Ultrasound Concave 2-D Ring Array for Retinal Stimulation

An ultrasound concave 2-D ring array transducer was designed for applications in visual stimulation of the retina with a long-term goal to restore vision in individuals with intact neurons but suffering blindness due to retinopathies. The array was synthesized and has a frequency of 20 MHz (0.075-mm wavelengths in water), 18-mm focal length (the curvature of the concave array), 1004 elements (with a pitch of 4.0 wavelengths), and inner and outer diameters of 9 and 14 mm, respectively. Wave patterns produced with the array at the focal distance were simulated. Results show that the wave patterns obtained can achieve a full-width-at-half-maximum (FWHM) resolution of 0.147 mm that is very close to the FWHM diffraction limit (0.136 mm). In addition, a scaled experiment at a lower frequency of 2.5 MHz was performed. The result is very close to those obtained with the simulations.


I. INTRODUCTION
M ANY people suffer from eye diseases, such as age-related macular degeneration (AMD) [1] and retinitis pigmentosa (RP) [2], which lead to blindness.Efforts of vision restoration for blind people have been studied for many years [3], [4], [5], [6].The methods used include include direct stimulation of the visual cortex of the brain [3] or implanting a [4] or an array of electrodes [5], [6] on retina to stimulate the retina cells that subsequently stimulate the corresponding neurons in the brain.For electrode array stimulation of the retina, optical images are converted into electrical signals and projected onto the electrode array.The electrical images are then mapped to the corresponding neurons in the brain to obtain visual perception.Retinal prostheses, such as Argus II, which is based on this principle, are now commercially available [7].Although the implant-based techniques are successful to restore vision, they are invasive since surgical procedures are needed to insert the chemical biomimetic chip or electrode array into the retina.The surgical procedures may cause medical complications and take time to recover.Also, there may be adverse reactions between the eye tissues and the implants.
To overcome the invasive nature and associated medical complications of the surgical procedure to implanted devices, such as biomimetic chip and electrode array, efforts have been made over many years to use ultrasound as an alternative method to stimulate the cells in the retina to restore vision noninvasively [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].Recently, focused ultrasound was used to stimulate the cells in the retina in vivo, and these cells subsequently stimulate the corresponding neurons in the brain of a rat [11].A 3.1-MHz ultrasound transducer of 10-mm diameter and 10-mm focal length was used to stimulate the vision center of a rat with a spatial resolution of 250 µm and temporal resolution of about 200 ms (5 Hz).The focused ultrasound stimuli were moved across the retina in a specific pattern, and a corresponding change in the pattern of the neuronal activities (electrical signals) was recorded by an electrode array placed on the visual cortex.The temperature rise was about 1.8 • C when the negative peak pressure produced by the transducer was 2.83 MPa.In addition, a helical transducer of 4.4 MHz

Highlights
• A 20-MHz ultrasound transducer was studied to produce beam patterns for vision restoration using a small number of square elements and less computation.
• Both computer simulation and a scaled experiment showed that the array can produce a high-quality beam pattern on retina.
• This study paved a way for the development of a practical ultrasound visual prosthesis for vision restoration.
(4-and 2-mm outer diameter (OD) and inner diameter (ID), respectively, with a 10-mm focal length and 1.2-mm vertical separation) was used to produce a static stimulation pattern of letter "C." Corresponding neuron activities were observed from the visual centers (including superior colliculus and the primary visual cortex) [11].Although the study is useful to demonstrate that ultrasound can be used to stimulate the retina that subsequently stimulates the brain, it does not stimulate multiple spots on the retina simultaneously.Also, the stimulation effect is slow and hence may not produce a natural vision of an entire object.
