensity in this case, in terms of Ω_rot, is

Therefore, it can be seen that now, the rotation rate of the pattern will be given by

However, notice that for the 2-spot pattern of an LG + LG superposition, where l = 1, we have from (8) that the rotation rate is Ω_rot, while for a 2-arm spiral in the case of an LG + plane wave superposition, where l = 2, we find by using (10) that the rotation rate is also Ω_rot. With this example, it is clear that even when the expressions for the rotation rates are different in each case, the final result for equivalent patterns (with the same number of tweezing sites) is the same.

The rotating half-wave plate can also be used to achieve extremely high-resolution movement of interference patterns including standing wave patterns. As an example: a 5^O turn of the half-wave plate can create a translation of 40 nm in the silica spheres trapped in linear fringes shown in Fig. 3.

4. Results

Figs. 3 (a)–(d) show 1 μm diameter silica spheres being transported from left to right across a sample slide as linear interference fringes are continuously scanned using the angular Doppler effect.

 

Fig. 3. 1 μm diameter silica spheres moving from left to right as the linear fringes of an interference pattern are scanned using the angular Doppler effect. The particles are trapped (a) along the bright fringes and then move to the right (b)-(d). Continuous motion of the pattern amalgamates all of these spheres on the right hand side of the pattern region (d).

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To produce a rotating interference pattern from an LG beam, a slightly adapted form of the set-up shown in Fig. 1 was used. A Gaussian TEM₀₀ mode was directed onto a hologram before being directed into the interferometer. In one arm of the interferometer a dove-prism is used to reverse the helicity of the LG mode in that arm. The rest of the set-up worked as with the Gaussian beams so that there are two circularly polarised modes (either both left-hand or both right-hand polarised), one LG beam with positive l and one LG with negative l and of slightly different frequency, overlap at the output of the interferometer. These overlapping beams interfere to form a pattern of |l ₁ - l ₂| spots rotating at an angular frequency dφ/dt (see Fig. 4). The sense of rotation of the pattern is simply reversed by reversing the sense of rotation of the wave-plate (shifting the frequency down rather than up). By focussing the rotating pattern produced from an l = 2 LG beam with a ×63 microscope objective, 5 μm long, 1 μm diameter glass rods were trapped in two dimensions and rotated continuously at up to 20 Hz (see Fig. 5). Other objects can be rotated and LG beams of different l can be used to adapt to the shape of a specific object^6,7,8. The non-invasive nature of the procedure also allows for biological matter to be manipulated in a similar fashion.

 

Fig. 4 Rotating interference patterns: patterns produced in order between l = 1 + l = -1, l = 2 + l = -2 and l = 3 + l = -3 [see video-clip: arlt1.m1v (1,711 KB)].

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Fig. 5 a 5 μm long glass rod trapped in the interference pattern between one Laguerre-Gaussian beam with l = 1 and one with l = -1, is rotated continuously in water using the angular Doppler-effect [see video-clip: arlt2.m1v (1,470 KB)].

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5. Discussion

We note that acousto-optic modulators in conjunction with highly stabilised synthesizers could be used to create this frequency shift. However, in addition to being significantly expensive this system is potentially difficult to align due to small input apertures in the AOMs. Typically this requires lenses to focus and recollimate the light in an arm of the interferometer. In turn this can create difficulties in matching the beam parameters in each of the interferometer arms. When using the Laguerre-Gaussian beams described above, the relative beam parameters play a pivotal role in the final form of the interference pattern obtained⁶ and thus this is an important issue to consider. In contrast using the angular Doppler effect is inexpensive, simple and requires very basic waveplates (with large apertures) making it a very powerful and practical method to create this frequency shift and thus slow motion of an interference pattern.

Mechanical rotation of a half-wave plate is especially useful for low rates of rotation from less than 1 Hz to 100s of Hz. At much higher rotation rates it becomes less suitable due to the demands on the mechanical components, though it could easily be used up to the kHz range with good quality components. For frequency shifts larger than a few KHz the rotating half-wave plate could be replaced by a Pockels’ cell that electro-optically reproduces the retardation effects of the rotating wave plate¹⁴.

6. Conclusion

We have shown the first practical application of the angular Doppler shift. By creating very small frequency shifts between two beams derived from the same source, the continuous scanning of linear interference fringes was used to transfer microscopic particles. 5 μm long glass rods were rotated continuously by creating a frequency shift between overlapping Laguerre-Gaussian beams of opposite helicity. This powerful technique could be extended to any instance where slow motion of an interference pattern is required such as for the deterministic delivery of cold atoms³. It presents a significantly simpler, more reliable and cheaper alternative to existing techniques for creating very small frequency shifts to create motion in interference or standing-wave patterns. The technique also lends itself to absolute positional control of particles and cold atoms trapped in such an interferometric pattern.