The Science of Phased Arrays

We take an in-depth look into the basics of ultrasonic array science. We talk about what an array is, different types of array, important design parameters and properties and how they apply to Ultrahaptics. We look at how array design is often a trade-off of these different parameters and how this may affect the design of an application using Ultrahaptics.

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What is an array?

When talking about Ultrahaptics we often refer to the array. Here we answer the question of what an array is and how it works, why it looks like it does and important considerations that go into its design. For a customer, this becomes important when making decisions about the size of your interaction zone, the space available in which to integrate your array, haptic feedback and any other compromises that must be made for your application.

In electronics, an array is an arrangement of elements. These may be input or output devices, such as loudspeakers and microphones, radio aerials, transmitters or receivers, ultrasound transducers or even cameras. Since the Ultrahaptics array is an output device, we will discuss arrays in this context. When we refer to an element, we mean an individual ultrasound output transducer.

Arrays can come in many shapes, as shown in Figure 1. Individual array elements can be driven in several ways all at once, one row at a time, individually etc. When each array element can be controlled individually, we refer to this as a phased array. By using a phased array, an array can be ‘steered’ and ‘focused’.

    • Array Steering emitted energy is concentrated in a given direction
    • Array Focusing emitted energy is concentrated in a small area

Array focusing is the core idea behind the Ultrahaptics technology.


Figure 1 – Various designs of arrays. Many of these designs will have specific applications where their performance is optimal. Systems such as the 2D matrix array are among the most flexible. Image reference:

We will discuss the difference between steering and focusing and how this affects the array design. We will also look at some of the mathematics of array design and how it affects array performance. Looking at Figure 1, you will realise that Ultrahaptics uses a 2D matrix array. To keep this article simple, we will restrict ourselves to one-dimensional arrays. However, the extension to 2D is fairly intuitive.

Array parameters

The key parameters of array layout are shown in Figure 2.


Figure 2 – Key parameters of one-dimensional, phased array design.

Each element in the Ultrahaptics array produces a single frequency sinusoidal signal. The signal frequency f, matches the resonant frequency of the transducer. This type of acoustic signal is referred to as harmonic and is used as it makes mathematical simulations more straightforward.

Steering the array

To steer our array in a given direction we must apply a pre-calculated time delay to each element. Since we will model the acoustic field using harmonic excitation, we also need to apply a phase shift. The steering angle θs gives the direction the array is being steered in. An angle of θs = 0° would give a beam steered normal to the array, i.e. directly forward. An angle of θs = –90° would be an array steered directly left.

Figure 3a shows a beam steered at θs = –45°. The thick, black line shows the line of equal phase, known as the ‘equiphase front’. Figure 3b shows the illustrative time delays required for each element.

(a) beam-steering-a-300x220.png (b) beam-steering-b-300x237.png

Figure 3 – Beam steering.

For steering, the constant time delay ∆t between adjacent elements is calculated by the equation below.

  equation-1.jpg (1)

where c is the speed of sound and p is transducer pitch (in air c = 343 m / s). The corresponding phase shifts are given in Eq. (2) & Eq. (3).

  equation-2.jpg (2)
  equation-3.jpg (3)

Focusing the array

Focusing allows the emitted wave fronts from each element to converge to a single point. At this point, wave fronts undergo constructive interference, resulting in a region of high pressure.

For the Ultrahaptics array, it is this we refer to as the focal or control point.

To achieve focusing, individual phases are calculated for each element. For this, we first calculate the distances from each transducer to the focal point position, as shown in Figure 4a. We then calculate the relative time differences between elements as given in Eq. (4).

  equation-4.jpg (4)

where dj is the distance between the jth element and the focal point, and do is the distance between the central element and the focal point. This is shown in Figure 4b:

(a) array-focusing-a-300x221.png (b) array-focusing-b-300x230.png

Figure 4 – Array focusing. (a) do – distance between the middle element of the array and focal point and dj -jth element to focal point distance. (b) Example time delays for each of the elements.

We then convert these time differences to phase differences using Eq. (5).

  equation-5.jpg (5)

The phase shifts are applied to the driving signals for each element. The resulting acoustic fields will converge at the desired focal point.

