Long-range frequency-domain optical delay line based on a spinning tilted mirror for low-cost ocular biometry

Optical biometers are routinely used to measure intraocular distances in ophthalmic applications such as cataract surgery planning or myopia monitoring. However, due to their high cost and reduced transportability, access to them for screening and surgical planning is still limited in low-resource and remote settings. To increase patients’ access to optical biometry we propose a novel low-cost frequency-domain optical delay line (FD-ODL) based on an inexpensive stepper motor spinning a tilted mirror, for integration into a time-domain (TD)-biometer, amenable to a compact footprint. In the proposed FD-ODL, the axial scan range and the A-scan rate are decoupled from one another, as the former only depends on the spinning mirror tilt angle, while the A-scan rate only depends on the motor shaft rotational speed. We characterized the scanning performance and specifications for two spinning mirror tilt angles, and compared them to those of the standard, more expensive FD-ODL implementation, employing a galvanometric scanner for group delay generation. A prototype of the low-cost FD-ODL with a 1.5 deg tilt angle, resulting in an axial scan range of 6.61 mm and an A-scan rate of 10 Hz was experimentally implemented and integrated in a dual sample beam optical low-coherence reflectometry (OLCR) setup with a detour unit to replicate the measurement window around the anterior segment and the retina. The intraocular distances of a model eye were measured with the proposed low-cost biometer and found to be in good agreement with those acquired by a custom swept-source optical coherence tomography (SS-OCT) system and two commercial biometers, validating our novel design.


Introduction
Ocular biometry [1] consists of the measurement of the intraocular distances of an eye, such as the eye axpuial length (AL), i.e., the distance between the posterior corneal surface and the retinal pigment epithelium [2]; the anterior chamber depth (ACD), i.e., the distance between the posterior corneal surface and the anterior crystalline lens surface; the central corneal thickness (CCT) and keratometry, i.e., the anterior and posterior corneal radii (K1 and K2), among others.Ocular biometry is an indispensable step in cataract surgery planning, as it allows to select the optical power of the intraocular lens (IOL) to be implanted, via an appropriate IOL power calculation formula [3][4][5].In order to accomplish target refraction for the patient, at least AL, ACD and K1 need to be accurately measured [6].Axial length is also the optimal biomarker for myopia development and monitoring of myopia control interventions [7].
Two main types of instruments can be used for ocular biometry: ultrasound biometers and optical biometers.For most of the aforementioned intraocular distances, optical biometers use low-coherence interferometry [8] as their working principle.This is the same working principle that Optical Coherence Tomography (OCT) [9] is based on.Optical biometers outperform ultrasound-based biometers due to their higher accuracy [10], and their non-contact nature, more comfortable for patients compared to applanation or immersion ultrasound biometers.Since their inception, two main categories of optical biometers have been developed [2,11]: time-domain (TD) low-coherence interferometry systems that produce a 1-D axial eye scan along the visual axis and, more recently, Fourier-domain FD-OCT systems that allow for faster 2-D cross-sectional eye scans, owing to axial scan speeds orders of magnitude larger than those of TD systems.
TD systems can be classified, depending on the interferometer configuration, in two main categories: partial coherence interferometry (PCI) [12][13][14] and optical low-coherence reflectometry (OLCR) [15][16][17].In PCI, two copies of the input beam with a variable delay, introduced by an optical delay line (ODL), are directed to the eye in a common-path interferometer with the tear film acting as a partial reference reflector and the remaining eye surfaces as the sample [18].In OLCR instead, a dual-arm interferometer is used, generally in a Michelson configuration where the eye is in the sample arm and the reference arm comprises an ODL [19].
Commercial and research TD optical biometers include various types of ODLs.The majority employ bulky mechanically scanning components, such as retroreflectors mounted on a linear translation stage actuated by a stepper motor with leadscrew [13,20], as the Zeiss IOL Master 500 [21].Others employ a 25-30 mm rotating cube [22,23], as the Haag-Streit Lenstar LS 900 [24].In terms of axial speeds and scan ranges, the speed of the leadscrew mechanism can reach up to 70 mm/s, resulting in full eye biometry scan at frequencies below 2 axial scans per second, while a cube rotation frequency of 3.2 Hz results in 12.8 axial scans per second.The axial scan range in the latter case is approximately 10 mm, therefore not large enough to cover the full axial length of the eye.However, a detour unit [23] based on two polarizing beam splitters, allows to replicate the 10 mm scan window at a distance corresponding to roughly 23 mm of ocular media from each other to cover both the anterior and posterior segments.The corresponding axial scanning speed considering both segment is therefore approximately 420 mm/s.This speed is ultimately limited by the size of the rotating cube and the rotational frequency that can be imparted to it.Greater speeds are required to minimize patient's motion artifacts, affecting the measurement precision, especially in OLCR.
Cost and transportability are factors of paramount importance in population screening for myopia, as well as in remote and low-resource settings, where, in many cases, the prevalence of cataracts is higher [35] and accessibility to ocular biometry is all the more important and yet currently limited [36,37].Expensive (FD)-OCT biometers are a financial burden for the ophthalmic or optometry practice/doctors, while TD biometers using bulky delay lines with slow scan frequency are unsuitable for portable or handheld devices.
Voice coil-based ODLs have been proposed as a compact and lower cost alternative that can provide fast axial scanning, either through a multi-pass geometry [38], or multi-reference OCT [39] for scan ranges up to 3.1 mm, and, for biometry, in combination with methods to reference and compensate eye axial motion during acquisitions [40].
Alternatively, diffraction grating-based frequency-domain optical delay lines (FD-ODL), also known as rapid scanning optical delay (RSOD) lines [41][42][43] have been used in TD-OCT mainly for rapid axial scanning up to a few kHz, as, to generate an axial scan, they only involve a tilt of a small mirror back and forth over a short angular range.Other uses include hardware-based dispersion imbalance compensation in OCT [44][45][46], fast depth tracking and adaptive ranging [47][48][49][50], and complex conjugate artifact removal in FD-OCT [51][52][53][54][55].In FD-ODL the mirror counter-rotation is typically actuated by a galvanometric scanner, for which a PID servo controller and the associated electronics including a power supply capable of supplying high currents are needed.Alternatives for the actuator include the use of a polygon scanner in place of the galvanometer [56], and for the optical components, all-reflective [57], transmissive [58], and double grating [59,60] optical designs.
However, FD-ODLs have also been used for long range scanning [61] in anatomical OCT (aOCT), scanning up to 26 mm for +/-3.6°galvanometertilt at 100 Hz (subsequently upgraded to 500 Hz), for measuring the size and shape of the lumen of the human upper [62] and lower airways [63] with an endoscopic probe.Thus, a FD-ODL would also be suited for robust long-range scanning in ocular biometry.However, the high cost of a galvanometric mirror scanner [64] makes its use impractical for low-cost applications.
In this work, we propose a novel low-cost FD-ODL employing a small mirror and an inexpensive stepper motor instead of a galvanometric scanner.In this case, the direction normal to the small mirror surface is set at a constant tilt angle to the stepper motor shaft axis so that the angle projection on the X axis (see Fig. 1) varies similarly to the variable tilt angle of a mirror in a galvanometric scanner.This way, the proposed novel low-cost FD-ODL consists of effectively spinning a slightly tilted mirror.The scan range and frequency are independently set by the mirror tilt angle and motor speed respectively, unlike in leadscrew-based linear mirror translation ODLs or galvanometer mirror-based FD-ODLs.We validate its use in a novel OLCR TD optical biometer prototype, capable of scanning long depth ranges, and compare its biometry readings for a model eye with those of commercial and custom built biometers.

