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Abstract
We propose and design a multi-stage cascaded scanning laser ophthalmoscope (SLO) for ultra-wide field (UWF), which uses conicoid mirrors, constructed by conjugation of pupil plane. The vergence uniformity and the angular magnification of a cascaded conicoid mirrors (CCM) system are analyzed recursively and optimized preliminarily to achieve high quality imaging with UWF, and the optimal system with the model eye are obtained by simulation and optimization. Two-stage and three-stage cascaded systems are designed with this method, and the formulas of beam vergence and angular magnification are obtained by theoretical derivation. As compared to the two-stage CCM system, the proposed three-stage cascaded UWF SLO has superior performance in imaging quality. Its average RMS radius of spot diagram is calculated to be 26.372 µm, close to the diffractive limit resolution. The image resolution of human retina can be up to 30 µm with 135° FOV in theory. The three-stage cascaded SLO is more suitable for UWF fundus imaging. This study will be helpful for early screening and accurate diagnosis of various diseases in the peripheral retina.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The early lesions of many fundus diseases such as high myopia retinopathy [1–4], diabetic retinopathy [5–7], retinal vascular disease [8–10] and retinal hole [11,12] generally occur in the peripheral retina. Wide field fundus examination is essential for the early screening and diagnosis of various fundus diseases. At present, existing fundus imaging technology mainly include fundus camera [13–15], scanning laser ophthalmoscope (SLO) [16–18], optical coherence tomography (OCT) [19–21], adaptive optics (AO) fundus imaging system [22–24]. The field of view (FOV) which is defined here as the external-angle or visual-angle [25] is limited within about 60° for one single frame, and unable to cover directly the tissue structure of peripheral retina. Therefore, an ultra-wide-field fundus imaging system will be significant especially in clinic.
There are several contact methods to extend the FOV of fundus imaging systems. Pomerantzeff et al. used trans-pupillary illumination and trans-scleral illumination to obtain fundus images with a FOV of 115° [25–30]. RetCam features a FOV of 88° for fundus imaging by using trans-pupillary illumination [31,32]. Gliss et al. proposed two miniaturized wide-field fundus cameras for the FOV larger than 60° [33]. Toslak et al. proposed a pediatric fundus camera with trans-pars-planar illumination for a FOV of 135° [34]. However, these contact methods have the disadvantages of corneal contact, poor comfort, and low image resolution.
The non-contact method is much more preferred. Toslak et al. developed a smartphone-based contactless fundus camera for 103° FOV [35]. Optos system features an ultra-wide FOV of 135° by using laser scanning and two-stage cascaded reflective system [36–38]. However, the limited image resolution and frame rate require further improvement.
In this paper, a novel confocal SLO which uses multi-stage off-axis cascaded conicoid mirrors (CCM), including ellipsoidal mirrors and hyperboloidal mirrors, is proposed for the fundus imaging with ultra-wide field (UWF) and high resolution. The system uses the focal point conjugation characteristics, the off-axis method and the multi-stage cascade method to achieve UWF imaging and high angular magnification. The vergence uniformity and the angular magnification of CCM system are analyzed theoretically by using recursive methods. Two-stage and three-stage cascaded systems are designed as examples. Optimal initial structure parameters of the systems are calculated by using software programming. The performance of the systems is optimized and comparatively analyzed to achieve clearly fundus imaging with UWF.
2. System structure and requirements
2.1 System structure
The configuration of UWF SLO is shown in Fig. 1. The collimated light from super-luminescent diode (SLD), passing through the reflective system, enters the eye and illuminates the retina. The reflected light from retina goes back through the eye and the reflective system, and focused on the pinhole (PH) before the photomultiplier tube (PMT). The PH conjugated with retina filters out the stray light. The two-dimensional scanning of the retina is controlled by the fast scanner (FS) and slow scanner (SS) which are conjugated with eye pupil. The movable lens L1 is used to compensate the refractive error of eye.
