Typically, holographic optical tweezers systems utilise a high numerical-aperture microscope objective to form an array of trapping sites in the objective’s focal plane. It is possible, using iterative computer algorithms, to calculate the phase-only hologram, orkinoform, that must be displayed on the SLM device to achieve the desired distribution of optical traps.
To generate my kinoforms I used a version of the adaptive-additive (AA) algorithm, as suggested by Soiferet al.[15]. The code I used was originally implemented by Dufresne [6] to generate templates for static kinoforms and was later adapted by Gabe Spalding for use on computer-operated LC SLMs. A mathematical analysis is included here for completeness and is drawn entirely from this earlier work.
For our analysis, we are interested primarily in the transverse intensity profile in the focal plane of the microscope objective. This is related to the electric field in the focal plane
2.2. Fourier optics and the adaptive-additive algorithm 40
which is given by
~
Ef(~ρ, t) =Af(ρ~)eiΦf(~ρ)
eiωteˆ (2.1)
whereωis the angular frequency of the incident light, and ˆeis its polarisation vector. Af(~ρ)
represents the amplitude of the electric field in the focal plane, while Φf(ρ~) describes the
phase. The intensity in the focal plane is given by
If(~ρ, t)∝E~f(~ρ, t)·E~f(ρ, t~ )∗=Af(~ρ)2 (2.2) and so we see that the intensity distribution in the focal plane is neither a function of the time nor the phase of the electric fieldE~f(~ρ). The electric field in the focal plane of a lens can be expressed by the Fourier transform of the electric field at the input plane [16].
Ef(~ρ) =F
Ein(~r) ≡2πfk eiθ(~ρ)
Z
d2~rEin(~r)e−ik~r·~ρ/f (2.3)
wheref is the focal length of the lens andkis the wavenumber of the light. θ(~ρ) is a phase term contributed by the presence of the lens, but it is not relevant to this discussion and can be ignored.
An inverse Fourier transform can be used to express the electric field at the input plane in terms of the field at the focal plane:
Ein(~r) =F−1
Ef(~ρ) ≡ k
2πf
Z
d2ρe~ iθ(~ρ)Ef(~ρ)e−ik~r·~ρ/f (2.4)
To begin with, we consider a two-dimensional array of identical Gaussian traps in the focal plane. The electric field describing such a distribution is given by
Ef(ρ~) = " X α E0f(~ρ−~ρα) # eiΦf(~ρ) (2.5)
Since we are considering a two-dimensional transverse distribution of traps, we can express the electric field in the focal planeEf as a convolution of the electric field profile for a single
focal plane Ef(~ρ) =hEf 0 ◦T(ρ~) i eiΦf(~ρ) (2.6) where T(~ρ) =X α δ2(~ρ−~ρα) (2.7)
The relationship between the electric fields at the input and focal planes of the objective lens can then expressed as
FnEin 0 (~r)eiΦ in(~r)o =hE0f(~ρ)◦T(~ρ) i eiΦf(~ρ) (2.8)
From Hecht [16] we can write
F{ab}= k
2πfF{a} ◦F{b} (2.9) Using this relationship means we can decompose equation 2.8 into two terms:
E0f =F E0in(~ρ) (2.10) and 2πf k T(~ρ)e iΦf(~ρ) =FneiΦin(~r)o (2.11)
The interpretation provided by Dufresne is that a phase hologram at the input plane of the microscope objective can control the positions of the traps in the focal plane, but not their actual structure. The nature of the traps themselves is determined by the nature of the electric field immediately before the phase hologram. This approach has been used, for example, to create arrays of optical vortices [7]. In reality, phase holograms can be used to control the trap structure in addition to the trapping position, and I will return to this aspect later in the thesis. For the work presented in this chapter however, we are concerned solely with arrays of standard Gaussian traps.
2.2. Fourier optics and the adaptive-additive algorithm 42
The adaptive-additive algorithm [15] is an iterative algorithm designed to calculate the phase profile at the input plane required to minimise the deviation of the derived intensity profileIf(~ρ) from the desired intensity profileIf
0(~ρ). Input plane Focal plane FFT FFT-1 YES NO
Figure 2.1: Flow chart showing the operation of the iterative adaptive-additive algorithm. The desired intensity profile in the focal plane is provided by the user as an input. Using a recursive FFT algorithm, the program iteratively calculates the phase profile for a kinoform that will generate the desired intensity distribution when placed at the back aperture of the microscope objective.
Figure 2.1 outlines the steps of the adaptive-additive algorithm. Each iteration begins with the description of the electric field in the objective input plane Ein
n . For the first
iteration, wheren= 0, Ain
0 is the beam profile before the kinoform, and Φin0 is a random,
discretised M ×M distribution, where M is the number of pixels along one side of the kinoform. The computational AA algorithm then applies a fast-Fourier transform (FFT) to the array to extrapolate the corresponding expression for the electric field in the focal plane of the objectiveEf
the expressionAf
n(~ρ) with a combination ofAfn(~ρ) and the desired electric field amplitude
Af0(~ρ) using the relationship
¯ Enf(~ρ) = h aAf0(ρ~) + (1−a)Afn i eiΦfn(~ρ) (2.12)
where the dimensionless mixing parameter a lies in the range 0 < a < 1. It has been suggested [6, 7] that setting a = 0.5 yields the optimum results using this method and a= 0.5 was the value used in the work presented in this chapter.
An inverse FFT then transforms the new expression for the electric field in the focal plane to gain a new expression for the electric field in the input planeEin
n . The description
for the amplitude in the input plane Ain
n is now no longer an appropriate description of
the real laser profile at the input plane, and must be replaced with the original description Ain
0 (~r). The iterative cycle runs until the error, defined as
ǫ≡M12 M2 X i=1 If(~ρ)−I0f(~ρ)2 (2.13) converges so that ∆ǫ/ǫ < X where X is a user-defined error threshold. For the work presented here we set X =e−5 and ran the whole algorithm 20 times, varying the initial
random phase profile each time and then keeping the result that produced the minimum final error.
The algorithm assumes the discretisation of both the input and focal planes of the objective into square grids of side M. It is worth noting that since we are using FFT algorithms, this approach necessarily deals with discrete arrays of pixels and so cannot be effectively used to create continuous landscapes, which while not a limiting factor for the work presented in this chapter, is relevant to the work presented in chapter 5.