1.2 Project goal
2.1.5 Computer generated holography
For manipulating acoustic fields, the inverse problem is of interest. That is, starting from a desired 3-D distribution of pressure p(x, y, z) and possessing the ability to control properties of the source such as the amplitude A(x, y) or phase φ (x, y) across a 2-D plane, how to derive a complex distribution A(x, y)eiφ (x,y) (or hologram) that will best generate the target field. In optics, the development of algorithms that address this inverse problem is known computer generated holography (CGH).
2.1.5.1 Forward models
To solve the holography inverse problem, models that can calculate the 3-D distribu-tion of acoustic pressure generated by a given hologram distribudistribu-tion are necessary.
In acoustics, due to the length scales involved, two models are most commonly used:
the angular spectrum approach (ASA) and the Rayleigh-Sommerfeld integral.
The ASA was originally developed in optics [50] for the rapid evaluation of the field generated by a known planar source distribution across other parallel planes.
Assuming the complex distribution of the pressure p(x, y, z0) is known across some plane z0, the pressure at a depth z is evaluated using the ASA by first taking the spatial Fourier transform of the pressure using
P(kx, ky, z0) = Z Z
p(x, y, z0)ejkxx+ jkyydxdy. (2.2)
Here kxand kyare the spatial frequencies in x and y respectively. This decomposes the field into a set of plane waves propagating at different angles defined by the spatial frequencies. Each of these plane waves is then multiplied by a propagator term H(kx, ky, ∆z) which accounts for the phase change introduced by propagation to the new target depth. This is given by
H(kx, ky, ∆z) =
e(−i∆z
√(k2−k2x−k2y))
, if k2x+ ky2< k2. e(−∆z
√(k2x+k2y−k2))
, if k2x+ ky2> k2.
(2.3)
Here k is the wavenumber, and ∆z = (z − z0). Where k2x+ k2y < k2 the waves are real and propagate, and where k2x+ ky2> k2, the waves are evanescent so decay exponentially. After applying this propagator, the field across z is reconstructed using the inverse spatial Fourier transform. In order to ensure numerical accuracy, there are additional requirements. When the field is back-propagated (i.e., ∆z < 0), evanescent waves are amplified exponentially so need to be cutoff. Additionally, the ASA is subject to undersampling for higher spatial frequencies which introduces aliasing. To eliminate this, it is necessary either to significantly extend the source plane or to restrict the spectrum that is propagated [51].
The ASA is useful for problems where the target distribution of pressure and source occupy 2 parallel planes, as the field can be rapidly propagated between the two using Fourier transforms. However, if the source is non-planar or the target distribution consists of a few discrete points, it becomes less efficient to propagate the whole field. Where this is the case, it is more convenient to use the Rayleigh-Sommerfeld integral. For a planar source p(x0, y0, 0) mounted in an infinite baffle,
this gives the pressure at a point elsewhere in the field as [47]
p(x, y, z) = Z
S
p(x0, y0) z 2πr
ik r + 1
r2
eikrdS. (2.4)
Here dS is a area element, S defines the area over which the source is non-zero, and r is the distance between the (x0, y0, 0) and (x, y, z) coordinates given by r = p(x − x0)2+ (y − y0)2+ z2. When numerically accurate, the Rayleigh-Sommerfeld and ASA are equivalent [50, 52], so the appropriate forward model depends on the problem geometry.
2.1.5.2 Complex modulation
The difficulty of calculating a computer generated hologram depends on the prop-erties of the transducer. In the ideal case of a dense 2-D array for which both the amplitude and phase can be freely modulated, the problem can be simple [53, 54].
For example, for a planar target distribution At(x, y) at a depth z, the field can be back-projected to the hologram plane using the ASA. Using the phase and ampli-tude values that result from this back-projection as driving signals generates a good realisation of the target distribution. This is illustrated in Fig. 2.3. For non-planar distributions, such as a set of arbitrarily placed focal points rj, the phase conjuga-tion approach described by Ibbini can be used [55]. This uses Eq. 2.4 to calculate the complex amplitude Ai jeφi j generated at each target point rj by each transducer position ri. The sum of the complex values is then taken for each transducer element to give a single phase and amplitude pi= Aieφi, and the complex conjugate of this value is used as a driving signal.