Working principle of the proposed FD-ODL
In this work, we present a novel, low-cost FD-ODL enabled by converting the alternate counterrotating motion needed for the standard galvanometer-based FD-ODL into a continuously spinning motion of a tilted mirror, driven by an inexpensive stepper motor.The main optical components of the proposed FD-ODL are a fiber collimator, a diffraction grating (DG), an achromatic doublet lens (L), a spinning tilted mirror (TM) attached to the shaft of a stepper motor (SM) and a perforated double-pass mirror (DPM).The optical group delay of a light beam through the proposed optical delay line varies during the rotation of the TM.A full TM rotation corresponds to a full forward and backward group delay scan.
A schematic representation of the proposed optical delay line is presented in Fig. 1.The FD-ODL is in a double-pass configuration that can be described in the following steps.A collimated broadband input light beam from a fiber collimator propagates through the central hole of the perforated double-pass mirror and is incident on the diffraction grating at an angle β to the normal of the diffraction grating (see step 1 in Fig. 1).The angle β is chosen such that the angle γ between the normal to the diffraction grating and the central wavelength component of the diffracted beam is zero, hence, the diffracted beam at the central wavelength propagates on-axis with the optical axis of the FD-ODL (see yellow beamlet after step 1 in Fig. 1).Unlike in the standard FD-ODL [62], the achromatic doublet lens L is centered on-axis with the optical axis of the FD-ODL and placed at a focal length f distance from the diffraction grating on one side, and from the spinning tilted mirror on the other side.This way, the lens L both focusses each wavelength component of the diffracted beam in close proximity of the tilted mirror TM surface and redirects the chief rays of each wavelength component parallel among themselves, creating a horizontally oblong spot on the TM (see step 2 in Fig. 1).The TM is tilted such that the normal to the mirror surface forms an angle α with the stepper motor shaft axis, which, in turn, it is aligned with the optical axis of the delay line.Therefore, the light impinging on the TM is reflected at an angle 2α (optical angle) from the optical axis.The reflected light passes again, this time off-axis, through the lens L, which collimates all wavelength components and redirects them to the same spot on the diffraction grating at a distance f tan(2α) from the original spot.Thus, light gets diffracted again into a collimated broadband beam parallel to the input light beam (see step 3 in Fig. 1).The beam propagates until the perforated double-pass mirror DPM, where it is reflected on its path (see step 4 in Fig. 1), and back-traces all previous steps until it couples back into the collimator.As the tilted mirror is rotated by the stepper motor about the FD-ODL optical axis (Z-axis in Fig. 1), the tilt angle α is maintained constant.This results in the reflected light tracing a circle on the lens L and on the diffraction grating plane during the rotation of the TM (see step 5 in Fig. 1).The circle radius is f tan(2α).However, after diffraction at an angle β in step 3 of Fig. 1, the circle becomes an ellipse on the perforated double-pass mirror, with the major axis (equal to the circle diameter) on the vertical direction, and the minor axis on the horizontal direction.We can also observe that the optical path length (OPL) inside the FD-ODL varies as the TM is rotated.Interestingly, the only direction of the light beam displacement that induces a difference in OPL is the one orthogonal to the diffraction grating grooves orientation, hence the horizontal direction, i.e., the X-axis in Fig. 1.The variation of the optical pathlength (∆OPL) after the double-pass as a function of the rotational angle θ of the stepper motor shaft follows the projection of the beam position on the X-axis at the diffraction grating, i.e., [1 − cos(θ(t))].Equation (1) puts it all together, by expressing ∆OPL as a function of time and all the aforementioned angles: From Eq. (1), we can notice that the maximum ∆OPL that can be induced is proportional to the diameter of the circle traced by the beam at the diffraction grating and that a complete rotation of the stepper motor shaft performs two scans of the group delay (over the full axial range), in opposite directions.Equation (2) develops Eq. ( 1) by expressing the incidence angle β on the diffraction grating as a function of the grating pitch p and the central wavelength λ o , as: From Eq. ( 2) we can observe that ∆OPL increases with longer central wavelengths λ o , shorter grating pitches p, longer focal lengths f, and larger tilt angle α of the spinning titled mirror.To maintain a small footprint, it is advisable to keep the focal length below 100 mm.Increasing the tilt angle α increase the radius of the circle traced on the lens L surface, therefore, the clear aperture of the lens L is ultimately a constraint to the maximum axial scan range of the FD-ODL.
As for a standard FD-ODL, the heterodyne modulation frequency f FD−ODL , i.e., the fringe carrier frequency, can be induced by using an off-pivot configuration, where x 0 is the offset between the rotation axis of the spinning TM, i.e., the stepper motor shaft axis, and the optical axis of the FD-ODL.The heterodyne modulation frequency, in the double-pass, is determined by the temporal rate of change of the phase delay ϕ(λ 0 , t) [41], i.e., where ]/∂t, is the angular velocity about the Z axis of the projection of the optical angle 2α of the tilted mirror on the X-Z plane, which, unlike in the standard FD-ODL, it is not constant in time (see Visualization 1, right side).Considering that the stepper motor rotation angle θ(t) varies as θ(t) = 2πf SM t, where f SM is the rotational frequency of the stepper motor, Γ Z ′ (t) = 4παf SM • sin(2πf SM t).Hence, Therefore, by increasing the offset x 0 one increases the maximum fringe frequency, as in the off-pivot configuration in a standard FD-ODL [43].However, the spinning tilted mirror variable angular velocity projection on the X-Z plane implies that the fringe carrier frequency varies sinusoidally across the axial scan range.
Moreover, as for a standard FD-ODL, group velocity dispersion can be controlled by modifying the distance between the diffraction grating and the lens L to compensate the potential dispersion imbalance with the sample arm and improve the axial resolution [45].
Visualization 1 shows a side-by-side comparison between a standard FD-ODL and the proposed FD-ODL based on a tilted mirror spun by an inexpensive stepper motor.The two designs lead to the generation of equal optical delays at the same frequency.However, the most striking difference is the need for alternating rotation in the galvanometer-based mirror rotation around the Y-axis for the standard FD-ODL versus the continuous rotation in the stepper motor-based titled mirror rotation around the Z-axis.As continuous rotation does not require the complex control electronics of a servo system nor the high currents needed for the start-stop motion, a much cheaper mirror actuator can be used in the proposed FD-ODL: a simple low-cost stepper motor.This is represented also in Fig. 2, where Figs.2(a), 2(c) show the variation of the angles θ and α with time for the standard galvanometer-based FD-ODL and the proposed spinning tilted mirror-based FD-ODL, respectively.Similarly, Figs.2(b), 2(d) show their respective actuator speed with time, where the standard FD-ODL requires regular abrupt changes of direction and speed, while the proposed FD-ODL requires simply a constant spin to provide the same duty cycle and axial scan range as that generated by the standard FD-ODL, if the TM is titled by the corresponding peak α.