As the frequency of the system goes up, and thus the time-period goes down, there is increased reliance on electronics capable of accurately tolerating the small variations in time and phase delay. Current Ultrahaptics arrays operate at 40kHz.

Steering and focusing compared

Note that while similar, focusing and steering differ. For beam steering we direct a plane of pressure of equal phase in a given direction; with focusing, we converge the pressures to a point. This is shown in Figure 5 below: (a) shows a beam steered at an angle of θs = – 15°. In (b) a beam is focused at x = -0.05m & z = 0.20m (highlighted by a blue circle).

Black points indicate the locations of each of the elements (in this example N = 16).

(a) beam-forming-a-300x225.png (b) beam-forming-b-300x225.png

Figure 5 – Beam forming (note the array is located at y = 0 and centred on x = 0). Note the higher peak pressure (≈ 130dB) for the focussing case.

With increasing focal point distance, the relative phase difference between neighbouring elements decreases so that eventually the two scenarios converge. This point is termed array far-field distance and is given by the equation below.

  equation-6.jpg (6)

where A is the overall width of the array and λ is the wavelength in air.

For a 16x16 element Ultrahaptics array, with f = 40kHz transducers and 10mm pitch, this distance is approximately 0.75m. This means that focusing up to this distance is theoretically possible while beyond this distance only beam steering can take place. We can consider Ztran to be a bounding distance around the array within which focusing, and thus haptic interactions can occur.

Array performance metrics

Many of the equations we will discuss provide us with important parameters of array performance, namely array directivity Harray, and array sharpness factor q.

Due to the complexity of accounting for focusing to an arbitrary location, these equations apply only to beam steering. For assessing focusing performance, numerical simulations must be used to obtain metrics such as focal point size and focal depth [Azar et. al. – 2000]. However, beam steering metrics are still very useful in understanding focusing performance.

Element Directivity

When discussing directivity, keep in mind that we may refer to either individual element directivity or whole array directivity. Element directivity, Hel(θ), is defined as the ratio of output pressure at an angle θ to the output pressure at the on-axis angle, e.g. θ  = 0 as shown in Figure 6(a) below. The directivity equation is shown below.

  equation-7.jpg (7)

Where a is the individual element width/diameter (a.k.a element aperture) and λ is the wavelength of the emitting signal. Figure 6(b) shows the output directivity profile of an example transducer. This is commonly referred to as the directivity sinc function, given below.

  equation-8.jpg (8)


(a) transducer-directivity-300x225.png (b) transducer-directivity-b-300x284.png

Figure 6: Example transducer directivity

The directivity profile is dictated by a and λ. The directivity profile is shown for different a/λ ratios below in Figure 7. Note that as element width increases for a given frequency the element becomes more directional and gives rise to smaller ‘side lobes’.


Figure 7: Directivity profiles versus element width to wavelength ratio

When modelling the acoustic field generated by the array, we can approximate the directivity of an element using the sinc function mentioned above. This model has been shown to be a good approximation of the ultrasound devices used by Ultrahaptics (see here for experimental data).

Note that with increased frequency for a fixed element width, the result is increased directivity. However, if an element is very directional, an array can only focus into directions in which its elements can project energy. Hence, there is always a compromise to be made

  • A directional element will concentrate its energy into a smaller area, increasing peak sound pressure, but
  • This will mean that no energy can be projected elsewhere.

An array built with highly directional elements will be less steerable/focusable than one built with low directivity elements.

Phase Directivity

We should also consider phase directivity when assessing array elements. This is measured experimentally by driving a known signal through a test transducer and measuring the output phase around a circle. An ideal source, or point source, will have a perfectly flat profile and a perfectly circular wave-front. In reality, transducers are not point sources and will have some phase variance. Our research has shown that the transducers used by Ultrahaptics have a relatively flat phase profile and can be treated as point-like in the element far-field (distances > approx. 1.4mm from transducer surface).

Beam Directivity

The Ultrahaptics array is an acoustic array with ultrasound transducer elements. Just as each element has directivity, so too does the array. Here we will cover how ‘array directivity’ is quantified and look at the factors that influence it.

Modelling an acoustic array

To quantify the properties of a phased array it is common to model the acoustic field mathematically. Two approaches are available: point source models based on the Huygen’s principle and Finite Element Analysis (FEA).