For many practical systems, however, it is only possible to control either the amplitude or the phase. Moreover, the values these can occupy are usually quan-tised, in the extreme case to binary values. Where the source has these constraints, the simple back-propagation approach no longer yields acceptable results. This is illustrated by Fig. 2.4, which shows the pressure distribution generated for the same problem as Fig. 2.3 with different constraints applied to the source after back-propagation. From the top row to the bottom it shows: phase and amplitude
modula-Target distribution
Phase Amplitude
Generated field
Figure 2.3: Generation of an arbitrary pressure distribution by using back-propagation.
First column is the target distribution which resembles a B. The ASA approach is used to back-propagate to the target plane giving the phase and amplitude values shown in the middle column. When these values are forward propa-gated with the ASA approach the distribution on the right is generated at the target depth. This closely matches the target distribution
tion, only phase modulation, binary phase modulation, only amplitude modulation, and binary amplitude modulation. In each case, the generated field is significantly distorted compared to the ideal case shown in Fig. 2.3. To design holograms for sources possessing these limitations, a wide variety of algorithms have been devel-oped. Two popular approaches are iterative Fourier transform algorithms, and direct binary search algorithms.
2.1.5.3 Iterative Fourier transform algorithms
Iterative Fourier transform algorithm (IFTA) is a general term for a group of algo-rithms that originated with the work of Gerchberg and Saxton in 1972 [56]. They addressed the problem of retrieving the phase distribution across two planes, the fields of which are related by a Fourier transform, from amplitude measurements on both planes. This is a relevant problem for several fields such as astronomy and electron microscopy. These methods have also been applied for the calculation of
Amplitude Phase Field +
+
+
+
+
+
=
=
=
=
=
Figure 2.4: Attempting to generate the same arbitrary pressure distribution as Fig. 2.3 with different constraints on the ability to manipulate the phase and amplitude. From top to bottom the constraints applied are: None, only phase modulation, binary phase modulation thresholded to 0 and π, only amplitude modulation calcu-lated taking the real component of the complex modulation, binary amplitude modulation calculated by thresholding the real component about 0.
phase only holograms eφh(x,y) that generate target planar intensity (or pressure) dis-tributions At(x, y) [57]. The basic algorithm is illustrated in Fig. 2.5. It starts by imposing a starting phase distribution (usually random) onto the target amplitude distribution At(x, y). This field is then propagated to the hologram plane where the phase φh(x, y) is kept and the hologram amplitude Ah(x, y) is replaced with a known distribution (i.e., the spatial profile of the beam). The new field is then propagated back to the target plane where the phase is retained and the amplitude is replaced by At(x, y). The process is iterated until the algorithm stagnates or the error,
eval-Initial condition
Propagate to hologram plane
Propagate to target plane Propagate to
hologram plane
Replace amplitude
Amplitude Phase
Converged solution
Amplitude Phase
Replace amplitude
Amplitude Phase
Amplitude Phase
Figure 2.5: Illustration of iterative Fourier transform algorithm for the same problem as Fig. 2.3. The converged solution is shown in the bottom right. The improve-ment compared to the solution from the initial iteration (directly above) can clearly be seen.
uated by comparison with the target amplitude, falls below a threshold. For planes related by a Fourier transform, the amplitude error in the reconstruction decreases or remains constant with each iteration due to the unitary property of the Fourier transform [56].
The model relating the two planes depends on their positioning. In the original implementations of the IFTA, holograms were designed to generate target distribu-tions in the far-field or in the focal plane of a lens [58], where the fields are related by a Fourier transform. However, they have also been designed to operate in the Fresnel region where planes are related by the Fresnel transform [59], and in the near-field where they are related by the ASA [60].
One drawback of IFTA is the difficulty of calculating quantised phase holo-grams. If the phase distribution φh(x, y) is quantised, in each iteration errors are introduced and the algorithm is prone to getting stuck in poor local minima [61].
To avoid this, the phase can be quantised gradually, either by defining limits around the discrete levels for which the values are quantised that gradually expand, or by slowly decreasing the number of available phase levels as the iteration number in-creases [62].