FD-ODL system components and specifications
We implemented an experimental FD-ODL prototype selecting off-the-shelf components.The specifications of the selected optical components are summarized on the left side of Table 1.We designed the FD-ODL for use with light from a Superluminescent Light Emitting Diode (SLED) centered at 840 nm with 50 nm of bandwidth (SLD-371, Superlum), leading to a theoretical axial resolution of ∼6 µm.The light is delivered to the FD-ODL by optical fiber to a collimator (F220APC-850, Thorlabs), producing a beam with 1/e 2 diameter of 2.41 mm.The beam impinges on a reflective diffractive grating (DG) (GR50-0608, Thorlabs) with a groove density of 600 lines/mm.For the diffracted angle γ to the normal of the DG at the central wavelength to be zero, the incident angle β is set to 30.2 deg.The subsequent lens is an achromatic doublet (AC508-075-B-ML, Thorlabs) with a focal length of 75 mm and a 2" aperture.The spinning titled mirror TM is a 12.5 mm diameter mirror (PF05-03-P01 Thorlabs).We tested the FD-ODL with the mechanical tilt semi-angle α set to either 1.5 deg or 2.5 deg.The constant tilt to the rotation axis is provided by a 3-D printed collar mount to the shaft of a stepper motor SM which houses the TM already at the set angle by design.The SM (12 V Bipolar NEMA 17, Longruner) spins the TM at a chosen revolution speed.For the sake of demonstration, we set this at ∼5 Hz in the remainder of the manuscript.The double-pass mirror (ME2-P01, Thorlabs) is a 2" circular mirror that has been machined cut to produce a central hole of 5 mm diameter.
The right column of Table 1 summarizes the expected system specifications for the proposed FD-ODL with the chosen components.The axial range of the FD-ODL, calculated from Eq. ( 1), was 7.89 mm for α = 1.5 deg, and 13.22 mm for α = 2.5 deg.The axial scan (A-scan) frequency of the ODL corresponds to twice the revolution speed of the SM, as both a forward and backward scan take place during a complete rotation of the TM.Thus, the A-scan rate is ∼10 Hz.Moreover, due to the continuous rotation of the TM, the duty cycle of the proposed FD-ODL could approach 100%.Considering that the root-mean-square (RMS) of a sine wave is 1/ √ 2 of its peak amplitude, the expected RMS fringe frequency is 19.2 kHz, as per Eq. ( 4).

FD-ODL simulation: optical pathlength and axial scan range variation with system parameters
An analysis of the optical pathlength variation ∆OPL and the corresponding axial scan range dependency on the system parameters was performed using Eq. ( 2) for the components and values shown in Table 1.The proposed FD-ODL system was also simulated in an optical design program (OpticStudio, Zemax) to validate the analytical results.As independent variables, we considered the stepper motor rotation angle θ and the tilt angle α of the TM.beam outside the clear aperture of the lens L, and therefore cannot be considered with the chosen components.We can also notice that the simulation results (dots in Fig. 3(a)-3(b)) are in good agreement with the analytical plots.While the maximum axial scan range is nearly 10 times larger than that reported in [41], it is not sufficiently large to cover the variability in AL from different human eyes, considering also that for an average human eye with AL = 24 mm [65], the corresponding OPL is 32.5 mm, assuming an average refractive index of 1.354 for the whole eye [66].Nevertheless, we solved this issue by employing a dual sample arm, as described in the next section.