FEA is an extremely powerful technique, capable of modelling the behaviour of each element as well as complex reflections, scattering and interactions with objects. However, it has a large computational cost. For this reason, it can only be applied to small volumes and is generally not suitable for the interaction zone size used by Ultrahaptics.

The Huygens model is much simpler and is based on optical ray-tracing and has the advantage of being able to model a large domain in a much quicker time. The Huygen’s model assumes that a propagating wavelet does not undergo any physical changes by reflection, refraction, etc. It also assumes that all interactions between wavelets are linear.

Each element is treated as a point source with the directivity profile discussed above applied. This significantly reduces the calculations required allowing us to approximate the effective acoustic field to a high level of precision.

Steered Beam Directivity

Array directivity for a steered acoustic beam is given below.

  equation-9.jpg (9)

Where p is element pitch and N the number of elements. For example, the Ultrahaptics STRATOS Explore (16×16 element rectilinear array), f = 40kHz, p = 10.5mm and N = 16. For our model we assume a single dimension array, i.e. a single row of transducers.

As an example, suppose we wish to steer our array to θs = 20°. The acoustic pressure of the array is measured in the far field in relation to the array centre. The beam directivity is shown in Figure 8.


Figure 8: Phased array directivity

Note the key features of Figure 8: (1) the main lobe at +20°, (2) a number of small side lobes next to the main lobe and (3) a relatively large grating lobe at -30°. It is important to note that side lobes and grating lobes are not the same phenomena

  • Side lobes are caused by acoustic energy radiation away from transducers at defined angles, given by the element directivity.
  • Grating lobes are caused by the summation of aliased sampling of element directivities. These can also be referred to as secondary maxima.

Grating lobes are a significant issue for phased arrays as their amplitudes can be large compared to the main beam. Much of array design methodology attempts to remove or reduce their occurrence. We will be looking at some of these methodologies later.

Steered Beam Sharpness Factor

An import measure of steering performance is ‘main lobe sharpness factor’ q, and is defined below.

  equation-10.jpg (10)

Where q has units of radians. Main lobe sharpness factor is a measure of beam directivity in the steering direction.

Smaller q indicates a ‘sharp’ or ‘narrow’ and well directed main lobe.

Note that q approaches zero as N→ ∞. That is, with an infinite number of elements the beam width in the steering direction will be zero. q is also affected by varying the steering angle and frequency. This is shown in Figure 9 below for (a) steering angle and (b) frequency (blue represents low q and yellow is high).

(a) array-sharpness.png (b) array-sharpness-b.png

Figure 9: Array sharpness factor versus steering angle for a 16 element array at (a) 40kHz and (b) a range of ultrasonic frequencies.

Note that in Figure 9(a), beyond 60˚ offset, lobe sharpness falls significantly.

Focusing Sharpness Factor

Unlike beam steering there are no equivalent functions for sharpness q or directivity Harray(θ) when focusing a beam. We, therefore, employ numerical techniques, such as the Huygen’s approach outlined above, to model the acoustic field and define related metrics.

A useful quantitative measure for assessing focusing quality is the Array Performance Indicator (API)

  equation-11.jpg (11)

Where Area_6dB is the area within which sound pressure is within 6dB of the peak focal point pressure. This can be extended to three dimensions by calculating the corresponding volume and dividing by wavelength cubed. A small API indicates a more focused and concentrated focal point.

Since the area cannot be calculated directly the acoustic field must be simulated for fixed array parameters. The effect of element number N, on the API for a frequency of 40kHz with an element pitch p = 10.5 mm is shown below:


Figure 10: Aperture Performance Indicator

As the number of elements increases the size of the API reduces, indicating a more focused focal point. In the above example, greater than approximately 18 elements the increase in focal point resolution is negligible.

Grating Lobes and Grating Foci

In our previous discussion on steered beam directivity, we introduced the terms main lobe, side lobe and grating lobe. We noted how the side lobes and grating lobes differ. Here, we go into more detail of the differences, how the effects of grating affect phased arrays for Ultrahaptics, how this can be mitigated and how this is always a design compromise.

In general, side lobes have lower amplitude than the main lobe and grating lobes. From a haptic point of view, using the Ultrahaptics array, these side lobes are generally not a problem as they are below the threshold required for tactile sensation.