2.1.5.4 Direct search algorithms
An alternative to IFTA are a group of algorithms that directly perturb the hologram amplitude Ah(x, y) or phase φh(x, y) in order to find a solution that best approximates a target amplitude. These are used most often to calculate holograms for which the available modulation is highly quantised (in most cases binary). The most basic version of this approach has been dubbed direct binary search [63], and is illustrated in Fig. 2.6 for the design of a binary amplitude hologram.
The hologram is initialised in an arbitrary binary state Ah(x, y) and a target amplitude distribution At(x, y, z) is defined. The amplitude A(x, y, z) generated by the current hologram state is calculated using a forward model. The quality of this solution is then evaluated by comparing the current amplitude to the target distribution using a cost function, for example
C=
∑
x
∑
y
∑
z
|At(x, y, z) − A(x, y, z)|. (2.5)
Here the value of C is the ‘cost’ or current solution quality, where lower is better.
From this initial condition, individual pixels on the hologram aperture are selected, the amplitude is perturbed, and the change in C is evaluated. If C is decreased the change is kept, otherwise it is reversed. This process continues until no further changes are accepted or a stopping criterion is fulfilled. One disadvantage of the direct binary search approach is that the algorithm is greedy. Since only positive changes are accepted at each step, it is inevitably trapped in the first minima it encounters. This is defined by the initial hologram state and the pixel selection
Initial condition
Evaluate change in `cost’, keep or revert change
Converged solution Permute hologram
structure
Compute change in field
Figure 2.6: Illustration of direct search algorithm for designing a binary amplitude holo-gram for the same design problem as Fig. 2.4. The converged solution in this case can be compared to the final row of Fig. 2.4 the improvement can again be clearly seen.
criteria. If the solution space contains large numbers of bad solutions, then the algorithm will perform poorly.
To overcome this drawback several works have proposed simulated annealing [64, 65, 66]. This introduces a probability of keeping bad changes which increase the cost into the optimisation. This is done by defining a temperature T associated with the optimisation. Each time a pixel is changed, as before if C decreases it is kept, however, if C increases then the change can still be kept with a probability calculated based on the current value of T and the magnitude of the change ∆C.
This probability P is usually calculated using
P= exp
−∆C T
. (2.6)
As the optimisation proceeds, the temperature is gradually decreased. This reduces the probability of accepting bad changes until the algorithm effectively becomes di-rect binary search. Adopting this method gives the optimisation a chance of escap-ing very poor solutions reached early in the optimisation. The actual convergence will be determined by the choice of initial temperature and the cooling rate.
Another proposed alternative are genetic algorithms [67, 68], which can in principle converge to the global minimum. These work by starting with a popula-tion of initial solupopula-tions Hn(x, y). The cost Cnof each is then evaluated and the
solu-tions are ‘bred’ and ‘mutated’. Breeding involves creating new solusolu-tions Hnew(x, y) by combining 2 or more existing solutions. The solutions that are bred are selected with a probability based on the cost. Mutation involves perturbing a random set of pixels on each solution. After these steps the cost of each solution is re-evaluated and a percentage with the greatest cost are eliminated. The process is repeated through hundreds of generations until a stopping criterion, based either on stagna-tion or a target solustagna-tion quality, is reached. Having a large family of solustagna-tions and introducing random mutations allows the optimisation to avoid stagnating in local minima. It is, however, extremely sensitive to tuning the various parameters. Addi-tionally, genetic algorithms are computationally intensive as the acoustic field of a large family of holograms needs to be repeatedly evaluated.
In comparison to IFTA algorithms, direct search approaches have several ad-vantages. For the design of binary amplitude or binary phase holograms they can perform better as the quantisation is directly included in the model. Furthermore, the choice of forward model is arbitrary, so either the target distribution or the hologram distribution can be three dimensional without additional complexity. The disadvantage is that these involve significantly more computational effort. A typ-ical IFTA algorithms requires ∼50-100 iterations to converge, while calculating a 128×128 pixel hologram with a direct search method requires hundreds of thou-sands of operations due to the nature of the algorithm.