OLCR experimental setup
To demonstrate the axial scan capability of the proposed low-cost FD-ODL in measuring the intraocular distances of a model eye, we integrated it in the reference arm of a Michelson-based TD biometer based on an OLCR configuration, as shown in Fig. 4. Light from the SLED is coupled into an optical fiber and split by a fiber coupler with a 75:25 ratio to reference and sample arm, respectively.Light returning from the sample and reference arm is coupled in the detection arm and interferes at the photodetector (Nirvana Model 2007, Newport).An analog signal processing electronic (ASP) circuit was designed to extract the envelope of the interferometric signal (see Sec. 2.5) and thus allow the acquisition of A-scans with a low-cost DAQ card (T4, LabJack).A low-cost microcontroller board (UNO, Arduino) is used to drive the FD-ODL stepper motor via a motor driver (A4988, Pololu) and to generate the trigger signal to synchronize the acquisition of the photodetector with the rotation of the TM.
To measure the AL, considering the large portion of humor vitreous without biometric features of interest, we implemented a dual sample arm system [23] as commonly employed in several TD biometers.A fixed delay detour unit based on two polarization beam splitters (PBSs) was used: 1) to create two orthogonally polarized beams focusing on the anterior segment and the retina, respectively, thus improving the signal to noise ratio with better backscattered light collection; and 2) to create two "measurement windows" with a range equal to the axial scan range of the FD-ODL at a set optical pathlength from each other, centered around the anterior segment and retina, respectively.The additional optical pathlength offset introduced between the two sample beams was selected as the average axial optical pathlength of an adult human eye.Light in the sample arm is delivered by optical fiber and a collimator (F220APC-850, Thorlabs).Collimated light splits in two orthogonally polarized components at the first of two PBSs (PBS052, Thorlabs).The beam focusing on the retina (green in Fig. 4) goes through a 1:1 magnification two-lens telescope, with the first lens L1 (AC127-075-B, Thorlabs) located between the two PBSs, and the second lens L2 (AC508-075-B, Thorlabs) located after the second PBS.The beam focusing on the anterior segment (blue in Fig. 4) instead, only goes through L2, and therefore it is focused on the cornea.Moreover, the beam focusing on the retina travels a shorter optical pathlength through the polarization beam splitters than the one focusing on the anterior segment.The sample beams have a power of 1.582 mW and 0.61 mW at the cornea plane, for the one focusing on the anterior segment and the one focusing on the retina, respectively.This way, the anterior segment, and the retina can be simultaneously measured, and the axial length is recovered by accounting for the fixed delay between the two sample beams.A schematic of the proposed FD-ODL, the dual sample beam system and the electronics to acquire and process the signal is shown in Fig. 4.
The variation in optical pathlength ∆OPL with the TM rotation angle θ, the axial scan range and the fixed delay between the two sample beams have been experimentally characterized by a calibration procedure consisting of axially displacing a flat mirror in the sample arm by equally spaced distances and recording the corresponding interferograms with an oscilloscope (DSOX2002A, Keysight Technologies).From the interferograms, the envelope peak positions, the Full-Widths at Half Maximum (FWHM), i.e., the axial resolution, and the heterodyne modulation frequency f FD−ODL were extracted.The resulting calibration curve, fitting the sequence of envelope peak positions, was then used to linearize the TD biometer A-scans, varying sinusoidally with time.
For benchmarking purposes, we compared the axial scanning performance of the proposed lowcost FD-ODL with that of a standard FD-ODL based on a galvanometric scanner (> 2000 USD), where we exchanged the stepper motor and tilted mirror with a galvanometer scanner (GVS012, Thorlabs).We ran the calibration procedure for both FD-ODLs for two axial scan ranges each, with the same A-scan rate.For the standard FD-ODL, the galvanometer mirror was driven with a triangular wave.
For demonstrating the ability of the proposed TD biometer to record ocular biometry measurements, we employed a model eye (OCT model eye, Modell-Augen Manufaktur) comprising air-spaced cornea, crystalline lens, and layered retina, which are mimicking the human eye biometric features.We acquired measurements of the CCT, ACD and AL of the model eye with the proposed FD-ODL-based OLCR biometer and compared the obtained results with those from a custom laboratory SS-OCT system, thoroughly calibrated and previously used for pathlength measurements [67][68][69], the Lenstar LS 900 (Haag-Streit), a prime example of OLCR TD 1-D biometer, and the Anterion (Heidelberg Engineering), an example of 2-D SS-OCT biometer.We also employed a phantom made of three mostly transparent slabs comprising a glass microscope slide, a methacrylate slab, and a fused silica slab, of known thicknesses, comparable to the ocular distances of interest.We independently measured the slabs separately with a caliper, with the aforementioned custom laboratory SS-OCT system, after using it to evaluate the refractive index of each slab material separately [70] and then we measured the phantom made of the three slabs pressed together with the proposed TD biometer.
As the model eye was made of different materials to those of a human eye, such as glass, polymers and air, it would be beneficial to compare the optical distances first before converting them to physical distances by dividing them by the model eye manufacturer's refractive indexes.For the Lenstar biometer, the intraocular distances are only reported in physical units, assuming a human eye as sample.In such a case, as the exact refractive indices used in the Lenstar were unknown to us, we used the approximations reported in [66] to convert the measurement back to optical distances before reconverting them to physical distances by using the appropriate model eye manufacturer's refractive indexes.

OLCR signal processing
To reduce the burden of digitizing the full fringe signal at frequencies over 100 kHz (twice the Nyquist limit for the maximum heterodyne modulation frequency), and the associated cost of the digitizer, we designed and implemented an analog signal processing (ASP) electronic circuit for signal envelope detection.
The ASP circuit takes in the interferometric signal from the photodetector and outputs its one-sided envelope for slower digitization.It comprises 3 stages: high-pass filtering, rectification with pre-amplification, and low-pass filtering.Figure 5 shows each of the stages of the ASP circuit.The output at each stage is shown for reference using the signal acquired from the model cornea and recorded at each step with the oscilloscope.The first stage of the ASP circuit includes a high-pass filter to suppress the DC and low-frequency components present in the interferogram.Unlike in previous ASP designs for linearly driven galvanometric scanner-based FD-ODL, where a band-pass filter is centered around the set heterodyne modulation frequency [42], here the fringe frequency varies along the scan, so we implemented a high-pass filter instead.This helps to eliminate the low-frequency modulation due to slight light back-coupling variations throughout the A-scan.The high-pass filter consisted of a 4 th order Butterworth filter with a 3-dB cut-off frequency of 4.5 kHz and a stopband of 52.3 dB at 1 kHz.The second stage comprises signal amplification and rectification of the high-passed signal.The high-passed signal was amplified with a gain of ∼10 dB.The rectification stage uses a diode-based full wave rectifier.To ensure no DC level limits the dynamic range of the subsequent digitizer, an additional capacitor-based DC suppressing stage has been implemented.The last stage uses a low-pass filter to remove the rectified fringes (with doubled frequency) and leave the envelope signal.The low-pass filter consists of a 2 nd order Butterworth filter with a 3-dB cut-off frequency of 0.5 kHz a stopband of 71.1 dB at 30 kHz.
The output of the ASP circuit, i.e., the A-scan envelope signal, is then digitized by the low-cost LabJack T4 DAQ card with a sampling rate of 15 kS/s.Lastly, the signal peak positions are digitally detected and the signal envelope at each sample surface is then fitted to a Gaussian curve.