To produce a vision perception that is similar to viewing optical images of moving objects for blind people, it is desirable to produce real-time and dynamic ultrasound patterns on the retina to stimulate multiple spots simultaneously [12], [13], [14], [15], [16], [17], [18], instead of the point-bypoint stimulation.A recent simulation study has used an ultrasound transducer with a concave ring design for the stimulation of retina [18].It used a circularly segmented arrangement of array element and a matrix inversion method based on Rayleigh-Sommerfeld diffraction formula to generate ultrasound patterns on the retina.This method required a large number of transducer elements (about 3000 elements at 10 MHz) to avoid grating lobes and large amount of computation due to the matrix inversion.At 20 MHz, the number of elements is expected to be more than 12 000, complicating the imaging system.
In this article, we designed a high-frequency concave twodimensional (2-D) ultrasound ring array transducer used as a prosthesis that can be mounted on the surface of an eyeball to stimulate the cells on the retina for vision restoration [19].One of the advantages of using a ring array transducer is to avoid the heavy attenuation of ultrasound by the lens of the eye, especially at a high ultrasound frequency, and avoid the heating of the lens.Although the lenses of blind people can be surgically removed with a minimally invasive procedure since they do not have functionality, it is still invasive to remove them and the holes created by the surgery need to be filled out with proper materials to allow ultrasound to transmit through them.Also, the ring array reduces the total number of transducer elements.The concave shape of the ring array transducer allows better fitting of the array with the shape of the eyeball.By selecting a proper curvature of the array transducer, ultrasound can be focused on the retina to produce real-time images.Also, the concave shape of the 2-D ring array transducer allows a smaller number of elements to be used for the array transducer to produce images on the retina since the phase shifts needed to focus ultrasound beams are obtained through the concave shape of the array.This reduces the number of electrical connections to the array and thereby simplified the driving circuits and the imaging system.Using high-frequency ultrasound (20 MHz), the size of the array transducer can be made small enough to fit on the eyeball and a higher image resolution can be achieved.Compared to the previous study in [18], each element of the current array transducer has a square shape on a rectangular grid and is smoothly curved to form a concave surface, which makes the array easier to produce (see Fig. 1).Both computer simulations and a scaled synthetic array experiments were conducted to verify the performance of the array.Arbitrary beam patterns can be produced through the Fourier transform relationship between the waves at the surface of the array transducer and its focal plane (or between two concave surfaces).Instead of using matrix inversion [18], Fourier-based algorithm [20] was used to produce the beam patterns.This reduces computation since the Fourier transform can be implemented with the fast Fourier transform (FFT).
This article is organized as follows.A brief theory for the concave ultrasound ring array transducer is presented in Section II.Then, computer simulations and a scaled experiment are given in Section III.The results of the simulations and the experiment are shown in Section IV.Finally, discussion and conclusion are given in Sections V and VI, respectively.

II. THEORETICAL PRELIMINARIES
A brief theoretical description of the method is given in the following [20].The isotropic/homogeneous wave equation is given by [ Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where 2 is the Laplacian, (⃗ r , t) is the acoustic pressure, ⃗ r = (x, y, z) is a point in the space, t is the time, and c is the speed of sound of the medium or the speed of light in vacuum.Using Fourier transform relationship, a solution to the wave equation can be expressed as where ˜ (⃗ r 0 ; ω) is the Fourier transform of (⃗ r 0 , t) in terms of time, ⃗ r 0 = (x 0 , y 0 , z) is a spatial point where ultrasound field is produced, ω = 2π f is the angular frequency, and f is the frequency.