Fig. 1. Schematic diagram of UWF SLO system. CM1, ellipsoidal mirror; CM2, strip-shaped ellipsoid mirror; CM3-CMn-1, strip-shaped conicoid mirrors; CMn, strip-shaped hyperboloidal mirror; F1-Fn, focal points of CM1-CMn; SS, slow scanner; FS, fast scanner; L1-L2, lens; L3, collimating lens; SLD, super-luminescent diode source; PH, pinhole; PB, pellicle beamsplitters; PMT, photomultiplier tube; r, retina conjugate plane; P, pupil conjugate plane.
The system uses multi-stage CCM for the purpose of ultra-wide-field scanning. The multi-stage CCM system is composed by an ellipsoidal mirror (CM1), a strip-shaped ellipsoidal mirror (CM2), a serial of strip-shaped conicoid mirrors (CM3-CMn), a SS, and a FS. The cascaded structure is constructed by overlapping the focal positions of the successive mirrors, and eye pupil, SS and FS are located at focal points of ellipsoid CM1 and conicoid CMn respectively, to achieve the strict conjugate relationship between their center points by using the focus conjugation characteristics of CCM. The reflective system CM1-CMn should have two extra functions: 1) The angular magnification, defined by the ratio of FOV to the scanning angle of FS, should be maximized to improve the system FOV and the frame rate of fundus imaging; 2) The vergence of light incident at eye pupil should be kept stable within the FOV.
2.2 System specifications
The requirements of system parameter are shown in Table 1. To reduce the influence of the system aberration, especially coma and spherical aberration whose values are related to the square of the beam diameter size, the light source with small beam diameter can be selected for illumination, which can greatly reduce coma and spherical aberration. Meanwhile, considering the system limiting resolution, the beam diameter at the light source is set to be 2 mm in the two-stage CCM, thereby only solving the problem of beam vergence uniformity and astigmatism at the retina under narrow beam imaging. To maximize image quality of the system and eliminate the interference speckles caused by coherent light, laser source with short wavelength, high retinal reflectivity, high eye comfort, and wide bandwidth need to used, and the SLD in 785 nm wavelength is used as the light source [39,40]. In addition, the system FOV is set to 135°, and the high-speed scanner can be selected, such as polygonal scanner whose speed is up to 55 krpm or more. Taking a 16 faces scanner as an example, the frame rate of system can be up to 5 fps for 3000 pixels per column, and the angular magnification at a 16 faces scanner needs to be greater than 3. According to Littmann-Bennett formula [41]
where x is ocular axial length in mm, q is retinal length per FOV in µm/°, the single pixel size is about 13 µm for normal axial length of 24 mm and 3000 pixels per column. According to Nyquist sampling law, the system resolution only needs to be greater than 26 µm. Meanwhile, in order to ensure image quality, the system resolution at the retina in practical application is generally 30 µm or less.
Table 1. Specifications of the optical system
3. System design method
In order to facilitate system analysis, imaging optical path is used for system theoretical analysis, and the system is analyzed by using parallel light emitting from eye. The main concern here is to analyze theoretically system astigmatism, scanning beam vergence uniformity and angular magnification in UWF SLO for aberration-free human eyes and narrow beam scanning.
3.1 System astigmatism
Cartesian coordinate system ZOY is established in the elliptical section with origin O at the left vertex of ellipsoid and positive direction of Z-axis at the right direction of major axis in Fig. 2(a). In the case of narrow beam passing through the ellipsoidal focal point, the meridional curvature at the intersection point D of the ellipsoid is
and the sagittal curvature isFig. 2. (a) Ray tracing diagram of ellipsoid and (b) its spot diagram at different image surface 1-7 by Zemax. F and F’, left focal point and right focal point of ellipsoid; D, intersection point of ray; d, distance from optical focal point.