System characterization and comparison to a standard FD-ODL
For the system characterization we followed the procedure described in Sec.2.4.We performed a system characterization for both the standard galvanometer-based FD-ODL and the proposed low-cost FD-ODL, for two cases each, corresponding to a maximum angle α equal to 1.5 deg and 2.5 deg, respectively.For the case of the tilted mirror angle α = 1.5 deg, Fig. 6(a) shows an overlay of 14 interferograms recorded at different axial positions of the sample mirror, each one displaced by 0.5 mm from the previous.The peak intensity was maintained constant over time for the SM rotation angle θ between 35 and 120 degrees and decreased outside that range, due to reduced coupling efficiency and slight misalignments.Due to the sinusoidal dependence of ∆OPL with θ, the axial positions of the mirror in Fig. 6(a) are not equally spaced in time, as θ varies linearly with time.By fitting a sinusoid to the timepoints at which the equidistant peak positions are found, we obtain an experimental distance calibration curve for the axial scan of the FD-ODL (see Fig. 6(b)).The experimental duty cycle, calculated from the distance calibration as the angular range where the interferometric signal was measurable, was around 90%.The measured axial scan range for the proposed FD-ODL (see Fig. 6(d)), with the TM at an angle α = 1.5 deg, was 6.61 mm, in agreement but smaller than the expected 7.89 mm from theory, probably due to effectively a smaller tilt angle than the nominal one caused by manufacturing tolerances.Increasing the tilt of the TM to α = 2.5 deg, lead to an axial scan range of 10.45 mm, again smaller than the expected 13.22 mm from theory.For the standard FD-ODL (see Fig. 6(c)), where α was varied between -1.6 deg and 1.6 deg, the corresponding axial scan range was 8.16 mm, while when α was varied between -2.8 deg and 2.8 deg, the corresponding axial scan range was 12.83 mm.Figures 6(e), 6(f) show the ∆OPL plotted against the actuator angle of interest, i.e., α and θ, for the standard and proposed FD-ODL, respectively.These results demonstrate that the angle α determines that axial scan range, as expected from theory, and that a continuous rotation of θ allows to complete both a forward and backward scan without the need of counter rotation, as required for the standard FD-ODL (compare to Fig. 2).
The axial resolution of the OLCR biometer was experimentally measured by analyzing the FWHM of the envelope of the interferometric signals after applying the distance calibration.Figure 7(a) shows, as an example, the interferogram for ∆OPL ≅ 5 mm for the case of α = 1.5 deg, with a FWHM of the signal envelope equal to 31.1 µm.The full characterization throughout the axial scan is presented in Fig. 7(b).The average axial resolution was 39.51 ± 5.94 µm, and it was roughly constant throughout the axial scan.We also characterized the heterodyne modulation frequency f FD−ODL as a function of the axial scan distance.The expected sinusoidal trend was observed, with an RMS value of 22.62 kHz, not far from the theoretical value of 19.2 kHz.
Lastly, we characterized the fixed delay between anterior and posterior beams in the sample arm, which was set at an optical pathlength difference of 33.3 mm.

Experimental axial biometry results
The implemented OLCR biometer was validated both for the case of α = 1.5 deg and for the case of α = 2.5 deg by measuring the intraocular distances of a model eye and comparing them with biometric values obtained from other biometers and the model eye manufacturer, and by measuring the phantom made of slabs of known thickness and comparing them with the measurements obtained with a custom SS-OCT system and a caliper, respectively, as described in Sec.2.4.
The experimental measurement of the three-slab phantom is shown in Fig. 8. Figure 8(a) shows a photograph of the phantom.Figure 8(b) shows the detected A-scan with the proposed OLCR biometer with the tilt angle α set to 2.5 deg, after optical pathlength calibration and digital signal processing for envelope extraction.Peaks from both the anterior and posterior sample beams appear simultaneously in the A-scan.These correspond, in order from left to right, to the front surface of the microscope slide, the interface between the microscope slide and the methacrylate slab, the back surface of the fused silica slab, and the interface between the back surface of the methacrylate slab and the front surface of the fused silica slab, which appear as separate peaks, separated by less than 100 µm, due to imperfect adhesion.The reflection from the back surface of the fused silica slab is acquired with the posterior segment sample beam.We calculated the physical thickness of each component of the phantom by working out the optical pathlength between the signal peak positions corresponding to each slab, and diving it by the corresponding refractive indices, which were measured previously with the custom laboratory SS-OCT system.The measured refractive indices were 1.52, 1.49 and 1.46 for the microscope slide, methacrylate, and the fused silica slabs, respectively.The measured thickness of the components of the phantom are summarized in Table 2 and compared to those directly measured with a caliper.The measured thickness with the proposed OLCR TD-biometer of each component was 1.02 mm for the microscope slide, 2.98 mm for the methacrylate slab, and 22.74 mm for the fused silica slab, all in good agreement with the direct caliper measurement with an error below 5%.The small discrepancy can be explained by the difference in refractive indices of each material for different wavelengths, as the custom laboratory SS-OCT system is centered at 1060 nm while the proposed OLCR TD-biometer is centered at 850 nm.The experimental measurement of the model eye is shown in Fig. 9. Figure 9(a) shows a photograph of the model eye during a biometry measurement with the proposed OLCR biometer.Infrared light scattered by the various surfaces of the anterior segment highlights a trace of the axis where the biometry measurement takes place.Figure 9(b) shows the custom laboratory SS-OCT system B-scan of the full-length of the eye (in the same region scanned by the proposed OLCR biometer, i.e., the cornea, the anterior part of the crystalline lens, part of the vitreous and the retina), and the corresponding central axis A-scan (in linear units) with the cornea, anterior lens, and retinal reflections clearly visible.Figure 9(c) presents the detected A-scan with the proposed OLCR biometer along the optical axis of the model eye, after the ASP processing, low-cost DAQ digitization, and optical pathlength calibration.Peaks from both the anterior and posterior segment appear in the A-scan.These correspond, in order from left to right, to the retina, anterior corneal surface, posterior corneal surface, and anterior crystalline lens surface.Each ocular surface signal envelope was fitted to a Gaussian.Figure 9(d) shows the fitted peaks accounting for the fixed delay between the two sample beams, i.e., a full-length A-scan in optical pathlength units, where zero pathlength has been assigned to the anterior corneal surface.To measure the physical axial biometric distances of the model eye, we worked out the peak positions of the aforementioned ocular surfaces and divided the corresponding distances by the associated refractive indices provided by the manufacturer.We repeated these measurements three times, after realigning the model eye, i.e., simulating different measurement sessions with the proposed OLCR TD-biometer, while a single measurement was acquired with the remaining instruments.The measured ocular biometric parameters of the model eye are summarized in Table 3, and compared to those measured with the custom laboratory SS-OCT system and the Lenstar and Anterion biometers.The measured CCT was 0.48 mm ± 0.04 mm, in good agreement with the values from all other instruments.The measured ACD was 2.97 mm ± 0.14 mm versus a value of 3.11 mm measured with the Anterion.The differences can be possibly due to slightly different alignment of the corneal apex among measurements and across systems.The Lenstar software did not identify the crystalline lens in the scan, probably due to the difference in the back-reflected intensity when compared to an average human eye.For the AL measurement, a good agreement between our TD biometer and the custom SS-OCT system was obtained at 24.00 mm ±0.03 mm and 23.96 mm, respectively.The Lenstar returned a value of 22.95 mm.This discrepancy can be explained by the difference in refractive indices used by default for human eye measurements, versus those required for an eye phantom where both the spaces for the aqueous humor and vitreous humor are filled with air.For the conversion of the CCT optical pathlength to the CCT physical pathlength, we used a refractive index of 1.416, as indicated by the manufacturer.For the conversion of the AL optical pathlength to the AL physical pathlength, we assumed that the eye model was full of humor-mimicking fluid, as the Lenstar system does by default, and we used a refractive index of 1.354 as the human eye average [66].However, this might not be the exact value used by the Lenstar system itself.