The following limited-diffraction array beam [22], [23], [24], [25] uv Array (⃗ r 0 , t) is a solution to the wave equation in (1): where the subscript "Array" means array beam, u and v are integers, k = ω/c is the wavenumber, A(k) is the electromechanical transfer function of a transducer, D u,v (ω) are the complex coefficients, H (k) is the Heaviside step function where k x u and k y v are the x and y components of the wave vector, respectively, and x u + k 2 y v for propagating waves and k 2 x u + k 2 y v > k 2 for evanescent waves.From (3), one obtains the spectrum of uv Array (⃗ r 0 , t) as follows: Assuming that the system is linear, the total field at ⃗ r 0 is a linear superposition of those produced by (4) (assuming that the summation exists) Now, let us determine the coefficients, D u,v (ω), for a 2-D array of a given aperture weighting function.After getting D u,v (ω), the ultrasound field for ultrasound stimulation of retina can be obtained from (5).Assuming that a planar 2-D array transducer is located at the plane z = 0 and the field produced at the transducer surface is given by where Q(⃗ r 1 ; ω) is the field at the surface of the transducer, ⃗ r 1 = (x 1 , y 1 , 0) is a point at the surface, and w x and w y are the half widths of the aperture size of the array transducer along the x 1 -and y 1 -axes, respectively.The array transducer is assumed to be surrounded by a rectangular frame with zero field amplitude, where R x and R y are the half widths of the outer frame.In addition, the aperture weighting pattern in ( 6) is periodically repeated outside of the frame bounded by with periods of 2R x and 2R y in the x 1 -and y 1 -directions, respectively.Apparently, as both R x → ∞ and R y → ∞, (6) represents a single array without a spatial repetition.Since ( 6) is a periodic function, it can be expanded as a Fourier series [26] with periods of 2R x and 2R y along the x 1and y 1 -axes, respectively.Letting z = 0 in (5), we obtain such a series x 1 e ik yv y 1 (7) where k x u = uπ/R x and k y v = vπ/R y , and u, v = ±1, ±2, ±3, . . .The coefficients of the Fourier series in (7) are given by [26] A Assuming that an array transducer consists of M × N rectangular elements and the spatially quantized driving or aperture weighting function of an element centered at and M and N are the integers, ( 8) can be precalculated and stored as known coefficients as follows: where w x m and w y n are the half widths of the transducer element centered at (x 1 m , y 1 n ) along the x 1 -and y 1 -axes, respectively, and sin c(•) is the sinc function.Using (9), the 2-D Fourier series in (5) can be evaluated with an inverse FFT or IFFT [27].
Notice that the aperture weighting function ˜ 1 (⃗ r 1 ; ω) in ( 8) can be approximately written as 2  1 } that is due to the curvature of the spherically curved concave ring array transducer with a focal distance z = F, where F is the focal length and r 1 = (x 2 1 + y 2 1 ) 1/2 .According to Goodman [28], ˜ 1 (⃗ r 1 ; ω) is approximately related to the ultrasound image ˜ (x 0 , y 0 , z = F; ω) at the focal distance F (at the retina of the eyes) of the transducer through the Fresnel diffraction integral, or ˜ ′ 1 (⃗ r 1 ; ω) can be written as the following spatial 2-D Fourier transform F x 1 ,y 1 {•} in terms of x 1 and y 1 (see (28) of [29]): Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where λ is the wavelength, k x 0 = kx 0 /F, and k y 0 = ky 0 /F.If the wave source has a dimension D × D, the integration limits of the Fourier transform will be from −a to a, where D = 2a.Assuming that the optical image formed at the retina of the eyes of blind people can be approximated with an ultrasound image ˜ (x 0 , y 0 , z = F; ω) that is used for a visual stimulation of the brain, the driving function of the concave 2-D ring array transducer can be calculated by the inverse Fourier transform of (11) as follows (see (38) of [29]): × e i(k x 0 x 1 +k y 0 y 1 ) dk x 0 dk y 0 .
If the images are formed on a spherical surface, such as the retina with a radius of curvature of F, the phase term exp 12) should be removed when doing the inverse Fourier transform.
After getting ˜ ′ 1 (⃗ r 1 ; ω) from ( 12), one can get ˜ 1 (⃗ r 1 ; ω) using (10).Inserting ˜ 1 (⃗ r 1 ; ω) into (8) and using (9), one obtains D u,v (ω).Ultrasound images can then be obtained with (5) to stimulate the retina (note that usually a sine wave with many cycles (such as a 3.5-MHz sine wave that last for about 200 ms in [11]) is needed in ultrasound visual stimulation of the retina, and thus, A(k) is approximately a delta function with a fixed k = ω/c or angular frequency ω due to a narrow bandwidth of the driving signal, and thus, (5) instead of (2) can be used to calculate the ultrasound field).