The incident angle of the ray passing through focal point at point D is
Therefore, the oblique meridional and sagittal powers at point D are respectively calculated by Coddington formula, which can be written as,
oblique meridional power:
3.2 Beam vergence
The beam vergence at the left focal point Fn of ellipsoid CMn is ${L_{\textrm{F}n}}$, which is expressed as the reciprocal of the distance from Fn to the beam intersection point, and ${L_{\textrm{F}n}}$ is negative in divergent beam otherwise is positive. By using Eq. (5) and vergence formula, the beam vergence at the left focal point of the n + 1st ellipsoid CMn + 1 (the right focal point of the nth ellipsoid CMn) can be expressed by recurrence formula:
3.2.1 Two-stage CCM
To facilitate the theoretical analysis, the system model can be simplified as a two-dimensional model by rotating SS and CM2 and taking a scanning state of slow scanner, which is shown in Fig. 3, and Cartesian coordinate systems 1 and 2 are established separately by setting the left vertices of ellipsoids CM1 and CM2 as the origins. Eye and slow scanner are located respectively at the focal point F1 and the confocal point F2 (${\mathrm{F^{\prime}}_1}$). By using the formula Eq. (7), the system beam vergence with the incident parallel light at the focal point ${\mathrm{F^{\prime}}_2}$ can be written as
Fig. 3. Ray tracing diagram of two-stage CCM. F1, left focal point of ellipsoid CM1; F2 and ${\mathrm{F^{\prime}}_2}$, left focal point and right focal point of ellipsoid CM2; D1, D1a and D1b, intersection points of ray, initial ray and ending ray with ellipsoid CM1; D2, D2a, D2b, intersection points of ray, initial ray and ending ray with ellipsoid CM2.
The formula can be rearranged as
3.2.2 Three-stage CCM
Taking the system of two ellipsoidal mirrors and a hyperboloidal mirror as the research object, a hyperboloidal mirror is added based on the above analysis to form a three-stage CCM. As shown in Fig. 4, the right focal point ${\mathrm{F^{\prime}}_2}$ of ellipsoid CM2 coincides with the right focal point F3 of hyperboloid CM3, and the fast scanner is located at the left focal point ${\mathrm{F^{\prime}}_3}$. Cartesian coordinate system 3 is established by setting the left vertex of hyperboloid CM3 as the origin. Hyperboloid CM3 rotates around the focal point F3, and the angle between Z-axes in the coordinate system 2 and 3 is β, which from coordinate system 2 to 3 is specified to be positive in the counterclockwise direction. By using the recurrence formula Eq. (8), the beam vergence at the focal point ${\mathrm{F^{\prime}}_3}$ can be written as
Fig. 4. Ray tracing diagram of three-stage CCM. ${\mathrm{F^{\prime}}_2}$ and ${\mathrm{F^{\prime}}_3}$, left focal point and right focal point of ellipsoid CM3; D3, D3a, and D3b, intersection points of ray, initial ray and ending ray with ellipsoid CM3.