Discussion
The optical delay line described in this manuscript is based on the working principle of a long range frequency-domain optical delay line (FD-ODL) [41,62] with a crucial difference that enables the use of a low-cost actuator to induce the displacement of the light beam at the diffraction grating, and therefore scanning the group delay.While in the standard FD-ODL a mirror on an expensive galvanometric scanner is used to linearly displace a light beam on the diffraction grating, in the proposed FD-ODL a light beam is circularly displaced on the diffraction grating by a mirror mounted on the shaft of an inexpensive stepper motor with a slight tilt, such that the mirror normal direction and the rotational axis are slightly angled between each other.We characterized its performance and compared it to that of a standard FD-ODL.We then employed it into an OLCR biometer to provide the intraocular distances of a model eye, and we benchmarked them against the one produced by a custom SS-OCT biometer and two commercial biometers.
In terms of cost, we evaluated that even with off-the-shelf components, the cost of the proposed FD-ODL optics and mechanical actuator did not exceed 750 USD, with the stepper motor itself only contributing to less than 50 USD, compared to an average of ∼2000 USD for a 1-axis galvanometer mirror scanner.This lower cost could be very advantageous for screening devices and in low-resource settings.In terms of transportability, the FD-ODL fits within a relatively small footprint of a 25 × 15 cm breadboard, with the remaining parts of the OLCR biometer prototype being amenable to a compact design.In terms of hand-held operability, and more in general, reduction of motion-artifacts during acquisition, the single most important factor is the A-scan rate and the axial scanning speed.Even though the A-scan rate used in this demonstration was only 10 Hz, the OLCR biometer exhibited an effective axial scanning speed of 295 mm/s, considering the scan window duplication via the detour unit in the sample arm.These values are still too low for hand-held artifact-free axial biometry, but they compare well with those from the standard of care in TD OLCR biometers, i.e., the Lenstar, with an A-scan rate of 12.8 Hz and an effective axial scanning speed of 420 mm/s.
Moreover, the stepper motor frequency could be reasonably increased to 15-20 Hz, leading to an A-scan rate of 30-40 Hz.Alternatively, for a slightly higher cost, but still below 100 USD, the stepper motor could be substituted by a DC motor which could lead to A-scan rates up to 100 Hz.
The effect of vibrations or small precession angles would need to be thoroughly evaluated in such a case.
In terms of scan range, we derived the analytical equations describing the relationship between the desired axial scan range of the proposed optical delay line and the tilt angle α of the TM.The experimental characterization of the scan range for different α showed that the axial scan range increases with α.Although angles up to 5.5 deg could be accepted by the proposed experimental FD-ODL, the alignment tolerances for proper light back-coupling become more stringent with larger angles α.Therefore, in the developed prototype, only angles up to 2.5 deg were tested, leading to a measured maximum scan range of 10.45 mm.The measured axial range was shorter than expected from theory by 15%.This could be attributed to a combination of the low manufacturing tolerances of the 3-D printed collar mount to the shaft of a stepper motor SM which houses the TM, and the TM not sitting exactly at the set angle inside the mount.A custom machined mount would be beneficial in reducing this discrepancy.Additionally, a shock-absorbing, sturdy case for the FD-ODL should be used to ensure alignment issues do not affect the device transportability.
The short axial scan range compared to the full axial length of a human eye is not considered a particularly limiting factor of the experimental implementation as it can be extended with the use of the detour unit in the sample arm to separately cover anterior and posterior segments, a common practice for most commercial biometers.Flawless identification and separation of the retinal signal from the anterior segment is utterly important in such a configuration.Signal processing strategies can help with it, by contrasting the characteristic prolonged scattered signal with depth from the retina versus the signal from the anterior segment surfaces, or, if needed, with the addition of hardware methods to encode a discriminating factor between the two segments.Such hardware methods could include either a phase modulator in one of the two sample beam paths, or simply a beam chopper to block periodically one beam and discern the corresponding signal.Nevertheless, the current implementation can already find good use in low-cost biometer by providing the intraocular distances of an eye.
We also compared the axial scan range performance of the proposed FD-ODL with that of a standard galvanometer-based FD-ODL, by substituting the spinning tilted mirror with a galvanometer mirror and changing the rotation axis from along the Z-axis back to the Y-axis.The experimental characterization of both FD-ODL implementations showed similar scan ranges for a given maximum angle α.Small differences are due to the fact that the galvanometer voltage to angle conversion was fully calibrated only after the measurements, resulting in slightly larger maximum α than desired.A significant difference that we expected and observed was in the effect that changes in the axial scan range have on the FD-ODL actuator for a set A-scan rate.To increase the axial scan range at a given A-scan rate, in the standard FD-ODL the galvanometer scanner is required to scan faster a larger angular range, increasing its power consumption, especially if driven with a triangular waveform.In the proposed FD-ODL instead, the axial scan range and scan rate are decoupled from one another.The stepper motor rotation only determines the axial scan rate, while the set tilt angle α determines the scan range, making our solution not only cheaper, but more versatile than the standard one.
Nonetheless, a drawback of the proposed FD-ODL is the sinusoidal variation of the optical pathlength in time.Thus, a distance calibration throughout the axial scan range is required to linearize the depth reflectance profile of the measured samples.Also, as the optical pathlength varies sinusoidally in time, the axial measurement precision is reduced at the center of the scan where the optical path variation per time unit is bigger.However, by ensuring a sufficiently fast data acquisition rate to adequately sample the interferometric signal envelope, this issue should not become a problem.An accurate distance calibration relies on a stable trigger signal and a highly repeatable stepper motor motion.Trigger position inaccuracies can be compensated by making use of the fact that the system acquires a forward and backward A-scan within a full rotation of the stepper motor shaft, such that the forward and backward A-scans should exhibit symmetry, for fast enough A-scan rates.This inherent symmetry can be utilized to refine the starting position for the distance calibration throughout the scan, further minimizing the effect of the nonlinear optical pathlength scan on the precision of the measurement.
The sinusoidal variation in optical pathlength also implied a sinusoidal variation of the interferometric fringe frequency, thus a highpass filter in the first stage of the ASP was implemented instead of a standard bandpass filter centered at a set fringe frequency.Because of this, the proposed FD-ODL provided a 89% duty cycle, instead of the theorized 100%.
Interestingly another use that holds the potential for larger axial ranges is the adaptation of the proposed FD-ODL for narrowband wavelength sweeping of a semiconductor laser for SS-OCT.This could be done by adding an extra lens after lens L and switching from a 2f to a 4f configuration with the titled mirror TM placed at the focal length of the extra lens, similarly to how the FD-ODL in [56] was converted into a fine wavelength filter in [71].
In terms of biometric measurement uncertainties, two factors are important: the system axial resolution and the corneal centration for measurement repeatability.Although sub-resolution precision can be obtained for distances between strongly reflecting surfaces as those in the anterior segment, the axial resolution is a good measure of the standard ceiling for axial biometry precision.In the proposed OLCR biometer, the measured axial resolution was significantly broader than expected from the theory, despite being roughly constant throughout the scan range.This discrepancy could be explained in part by the residual dispersion imbalance between the sample and reference arm, and possibly by some spectral vignetting at the TM, due to an offset, for the heterodyne frequency generation, as large as 3.5 mm, with a mirror radius of only 6.25 mm.This could have caused a spectral bandwidth reduction, leading to a worse axial resolution.Additionally, the tilt of the TM mirror does not allow perfect focusing of every wavelength component on the titled mirror surface, resulting in increasing spectral bandwidth reduction with larger tilt angles.Nonetheless, the choice of a slightly larger mirror and careful dispersion imbalance compensation including between the two sample beams, by means of sliding prisms, for example, should restore the theoretical resolution of 6 µm.
Nevertheless, corneal centration is far more critical to increasing measurement repeatability, hence precision, and likely the most important factor for explaining the detected discrepancy in the ACD measurements.For eye alignment, our system only used a single pupil camera (see Fig. 9(a)) not fully integrated in the sample arm of our OLCR biometer.The pupil camera could be integrated either collinearly with the OLCR sample arm or in a stereoscopic setup and Purkinje reflection detection could be applied to ensure repeatable eye alignment [40,72].
For the sake of demonstrating the capability of an OLCR biometer based on the proposed FD-ODL, we measured the intraocular distances of a model eye and compared them to the ones obtained with a SS-OCT laboratory system and commercial optical biometers.The relatively high intra-system standard deviation of 141 µm for the ACD of the model eye considering repeated measurements after model eye realignment can be attributed to the absence of the corneal centration setup in our OLCR biometer.Similarly, the difference in the ACD measurement with that by the custom SS-OCT laboratory system and the Anterion system points to the fact that, for example, for the scan shown in Fig. 9(d), the measurement axis of the proposed OLCR biometer was likely slightly off the corneal apex, therefore resulting in a slightly smaller ACD measurement, while the CCT measurement was essentially the same for all systems.Differences in the intraocular distance measurements from the commercial biometers can also stem from the different refractive indexes used by each instrument, affecting biometry accuracy.While the Lenstar uses refractive indexes compatible with those of an adult eye [66], they are hidden from the user and kept proprietary.Moreover, the intraocular distances are only reported in their physical units.As the exact refractive indices used in the Lenstar were unknown to us, we use the approximations reported in [66] to reconvert the measurement back to optical distances for further processing.In the Anterion system instead, the optical distances were available, so we converted them into physical distances by dividing them by the model eye manufacturer's refractive indexes.An error in the used approximation could be responsible for the difference in the AL measurement for the Lenstar system.
Lastly, to test the biometer on a human eye the 75:25 fiber coupler could be exchanged with a 90:10 fiber coupler, to ensure that the sample beam collective power is well below the ANSI laser safety standards for maximum permissible eye exposure [73,74].This system modification should not affect the ability to detect the ocular surfaces, as the reference power will receive 20% more power, boosting the heterodyne gain.