A. Computer Simulation
Based on the above theory, an ultrasound concave 2-D ring array transducer suitable for the study of larger animals, such as rabbits, was designed, which is capable of producing a high-resolution image on retina for visual stimulation of the brain.The curvature of the array is the same as the focal length of the array and is given by F = F 1 = 18 mm.The center frequency of the array is f = f 1 = 20 MHz.The ID of the array is set to 0, 6, and 9 mm to investigate how image quality is affected by these changes, and the OD of the array is D = D 1 = 14 mm.Assuming that the medium where the ultrasound propagates is similar to water that has a speed of sound of about c = 1500 m/s, the wavelength of the ultrasound is λ = λ 1 = 0.075 mm.A summary of the parameters of the concave 2-D ring array transducer is given in Table I.All simulations were performed using the C programming language in a Linux operating system.In the simulations, the grid size was about 0.029 mm for aperture quantization.
To evaluate the performance of the array, computer simulations were conducted.In the simulations, a digital phantom consisting of 30 point objects with a height h = h 1 = 2.5 mm between the top-most and bottom-most rows of point objects was assumed (see the "Desired Pattern" on the upper-left corner of Fig. 2).For people with healthy eyes, an optical image of the digital phantom formed on the retina will vision.For blind people, the digital phantom has to be converted to an ultrasound beam pattern on the retina to stimulate cells that subsequently stimulate the neurons in the brain for vision perception.To produce an ultrasound image on the retina, an inverse spatial Fourier transform [26] of the digital phantom ˜ (x 0 , y 0 , z = F; ω) was performed using (12) to get ˜ ′ 1 (⃗ r 1 ; ω) (implemented with an IFFT [27]) based on the Fresnel approximation [28], [29].Then, ultrasound image on the retina was obtained from (5) using the relationship in ( 9) and ( 10) [20], [29], where w x m and w y n in (9) were the half width of the pitch (distance between the centers of two adjacent elements) of the concave 2-D ring array transducer and were assumed to be equal for all array elements.

B. Experiment
In addition to the simulations, an experiment was performed [30].In a linear system, the total ultrasound field produced by a 2-D array transducer can be viewed as a summation of the field produced by each individual array element.Also, the field of each element is a superposition of the fields produced by point sources within the element and each point source produces a spherical wave.In practice, it is not possible to get a point source.However, if the size of a transducer element is small as compared to the wavelength and the distance from the element is much larger than the size of the element, the field produced by the element will be close to a spherical wave.Combining the fields of such small elements, the field of a larger element of a 2-D array transducer can be produced.Thus, an experiment that measures the spherical wave can be performed to produce the field of a concave 2-D ring array transducer to obtain images on the retina for visual stimulation of the brain.
In the experiment, the center element of an existing 10-ring and 50-mm-diameter Bessel annular array transducer that was used to produce limited-diffraction beams was used to produce a spherical wave [31], [32], [33].The transducer was made of 1-3 ceramic/polymer composite, had a center frequency of f = f 2 = 2.5 MHz, and was broadband with its −6-dB fractional bandwidth of about 81% of the center frequency.The diameter of the center element of the annular array was about 4 mm.To reduce the size of the center element to produce a good spherical wave, a 1-mm-thick foam tape (3M Microfoam, 3M Commercial Office Supply Division, St. Paul, MN) with a 1-mm-diameter hole was attached in the front Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 2. Ultrasound beam pattern of a digital phantom (see the "Desired Pattern" on the upper-left corner) produced at the focal distance with a concave 2-D ring array transducer of 20-MHz frequency (0.075-mm wavelengths in water at a speed of sound of about 1500 m/s), 18-mm focal length, 14-mm OD, and a diffraction limit of about 0.12 mm (the radius of the Airy pattern, see (37) of [29]).The height between the top-most and bottom-most point objects in the digital phantom is 2.5 mm and the size of each point object is about 0.03 mm (smaller than the wavelength of 0.075 mm).There are 30 point objects in the digital phantom.For panels (a)-(i), from the top to the bottom rows, the pitches of the 2-D array are 1.5 (7137 elements when ID = 9 mm), 4.0 (1004 elements when ID = 9 mm), and 5.6 (512 elements when ID = 9 mm) wavelengths, respectively.From the left to the right columns, the IDs of the ring array are 0, 6, and 9 mm, respectively.It is seen from the images in the bottom row that artifacts are produced due to aliasing caused by a large size of the array elements (inadequate spatial sampling rate).The dimensions of each image are shown in the figure, and the grayscale bar represents the normalized magnitude of ultrasound pressure.(Modified from [19] with permission.) of the center element and aligned with the center of the element [30].
