Dynamic methods are those deemed to be easily changeable over time but tend to be more expensive than static methods. Yet they hold a great advantage to the experimenter in the quick and easy ability to test and evaluate ideas and designs, without the need to undertake complex production processes.
There are two main methods of approach for dynamic beam shaping; either a single beam is re-positioned continuously and is ‘shared’ between desired directions or the single beam is ‘split’ between all directions simultaneously.
In this thesis I design holograms to work with spatial light modulators that distribute the incident beam simultaneously into multiple beams. Next I will place this work in context by considering a number of alternate time-sharing techniques.
Deformable mirrors
Deformable mirrors provide wavefront modulation by physically altering the optical path length the light takes. The path length is usually modulated by some form of actuator placed behind the mirror to deform it. Either multiple actuators support a whole single mirror or the mirror is made from multiple segments each supported by an individual actuator.
When dealing with optical tweezers, beams more complex than a simple Gaussian are often desired, for example Laguerre-Gaussian, Bessel, or Matthieu beams. These, however, require a discontinuity in the phase wavefront of the beam which deformable mirrors cannot achieve unless they are segmented133.
Although the mirrors have reasonably high speeds ∼1 kHz134, their magnitude of beam deflection is small putting limits on their usefulness. However, they are ideally suited to correction of aberrations within optical tweezers135.
Scanning mirrors
Galvanometer and piezo-electric driven mirror techniques are relatively low cost with negligible optical power loss and allow large beam deflection angles at quite high speeds of
∼2 kHz, but they do suffer from wobble and jitter stability problems38. They have been
easily integrated into optical tweezers enabling the spatial patterning of particles136–138. They also enable the trapping of particles where the ratio of medium and particle refractive
indices is opposite to that required for trapping in a normal Gaussian beam139,140.
Acousto- and Electro-Optic Deflectors (AODs and EODs)
AODs have a reasonable optical transmission of ∼ 80% but for both x and y deflection this decreases to∼64%. In the context of optical tweezers where the time duration of the potential is important their high speed,∼20 kHz, allows the time sharing of more traps. One disadvantage is the variation of diffraction efficiency over the acoustic bandwidth of the device resulting in variation in powers of ∼10−15% for different diffraction angles. EODs can perform at much higher speeds,∼10 MHz, have a higher optical transmission of ∼ 90% (∼ 81% for x and y together), a far more uniform diffraction efficiency as a function of angle and increased accuracy38,141.
AODs and EODs have been used to perform some of the most stunning experiments with optical tweezers such as studying certain biomolecular processes at the single molecule level with no ensemble averaging142.
3.2.1 Spatial Light Modulators (SLMs)
Spatial light modulators (SLMs) are relatively slow in comparison to the previous tech- niques operating at ∼ 75 Hz but have the major advantage of diffracting beams into multiple directions simultaneously although this does decrease the peak power per beam. There are ferro-electric SLMs available capable of speeds of tens of kilohertz but these can only produce two levels of phase retardation, significantly reducing their efficiency90. Nematic spatial light modulators can achieve efficiencies of∼50%143, significantly below that of its competitors. However, two more major advantages contribute to making them useful versatile devices. Firstly, they can perform mode conversion on propagating beams, for example they can convert a Gaussian into a Laguerre-Gaussian beam. Secondly, they can also change the phase profile of a beam along its axial direction enabling the focussing of beams with no physical optical element.
Finally, at the time of purchase, the cost and ease with which the device could be plugged in and up and running (its ‘plug and play nature’) was far superior to its then competitors. At the time of writing other methods are becoming more user friendly but there are still challenges to overcome such as limits on resolution set by digital to analogue conversion on computer hardware144.
phase-only diffractive optical elements. Within the literature there are slight variations on the apparatus and design, for example it is possible to use Fresnel based systems91, amplitude modulation145,146 and ferroelectric SLMs90,147.
Having established my use of SLMs a background of their operation will now be given.
How the phase is modulated
Rather than varying the thickness of a glass substrate, as for phase-only DOEs, materials can be used that alter their retardation properties without changing size, namely liquid crystals. Liquid crystals are an additional state of matter lying between liquid and solid such that they do not possess positional order but do exhibit orientational order. On average, over time, the elongated liquid crystal molecules tend to point in a given direction called the director,n, of the liquid crystal as shown in figure 3.1.
Figure 3.1: Image showing the elongated nature of liquid crystal molecules along with their orientational, but not positional, order. The arrow and vectornindicate the di- rector which on a time average points along the direction of orientation of the molecules.
The basic design of all spatial light modulators is similar; they consist of a thin layer of liq- uid crystal between either two transparent electrodes or one reflective and one transparent electrode as shown in figure 3.2.
The nematic phase possesses a dielectric anisotropy, ∆ǫ, governing its response to an electric field, defined as
∆ǫ=ǫk−ǫ⊥, (3.1)
whereǫkandǫ⊥are the dielectric permittivity measured parallel and perpendicular to the
director respectively. Under an applied electric field the molecules will try and minimise their electric energy density by aligning perpendicular or parallel to the electric field given
Figure 3.2: A Liquid Crystal over Silicon (LCoS) spatial light modulator. The indi- vidually addressed pixels, which act as one electrical contact are shown at the bottom and a transparent second electrical contact layer is placed above the liquid crystal. When an electric field is applied the liquid crystal molecules, hence director, rotate and change the local extraordinary refractive index, thus altering the retardation im- parted to the wavefront in that pixels area.
that the dielectric anisotropy is either negative or positive respectively. They also compete against an opposing force due to the elasticity of the liquid crystal, hence the molecules also attempt to minimise their elastic energy density148.
As an electric field is applied across individual pixels the molecules, wishing to align with the electric field, begin to rotate until they reach an equilibrium with the elastic energy density (figure 3.2). With larger fields the molecules, hence local director, will rotate further. With no voltage applied across the liquid crystal polarized light entering along the extraordinary axis, will experience a refractive index ne. As a voltage is applied and the director rotates through an angle,θd, the light will experience a modified extraordinary refractive indexne(θd) giving an effective birefringence
∆n=ne(θd)−no, (3.2)
where no is the refractive index along the optical axis. Therefore the phase of the light incident on a specific pixel area is retarded by
δ= 2π
λo
d|∆n|, (3.3)
where d is the thickness of the liquid crystal layer and λ0 is the wavelength of light.
So, increasing the electric field across a nematic liquid crystal varies the extraordinary refractive index of the material, hence produces a shift in the phase of the incident light. The fact that the voltage alters the refractive index along the extraordinary axis of the material means the amount of retardation is polarisation sensitive and it must be ensured
that the incident light’s polarisation is aligned with this axis. Either a rotatable polariser or half-wave plate can be used.