Conclusion
We presented a novel low-cost FD-ODL employing a spinning mirror tilted by a fixed angle to the shaft of an inexpensive stepper motor, with the rotation axis parallel to the optical axis of the beam incident on the tilted mirror.We derived the analytical equation for the variation of the optical pathlength and the heterodyne modulation frequency as a function of the stepper motor rotation angle and the angle of the tilted mirror.We experimentally characterized the former throughout the scan range for two tilt angles and compared it to the variation of optical pathlength over time for a standard FD-ODL with the galvanometer scanning up to the same angles, both at a nominal frequency of 5 Hz, resulting in an A-scan rate of 10 Hz.The measured scan ranges for the proposed FD-ODL were 6.61 mm and 10.45 mm, for a nominal tilt angle of 1.5 deg and 2.5 deg, respectively.They agreed (up to a 15% difference) with both the theory and the equivalent scan range for the standard FD-ODL.We then characterized the heterodyne modulation frequency, the axial resolution and the time-to-optical pathlength calibration curve for the smaller tilt angle in the proposed FD-ODL, obtaining an RMS fringe frequency of 22.62 kHz, an average axial resolution of 39.51 µm, and an evident sinusoidal calibration curve.We integrated the proposed FD-ODL in a dual beam OLCR system with the central wavelength at 840 nm, delivering a power of 1.582 mW and 0.61 mW on the cornea, for the beams focusing on the anterior segment and retina, respectively.We built a custom ASP circuit board to extract the interferometric signal envelope and ease the digitizing task for low-cost integration.Finally, we acquired repeated biometry measurement of a model eye, showing readings of the central cornea thickness, anterior chamber depth and axial length, and compared those to the ones acquired with a custom SS-OCT laboratory system, and two commercial biometers, showing good agreement among each other.
In summary, the proposed optical delay line reduces the cost of the standard galvanometer-based FD-ODL.Additionally, it allows to independently select the A-scan rate, by changing the stepper motor speed, and the axial scan range, by simply changing the set tilt angle of the rotating mirror.Sourcing a faster motor will enable to overcome the current limitations on the A-scan rate and expand its use to diagnostic and scientific applications.Integration of the proposed FD-ODL in an OLCR biometer design with a compact form factor with a more conservative coupler splitting ratio for eye safety and a corneal centration setup shall provide a robust portable device for biometry testing in remote areas.