The center element of the transducer was driven by a 1.5-cycle and 2.5-MHz sine-wave pulse, and the acoustic pressure of the pulse was measured at distance z = 100 mm with a 0.5-mm-diameter broadband PVDF needle hydrophone (NTR 1000, NTR System, Inc., Seattle, WA) in water [30].The signal measured was amplified by a homemade preamplifier and digitized with an 8-bit analog-to-digital (A/D) converter at a sampling rate of 40 MS/s for 300 samples [30].Since the effective size of the center element was about 1.7 wavelengths at 2.5 MHz (the wavelength was about λ = λ 2 = 0.6 mm at the speed of sound of about c = 1500 m/s in water), in the far-field at z = 100 mm, a good approximation of the spherical wave was obtained.The hydrophone was scanned in the direction that was perpendicular to the z-axis (i.e., the axis from the point source on the transducer surface toward the starting point of the hydrophone) over a distance of 50 mm (starting from the axis), with a step size of 0.2 mm for 250 steps.At each step, the transducer was excited and 300 samples of the signal mentioned above were acquired after a fixed delay time and stored on a hard disk.This process was repeated until data were acquired from all steps.
For ultrasound visual stimulation, a long tone burst with many cycles is needed to deliver enough power to the cells on the retina.In this case, the signal has a very narrow bandwidth and it can be approximately viewed as containing a single frequency.To get single-frequency (also called continuous wave or CW) signals, the broadband signals measured from the experiment above were Fourier transformed first, and then, the 2.5-MHz signals were extracted for all 250 scanning steps.Since the spherical wave is axially symmetric, the ultrasound wave on a plane that is perpendicular to the z-axis will also be axially symmetric.Thus, to get the signals on 2-D rectangular grids with 0.2-mm pitch over a planar surface of 100 mm diameter, the data from the 250 steps were rotated around the z-axis using one-dimensional (1-D) interpolations along the radial direction (the rotation angle was continuous).
Because of the reciprocal principle, the pulse field measured above is the same as that when ultrasound is transmitted from each of the positions of the hydrophone across the planar 2-D array surface and the signal is measured at the fixed position of the point transmitter.This allows to weight the measured signals in the area between the inner and outer rings of the 2-D grids with the function ˜ 1 (⃗ r 1 ; ω) obtained from ( 12) and (10) to form an effective 2-D ring array transducer to produce an ultrasound image of the object (digital phantom) at the focal distance F = F 2 = 100 mm.
Since the synthesized low-frequency (2.5 MHz) ultrasound ring array transducer with 100-mm focal length and D = D 2 = 46.67-mmOD was used in the experiment, a proper scaling of the digital phantom is needed, so that the result of the experiment is comparable to that obtained with the highfrequency (20 MHz) ring array transducer of a smaller focal length of 18 mm and a smaller OD of 14 mm.Because image resolution (a higher resolution means a smaller beamwidth) at focus is inversely proportional to the wavelength and the f -number (focal length divided by the diameter of the aperture or the OD of the ring array transducer) (see (37) of [29]), the scaling factor s of the digital object can be calculated with the following formula: where  12) to produce ultrasound images at the focal distance F 2 .All experimental data were processed with the C programming language in a Linux operating system.