3.3 Angular magnification
In the case of ellipsoid, based on the geometric configuration show in Fig. 3, the angle ${\theta ^{\prime}_n}$ between Z-axis and ray $\overline {{{\mathrm{F^{\prime}}}_n}{\textrm{D}_n}} $ reflected by ellipsoid CMn can be expressed by recurrence formula:
In the case of hyperboloid, the angle ${\theta ^{\prime}_n}$ of hyperboloid CMn can be expressed by recurrence formula:
3.3.1 Two-stage CCM
As shown in Fig. 3, the angles between Z-axis and rays $\overline {{{\mathrm{F^{\prime}}}_2}{\textrm{D}_{\textrm{2a}}}} $, $\overline {{\textrm{F}_1}{\textrm{D}_{1\textrm{b}}}} $ and $\overline {{{\mathrm{F^{\prime}}}_2}{\textrm{D}_{\textrm{2b}}}} $ are ${\theta ^{\prime}_{\textrm{2a}}}$, ${\theta _{1\textrm{b}}}$ and ${\theta ^{\prime}_{\textrm{2b}}}$. By using the recurrence formula Eq. (14), the angle ${\theta ^{\prime}_2}$ reflected by ellipsoid CM2 can be written as:
Then angular magnification of two-stage CCM can be written as
3.3.2 Three-stage CCM
As shown in Fig. 4, the angles between Z-axis of coordinate system 3 and rays $\overline {{{\mathrm{F^{\prime}}}_2}{\textrm{D}_\textrm{2}}} $, $\overline {{{\mathrm{F^{\prime}}}_3}{\textrm{D}_\textrm{3}}}$, $\overline {{{\mathrm{F^{\prime}}}_2}{\textrm{D}_{\textrm{2a}}}} $, $\overline {{{\mathrm{F^{\prime}}}_3}{\textrm{D}_{\textrm{3a}}}} $, $\overline {{{\mathrm{F^{\prime}}}_2}{\textrm{D}_{\textrm{2b}}}} $, and $\overline {{{\mathrm{F^{\prime}}}_3}{\textrm{D}_{\textrm{3b}}}} $ are ${\theta _\textrm{3}}$, ${\theta ^{\prime}_\textrm{3}}$, ${\theta _{\textrm{3a}}}$, ${\theta ^{\prime}_{\textrm{3a}}}$, ${\theta _{\textrm{3b}}}$ and ${\theta ^{\prime}_{\textrm{3b}}}$. By using the recurrence formula Eq. (15), the angle ${\theta ^{\prime}_3}$ reflected by hyperboloid CM3 can be written as:
4. Preliminary design optimization
Taking two-stage CCM and three-stage CCM as examples, based on the above system theoretical analysis, the system vergence uniformity at exit pupil is related to the ratio of the semi-major (or semi-real) axes of adjacent two conicoid surfaces (${{{a_2}} / {{a_1}}}$ and ${{{a_3}} / {{a_2}}}$), the eccentricities of conicoid surfaces(${e_1}$, ${e_2}$ and ${e_3}$) and the position parameters (${\theta _{1\textrm{a}}}$, ${\theta _{\textrm{ss}}}$ and $\beta$), while the angular magnification of the systems is only related to the latter two. The vergence uniformity and angular magnification are taken as the objective functions, optimal initial structure parameters of the systems are obtained by setting some constraint conditions and using custom-made MATLAB (R2018a, MathWorks) program to traverse each variable.
4.1 Two-stage cascaded system structure parameters analysis
According to ellipsoidal mirror processing technology and system placement space, the semi-major axes of two ellipsoidal mirrors can be a1 = 150 mm and a2 = 100 mm∼200 mm, and the ellipsoid eccentricities e1 and e2 are 0.3∼0.5 and 0.5∼0.7 respectively. Judging from the analysis process of angular magnification, the larger the ellipsoid eccentricity, the larger the angular magnification of the system. In order to reduce the beam overflow, the beam occlusion and the mechanical interference, the angle ${\theta _{1\textrm{a}}}$ of incident light and the pitch angle ${\theta _{\textrm{ss}}}$ of slow scanner are 140°∼165° and 10°∼30° respectively, and the vertical coordinate values of intersection point of marginal light on ellipsoids CM1 and CM2 are y1a ≥ 15 mm, y2a ≥ 40 mm, respectively. Its main constraint conditions are shown in Table 2.
Table 2. The ranges of structure parameters of two-stage CCM
According to Table 2, optimal initial structure parameters of the system and ray tracing diagram are calculated by MATLAB numerical analysis in Table 3 and Fig. 5(a). It can be seen from Table 3 and Fig. 5(b) that the vergence uniformity at point F3 of the optimal initial system with two-stage CCM is 0.624, and the beam vergence ranges at points F2 and F3 are -29.07 D ∼ -5.97 D and -5.89 D ∼ -15.62 D, respectively, which shows that the system vergence distribution is not uniform enough, and an increase in the number of cascades has a certain effect on the system vergence uniformity. In addition, the system vergence decreases as the incident light angle increases. Therefore, it is necessary to design a three-stage CCM system, and appropriate surface type and spatial position of conicoid surface can be selected to achieve uniform distribution of system vergence.