Fig. 1 .
Fig. 1.Schematic representation of the proposed FD-ODL showing the optical path and circular displacement of a light beam resulting in group delay scan.DG: diffraction grating, L: achromatic doublet lens, TM: spinning tilted mirror, SM: stepper motor, and DPM double-pass perforated mirror.

Fig. 2 .
Fig. 2. Representation of the variation with time of the angles of interest α(t) and θ(t), as defined in Fig. 1 and Visualization 1, driven by the actuator responsible for delay generation in (a) a galvanometer (GM)-based FD-ODL and (c) the proposed stepper motor (SM)-based FD-ODL, for two scans (α 1 and α 2 ) with different axial ranges (∆OPL α 2 > ∆OPL α 1 ).Variation with time of the corresponding angular velocities for (b) the galvanometer-based FD-ODL, where the repeated abrupt change of speed and direction is evident, and (d) the proposed stepper motor-based FD-ODL, where the continuously rotating nature of the actuator enables its low-cost implementation, regardless of the mirror tilt angle α.

Figure 3 (
a) shows the ∆OPL induced during a complete rotation of the TM for different tilt angles α, plotted according to Eq. (2) in solid lines, and according to the corresponding Zemax simulation in the dotted line.The axial scan range, i.e., the maximum ∆OPL, is shown in Fig.3(b) with a solid line as a function of the tilt angle α.As we can observe, ∆OPL follows a sinusoidal function with a period equal to that of a full rotation of the TM and an amplitude set by the tilt angle α of the TM.
Figure 3(b) shows that the axial scan range of the proposed FD-ODL increases linearly with the tilt angle α, as expected from Eq. (2).With a tilt angle α = 5.5 deg, the axial scan range would be 29.74 mm.Larger tilt angles would redirect the

Fig. 3 .
Fig. 3. (a) ∆OPL variation as a function of the TM rotation angle θ for different tilt angles α.Solid lines represent the analytical plots while dotted lines represent Zemax simulations.(b) Axial scan range as a function of the tilt angle α of the TM.The solid line represents the analytical plot while the dots represent Zemax simulations.

Fig. 4 .
Fig. 4. Schematic of the OLCR-based TD biometer including the proposed FD-ODL in the reference arm and a dual sample beam in the sample arm.

Fig. 5 .
Fig. 5. Representation of the OLCR signal processing steps.(a) An example of the raw data of a segment of an A-scan, including anterior and posterior corneal reflections, obtained with the proposed FD-ODL at the photodetector (PD) stage.(b) A photograph of the electronics of the ASP system with white boxes around the components of its main stages.(c)-(f) The corresponding signal at each stage of the APS fed with the raw data displayed in (a).

Fig. 6 .
Fig. 6.Distance calibration and benchmarking scan performance of the proposed FD-ODL versus a standard galvanometer-based FD-ODL.(a) Overlay of 14 interferograms recorded at different axial positions of the sample mirror, each one displaced by 0.5 mm from the previous.(b) Calibration curve of ∆OPL versus time.Full forward and backward variation of ∆OPL with time for two axial scan ranges for (c) the standard FD-ODL and (d) the proposed FD-ODL.Experimental datapoints are shown as dots.Solid lines represent fits.Corresponding experimental fits to the variation of ∆OPL with the actuator angle of interest, i.e., α for (e) the standard FD-ODL, and θ for (f) the proposed FD-ODL.

Fig. 7 .
Fig. 7. Characterization of the axial resolution and heterodyne modulation frequency.(a) Interferogram close-up on a reflector point-spread-function for the case of α = 1.5 deg.(b) Axial resolution characterization throughout the axial scan.(c) Heterodyne modulation frequency f FD−ODL as a function of the axial scan distance.

Fig. 8 .
Fig. 8. Three-slab phantom thickness measurement.(a) Photograph of the phantom made of a microscope slide (left), a methacrylate slab (middle) and a fused silica slab (right).(b) Detected A-scan with the proposed OLCR TD-biometer after optical pathlength calibration and digital signal processing.

Fig. 9 .
Fig. 9. Model eye biometry.(a) Photograph of the model eye.(b) B-scan of the model eye acquired with a custom laboratory SS-OCT system and the central A-scan overlaid in blue.(c) Detected A-scan with the proposed OLCR TD-biometer after the ASP processing and optical pathlength calibration.(d) Reconstructed A-scan with Gaussian fitted peaks accounting for the fixed delay between the anterior and the posterior sample beams.