A. Computer Simulation
The results of the computer simulation [20] are given in Fig. 2, where the image dimensions are given in the figure .As mentioned before, the digital phantom (30-point object) is at the upper-left corner of the figure and is labeled as "Desired Pattern."This is an ultrasound pattern to be projected onto the retina.The height between the top-most and the bottom-most point objects of the digital phantom is 2.5 mm.Fig. 2(a)-(i) shows the simulated [20] images of the digital phantom obtained with the 20-MHz concave 2-D ring array transducer (see Table I except that the ID is varied for comparisons) at its focal distance of 18 mm.Images in the top, middle, and bottom rows of Fig. 2(a)-(i) were obtained with the 2-D ring array transducer of pitches of 1.5λ 1 , 4.0λ 1 , and 5.6λ 1 , corresponding to the number of array elements of N = 7131, 1004, and 512 when the ID was 9 mm.From the left to right columns of Fig. 2(a)-(i), images were obtained with an ID of 0, 6, and 9 mm, respectively.It is clear from the results that as the number of transducer elements increases, image quality increases (i.e., it has less artifacts and a higher sound pressure near the edges).Also, as the ID of the concave 2-D array transducer increases, the image quality decreases.At the 5.6λ 1 pitch, aliasing artifacts due to an inadequate spatial sampling rate of the drive signals can be clearly seen in the images.To exclude the lens of the eye, it is necessary to have a large enough ID (9 mm for rabbits) since ultrasound attenuation of the lens is high.Thus, as a compromise, the image in Fig. 2(f) with a pitch of 4.0λ 1 and an ID of 9 mm was used to determine the parameters of the concave 2-D ring array transducer (see Table I).
For quantitative comparison of resolution and sidelobes of the ultrasound images obtained by the 20-MHz concave 2-D ring array transducer, the line plots of the normalized magnitude of ultrasound pressure through the center of the bottom five point objects of the images in Fig. 2(d  Fig. 2(f) (ID = 9 mm, dashed line) were obtained and shown in Fig. 3.The full-width-at-half-maximum (FWHM) value of the line plot of the center point object when ID = 0 mm [Fig.2(d)] is about 0.147 mm, which is very close to the theoretical value of the FWHM of 0.141λ 1 F 1 /D 1 = 0.136 mm calculated with (36) of [29].It is clear from the plots that as the ID value increases (more central elements are removed), the sidelobes of the images will also increase, although the resolution is increased slightly with larger IDs.Higher sidelobes reduce image contrast and thus lower the image quality.
Fig. 4(a) and (b) shows the magnitude and phase of the driving function for each element of the concave 2-D ring array transducer to produce the image in Fig. 2(f) (pitch = 4.0λ 1 , N = 1004, and ID = 9 mm).It is clear that with the concave design of the array transducer, the phase change is relatively smooth across the array surface, which allows a bigger pitch (4.0λ 1 ), and thus, a smaller number of elements (1004) to be used for getting a reasonably good image.Without the concave design, electronic focusing is required, and thus, a smaller size of the array elements is needed to avoid spatial aliasing, increasing the number of elements and thus the complexity of the imaging system.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(13.33 times larger than that in Fig. 2).The image produced with a computer simulation using the same parameters as those of the experiment is shown in (b).These images are comparable to that in Fig. 2(f) obtained at a higher frequency and the same number of elements (1004).However, the pitch of the array used here is 1.655 instead of 4.0 wavelengths to maintain the same number of array elements (notice that the wavelength here is eight times larger than that used in Fig. 2).The dimensions of the images and the grayscale bar are given in the figure .B. Experiment Fig. 5(a) is the result of the scaled experiment as explained in Section III-B.The resolution of the image was lowered in Fig. 5 since f − number was increased from 1.286 to 2.143, and the frequency was reduced from 20 to 2.5 MHz.Thus, the size of the digital phantom was increased by a scaling factor of s = 13.33 [see (13)] (the height between the top-most and the bottom-most point objects was increased from 2.5 to 33.33 mm) to make the images in Fig. 5 comparable with the image in Fig. 2(f) that was obtained with the 20-MHz transducer.However, since a larger f − number was used in Fig. 5, the image is magnified more relative to the transducer aperture, and thus, more distortions can be seen around the edges of the images.Also, due to the size of the point source (1 mm in diameter or about 1.7 wavelengths) used in the experiment, the ultrasound field produced was not a perfect spherical wave.This causes additional distortions around the edges of the images.To be comparable with Fig. 2(f), the same number of elements (N = 1004) of the 2-D ring array (with an OD of 46.67 mm and an ID of 30 mm) was used in Fig. 5, which gives a pitch of 1.655λ 2 .