Fig. 5. Ray tracing diagram (a) and beam vergence distribution (b) of optimal initial structure system with two-stage CCM. Blue circles and red diamonds in (b) are beam vergences at points F2 and F3 in (a), respectively.
Table 3. Optimal initial structure parameters of two-stage CCM analyzed by MATLAB
4.2 Three-stage cascaded system structure parameters analysis
Based on the above optimization, a hyperboloidal mirror is added to form a three-stage CCM system. Considering system placement space and beam occlusion, the semi-real axis of hyperboloid can be a3 = 30 mm, and the eccentricity of hyperboloid can be e3 = 1.5∼3. According to the above vergence distribution law, β can be 10°∼30°. Its main constraint conditions are shown in Table 4.
Table 4. The ranges of structure parameters of three-stage CCM
According to Table 4, optimal initial structure parameters of the system and ray tracing diagram are calculated by MATLAB numerical analysis in Table 5 and Fig. 6(a). It can be seen from Table 5 and Fig. 6(b) that the vergence uniformity at point F4 of the optimal initial structure system with three-stage CCM is 0.048, and the beam vergence at points F3 and F4 are -5.89 D ∼ -15.62 D and -10.79 D ∼ -10.06 D, respectively, which shows that the system vergence distribution is almost uniform. The vergence uniform of three-stage CCM system is improved significantly as compared to two-stage CCM system.
Fig. 6. Ray tracing diagram (a) and beam vergence distribution (b) of optimal initial structure system with three-stage CCM. Black circles and red diamonds in (b) are beam vergences at point F4 and F3 in (a), respectively.
Table 5. Optimal initial structure parameters of three-stage CCM analyzed by MATLAB
5. Optimization, simulation and comparison
5.1 Illumination system
The spot size of fundus illumination determines system resolution, therefore the illumination systems are simulated accurately and optimized by using Zemax software (13 R2 SP4, Radiant Zemax). The Liou model eye [42] is applied based on optimal initial structure systems obtained by the above numerical analysis, in which the lens is replaced by a refractive index of 1.413, radii of 10 mm and -6 mm, and a thickness of 3.6 mm. Different scanning angles are simulated by a multi-configurational method. The imaging quality of the systems are analyzed by comparing two-stage and three-stage CCM systems. In order to facilitate analysis and contrast, the entrance pupil diameters in the above both CCM systems are 2 mm and 6 mm, respectively, for the similar beam diameters entering the pupil. The structural parameters and the optical structure of the optimized system are shown in Table 6 and Fig. 7.
Table 6. Optimal structure parameters of CCM optimized by Zemax
In the point-by-point laser scanning imaging system, the light intensity of each image point is collected, and stray light around the image point is filtered out by using the pinhole. When Strehl ratio is higher, information loss is less and image is clearer. Table 7 and Fig. 8,9 show Strehl ratio, spot diagram and modulation transfer function (MTF) at different FOVs in the illumination system, respectively. The specific configurations corresponding to FOVs are shown in Table 7.
Fig. 8. Spot diagram of illumination systems at different FOVs, and black circles represent Airy spots. (a) Two-stage CCM, (b) three-stage CCM.
Fig. 9. Modulation transfer functions of illumination system in three-stage CCM at different FOVs. Dashed lines represent sagittal direction, and solid lines represent meridional direction.