The main lobe is the desired lobe in the steering direction, while a grating lobe is unwanted. When discussing the Ultrahaptics arrays, we are usually talking about focusing. For this reason, we introduce the term ‘grating foci’ to refer to an unwanted focal point. Grating effects – lobes and foci – are a potential problem for phased arrays as they are high amplitude areas of pressure in unwanted locations. For Ultrahaptics this is a concern as it can create haptic regions outside of our desired interaction zone.

Grating effects are caused by aliasing of phase distributions in the acoustic field. While grating lobes and foci are similar, there are subtle differences.

Determining Grating Foci

There is no analytical method for predicting the location of a grating focus. However, while our directivity equation (9), is derived from beam steering, it can still be used in the case of beam focusing to predict the angle of a grating foci. Combined with numerical modelling techniques it can then be used to assess the whole acoustic field and locate the exact location(s) of a grating focus.

As an example, a model of the acoustic field produced when a control point is created at θs = -14°, x=-0.05m and z=0.20m is shown in Figure 11(a).

(a) Screen-Shot-2019-01-08-at-12.33.40.png (b) Screen-Shot-2019-01-08-at-12.33.47.png (c) Screen-Shot-2019-01-08-at-12.33.55.png

Figure 11: Grating focus calculated from beam forming equation and modelled acoustic field.

The dashed-red line has been calculated using the steered beam array directivity function; Figure 11(b) shows the pressure through a vertical slice of the array through the focal point (dashed yellow line in (a)) with pressure in decibels, while (c) is the beam directivity function predicted for a steered beam. This demonstrates that even without analytical functions for focused beam behaviour we can still make useful predictions about the direction of a grating foci.

Desired and grating foci strength

While in most cases the desired focal point strength exceeds that of the grating foci, in certain cases this may not be true. This is due to element directivity. As an example, if the desired focal point is away from the footprint of the array, the peak pressure achieved there will be relatively low, since the element directivity results in a lower intensity. However, at the same time, a grating focus may appear within the footprint of the array, where the element directivity leads to a higher-pressure focal point. This is shown in Figure 12 for two cases, where the blue line indicates the direction of the desired focal point, with the red dashed line predicting the grating foci.


Figure 12: Grating focus strength, circled, larger than desired focus (dashed blue line).

Of course, spurious focal points lead to confusion for any user of an Ultrahaptics array. For this reason, careful consideration of element directivity is critical in array design. It should also be taken into account in application design when defining interaction zone dimensions.

Mitigating the effects of grating foci

Element pitch

The element pitch p is the distance between the centres of neighbouring elements. This has a large effect on the radiated acoustic field. It affects the peak pressures generated and the shape of the focal point as well as the position and strength of grating foci. This is shown in Figure 13 below.

pressure-field-a.png pressure-field-b.png
pressure-field-c.png pressure-field-d.png

Figure 13: Acoustic field for a fixed focal point and varied element pitch p.

Critical pitch, at which grating lobes do not occur, can be defined for the beam steering case and is given by the equation below.

  equation-12.jpg (12)

This equation, while derived for beam steering, still has application to the focused beam case. The equation is often overly simplified to λ/2 for an ideal case of steering angle up to θs =±90° and large N. This corresponds to a planar array capable of steering up to a maximum right-angle without any grating lobes. This can be seen in the figure above where the element pitch values at which grating foci do not appear is when p ≲ 0.5λ.

With reduced steering angle, the critical pitch constraint is reduced. This is an important consequence, since manufacturing arrays with critical pitch constraint is difficult to achieve. The critical pitch for a 40kHz array (λ = 8.5mm) is approximately 4.25mm. The current Ultrahaptics transducer diameter is 10.0mm, with a pitch of 10.5mm (1.22λ). In a rectilinear array this is a sub-critical spacing and will result in grating effects. Below we will discuss how this can be mitigated by modifying element position.

Number of elements

The total number of elements used in an array has a significant effect on grating lobes and foci. For a given array size, the number of elements is determined by the individual element size. This physical limitation means there is an obvious limit to the number of elements per unit area, and hence, their spacing (pitch). Below critical pitch spacing the element number, while important to properties such as overall acoustic power, is less of a factor in the occurrence of grating effects.