As a comparison, a computer simulation [20] using the scaled digital phantom with parameters corresponding to those used in the experiment was performed.The result is shown in Fig. 5(b).It is clear that the simulation data are very close to that of the experiment except near the edges of the image.The better quality of the image around the edges in Fig. 5(b) is due to a smaller point source (about 0.098 mm diameter) that was used, and thus, a better spherical wave was produced.
Based on a comparison of Fig. 5(a) and (b) with Fig. 2(f), it is clear that the quality of images is very similar near the center of the images.The outer edges of the images in Fig. 5 have a lower quality due to the reasons explained above.

V. DISCUSSION
The concave 2-D ring array transducer studied in this article can be used to produce images to stimulate multiple points on the retina simultaneously.This is a natural way of stimulation as opposed to the point-by-point method since the optical image of the entire object is formed on the retina to produce vision as people seeing an object.However, the use of images to stimulate the cells on the retina has the advantage that a dynamic vision can be perceived by the blind people.To achieve this, the images of a moving object can be captured by a small video camera mounted on a frame of a wearable device that is similar to the eyeglasses, and then, the optical images are Fourier transformed with (12) to produce electrical drive signals to produce ultrasound images on the retina.The blind people can then "see" the moving object through the visual stimulation of their brain areas.
In terms of image quality, as is seen in Fig. 2, the best ultrasound image is obtained when the ID of the 2-D ring array transducer is zero.This is expected since lower sidelobes of point objects can be obtained when the entire 2-D array surface is utilized.However, as mentioned before, a large enough ID is needed to avoid the lens area that has a high ultrasound attenuation, which can cause heating of the lens material and distort the ultrasound beams.In this study, we chose ID = 9 mm for experiments on larger animals, such as rabbits.Also, a larger ID value will reduce the number of array elements.
The concave shape of the 2-D ring array transducer is important for two reasons.One is to have a better fit of the array with the geometry of the eyeballs.The second is to focus the ultrasound beam, which removes the need of electronic focusing and thus allows a larger size of the array elements to be used without causing severe aliasing artifacts.From the images in Fig. 2, it is clear that a pitch that is as large as 4.0λ 1 (a total number of elements of about N = 1004 when ID = 9 mm) still produces good images.A smaller number of elements for the 2-D ring array transducer will reduce the complexity of the transducer and the number of interconnections and thus simplify the drive electronics of the imaging system.
Since the Fourier transform relationship in ( 12) is based on the Fresnel approximation [28], i.e., the size of images cannot be too large in order to satisfy the paraxial condition, larger distortions are expected near the edges of larger images produced.
The images in Fig. 2 were obtained on a flat surface.However, an actual retina is on a curved surface.In this case, the phase term exp[−ikr 2 0 /(2F)] in ( 12) can be partially removed (if the curvature of the retina surface is equal to F, the phase term can be completely removed).Also, since the surface of the retina is curved, the distance from the center of the transducer to any points on the retina will be closer to the geometrical focal distance of the transducer, and thus, the quality of the images near the edges will be better than those obtained in Fig. 2 [29].
It is worth noting that another way of producing images on the retina is to use limited-diffraction beams, such as Bessel beams [31] and X wave [32], [33] (also see (25) and ( 26) of [29]).Ultrasound patterns can be designed and the patterns can stay in focus over a large depth of field using the method in [34].It is also possible that a zeroth-order Bessel beam (n = 0 in (25) of [29]) can be shifted and then coherently added to form an arbitrary beam pattern.However, because limited-diffraction beams have larger sidelobes than conventional focused beams at their focuses, the quality of images obtained with the limited-diffraction beams may be lower as compared to the method developed in this article.Also, the large ID of the 2-D ring array and the concave shape of the array will distort the limited-diffraction beams.

VI. CONCLUSION
A high-frequency (20 MHz) ultrasound concave 2-D ring array transducer was designed for the stimulation of retinal cells to subsequently stimulate the corresponding neurons in the visual cortex of the brain for vision perception in blind people.The high-frequency ultrasound produces a high image resolution and allows the transducer to be made small enough to fit the human eyeballs.The ring array has a large enough ID (9 mm) to avoid the lens area, where the ultrasound attenuation is high and thus may cause heating of the lens material and distortions of the beam when conducting experiments in large animals, such as rabbits.Although the lens can be removed by minimally invasive surgery, the holes created by the surgery need to be filled out with proper materials that allow ultrasound to penetrate without causing beam distortions and/or heating.The concave shape of the array will easily fit with the geometrical shape of the eyeball and allows the array to have smaller number of elements but still producing good images without aliasing artifacts caused by an inadequate spatial sampling rate.The smaller number of elements will reduce the complexity of the array transducer and the number of interconnections and simplify the driving electronics of the imaging system.Both computer simulations and a scaled experiment were performed, and the results show that good ultrasound images can be produced with this concave 2-D ring array transducer for the restoration of vision in blind people.

Fig. 1 .
Fig. 1.Concave ultrasound ring array for visual stimulation of retina.Each element of the array has a square shape on a rectangular grid but is smoothly curved to form a concave surface with a curvature of 18 mm (focal length F ).The ID can be changed for different studies.The frequency of the array is 20 MHz.
143 and f − number1 = F 1 /D 1 = 1.286 are the f − number of the 2.5-and 20-MHz ring array transducers, respectively, and λ 2 = 0.6 mm and λ 1 = 0.075 mm are their respective wavelengths.This gives s = 13.33, and thus, the height of the scaled digital phantom is given by h = h 2 = sh 1 = 33.33 mm.Using the scaled digital phantom and taking into consideration of the focal length of F 2 = 100 mm and the diameter of the outer ring D 2 = 46.67 mm, the weighting function of the concave 2-D ring array transducer can be obtained from ( ) (ID = 0 mm, solid line), Fig. 2(e) (ID = 6 mm, dotted line), and

Fig. 3 .
Fig. 3. Line plots through the center of the bottom five point objects of the three images obtained with the 20-MHz concave 2-D ring array transducer of IDs of 0 (solid line), 6 (dotted line), and 9 mm (dashed line) in the middle row (4.0-wavelength pitch) of Fig. 2. Vertical axis represents the normalized magnitude of the ultrasound pressure.It is clear that as the ID is increased, the sidelobes of the point objects in the images also increase.(Modified from [19] with permission.)

Fig. 5 .
Fig. 5. (a) Ultrasound image experimentally produced with a concave 2-D ring array transducer of a lower frequency of 2.5 MHz, 100-mm focal length, 46.67-mm OD, and 30-mm ID using a scaled digital phantom (13.33 times larger than that in Fig.2).The image produced with a computer simulation using the same parameters as those of the experiment is shown in (b).These images are comparable to that in Fig.2(f) obtained at a higher frequency and the same number of elements (1004).However, the pitch of the array used here is 1.655 instead of 4.0 wavelengths to maintain the same number of array elements (notice that the wavelength here is eight times larger than that used in Fig.2).The dimensions of the images and the grayscale bar are given in the figure.

TABLE I PARAMETERS
OF THE 20-MHZ CONCAVE 2-D RING ARRAY TRANSDUCER