Element positioning

So far, we have considered only uniform inter-element spacing. It is this very uniformity that gives rise to grating lobes and foci. As mentioned, grating effects are due to the aliasing of phase distributions in the acoustic field. The grating effect phenomenon can be thought of as unintentional steering or focusing of the array due to the even spacing of elements when p < pcritical. However, elements can be spaced in a non-uniform manner to mitigate the effects of grating.

Here, we model a 16-element array with f = 40kHz and an initial pitch p = 1.2λ. The location of each element is varied by an increasingly random amount up to 5mm (to avoid overlapping). The effect of this perturbation is to ‘break’ the uniformity of the element spacing. The figures below show results for element perturbation factor pert, between 0 and 1, where 0 represents no perturbation (uniform) and 1 corresponds to a maximum element perturbation of  5mm. The black dots at z=0 indicate element positions. Results are shown for a focus point at x = –0.05m, z = 0.20m indicated by the blue circle.

pressure-a.png pressure-b.png
pressure-c.png pressure-d.png

Figure 14: Acoustic field for a fixed focal point and randomly perturbed element spacing. pert = 0 is uniform, pert = 1 indicates maximum 5mm perturbation.

This result shows that grating foci are reducing in intensity by splitting them across a more complex field. In this way, grating can be reduced and even removed using this approach. This method also reduces the ‘fill factor’ – number of elements per unit area – resulting in lower output per unit area.

Sunflower arrays

The most efficient layout used to mitigate grating effects and enhance focal points is actually found in nature – the sunflower array. This arrangement has the property that no two distances between any two elements are equal, thus not requiring critical spacing between neighbouring elements. This arrangement is related to the golden ratio and gives the optimum elements per unit area (fill factor).

This layout can be found in the Ultrahaptics STRATOS Inspire haptic module.

Glossary of terms

Acoustic centre The apparent source of acoustic radiation from a transducer. Will not always be on the surface of the radiating structure.
Array directivity The relationship between the propagation direction of an array and the resulting amplitude and phase.
Array performance indicator (API) A metric of acoustic imaging/focusing, related to the connect of the point spread function.
Array A collection of transducers. In acoustics the elements are often individually addressable and referred to as a phased array. Arrays can come in many arrangements.
Critical pitch The element pitch at which grating lobes do not occur. Depends on the steering angle and is strictly only valid for beam steering.
Element directivity The relationship between the propagation direction of an element and the resulting amplitude & phase.
Element A single transducer which is able to convert electrical energy into vibration producing acoustic pressure. Elements are also referred to as sources or transducers.
Far-field distance For a transducer this is the distance at which the pressure follows a  relationship. For an array this is the distance at which the array can no longer create a focal point but only beam steering is possible.
Fill factor The number of elements per unit area.
Finite element analysis, FEA A powerful numerical technique for modelling complex physical processes.
Focusing To direct a coherent acoustic field to a given focal point.
Frequency The number of cycles/vibrations of an oscillator per second.
Geometric spreading As acoustic waves propagate, they spread out. The pressure follows a  relationship where  is the distance from the source.
Grating foci Similar to a grating lobe but the result of aliasing of the acoustic field when focusing a beam.
Grating lobe An unwanted high amplitude lobe caused by aliasing of the acoustic field when steering a beam.
Huygens principle A mathematically efficient method of modelling acoustic fields, assumes waves are independent.
Phase The position of a point in time (an instantaneous value) on a waveform cycle.
Pitch The distance between two element centres.
Point source An ideal source of sound which has equal amplitude and phase in all directions.
Sharpness factor The width of a steered acoustic beam. Smaller values of  occur with narrow well directed beams.
Side lobe Local maxima in the far-field radiated by a transducer.
Sinusoidal excitation Elements may be driven using a wide range of excitation signals. Common ones are sine, square and pulsed excitation. A sine excitation is continuous and may be combined with a DC offset.
Steering angle The angle of a steered/focussed beam.
Steering To direct a coherent plane of acoustic energy in a given direction.
Sub-aperture A reduced number of elements of an array.
Time/phase delay A pre-calculated delay in the excitation of a single element.
Wavelength The physical length of a complete wave cycle.
Wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero.


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