Table 7. Strehl ratio analysis of illuminations systems at different FOVs, from -67.5° to 67.5°
It is seen from Fig. 8 that the average RMS radii of spot diagram are 56.825 µm and 26.372 µm in the illumination systems of two-stage and three-stage CCM, respectively, and the average Airy spot radii are 25.158 µm and 24.732 µm, respectively. In addition, three-stage CCM system produces a small amount of coma in several FOVs, which is caused by the slightly larger beam diameter of beam entering CM3. However, these coma values are smaller than Airy spot radii, which basically does not affect the imaging quality. Imaging quality of three-stage CCM system is improved significantly as compared to two-stage CCM system, and the average RMS radius of spot diagram decreases from 56.825 µm to 26.372 µm, close to the diffractive limit resolution. Thus, the average resolution in the retina of three-stage CCM system is less than 30 µm, and the imaging quality is good. In particular, the RMS radii of system spot diagram are less than Airy spot radii within the FOV of ±47.5°, and the resolution in the retina of this system is close to the diffractive limit resolution (DaWes criterion: 21.89 µm∼30.07 µm) within the FOV of ±47.5°, which is basically less than 30 µm.
It is seen from Table 7 that Strehl ratio of three-stage CCM system is improved significantly as compared to two-stage CCM system. The average Strehl ratio increases from 0.090 to 0.398, especially for Strehl ratio within the FOV of ±47.5° which is greater than 0.5, and the imaging quality is good. Figure 9 shows MTFs of three-stage CCM system at FOVs of ±67.5°, ± 47.5° and 0°. MTF of this system is basically over 0.026 (visual resolution limit) at 16.7 lp/mm, especially for MTF of the system within the FOV of ±47.5° which is up to 0.1, and the resolution in the retina of the system is up to 30 µm. It can be determined that three-stage CCM system is basically ideal for imaging within the FOV of ± 67.5°, close to the diffractive limit resolution, and the imaging quality is good.
5.2 Imaging system
The spot size of fundus imaging determines PH size, which affects the signal-to-noise ratio for imaging. Imaging system simulation is used to compare and analyze the spot size at the image surfaces of the two-stage and three-stage CCM systems. In the Zemax simulation, the focal length of lens L2 can be 30 mm at the receiving end, and eye pupil can be 2 mm. The optical structures of established systems are shown in Fig. 10.
It is known from Fig. 11 that the average RMS radii of spot diagram are 151.349 µm and 25.311 µm in imaging systems of two-stage and three-stage CCM, respectively. In addition, three-stage CCM system produces some coma, which is related to the diameter of beam entering CM3. In the point-by-point laser scanning imaging system, the light intensity of each image point is collected individually, which represents a pixel, and does not affect the surrounding pixels. The size of the coma can be changed by changing the pinhole size. The spot quality at the image surfaces of three-stage CCM system is improved significantly as compared to two-stage CCM system. The average RMS radius of spot diagram decreases from 151.349 µm to 25.311 µm, and its RMS radii are all less than 35 µm, especially for RMS radii of the spot within the FOV of ±47.5° which are less than 30 µm. In addition, the Airy spot radii are all 1 µm. Meanwhile, consider the system error, the pinhole of 75 µm∼100 µm is selected to filter out more stray light, and improve image quality.
6. Tolerance analysis
According to the system characteristics and the requirements of optical element processing and assembly, the specific tolerances of system structure parameters are shown in Table 8. To simplify the analysis, the tolerances of model eye are neglected. The RMS spot radius of the system at the ±47.5° FOVs is used as an evaluation index. The independent analysis of tolerance sensitivity founds that the main influence of the system performance is surface radius of curvature and surface irregularity of slow scanner, and the system RMS spot radius increases by 9.40 µm and 6.11 µm. By using Monte Carlo tolerance interaction analysis, the system RMS spot radii of 90% probability are less than 34.89 µm under 50 random perturbations, which basically meets design requirements.
Table 8. Tolerance data of structure parameters in UWFSLO system
These tolerances bring new challenges to optical element processing and assembly: 1) The flatness requirement of scanning mirror is high; 2) The processing and assembly of the large aperture ellipsoidal mirror is difficult. These will affect the subsequent system building, and are the limitations in this study that could be addressed in future research. The ultra-precision processing technology which features high precision and large stroke brings hope to the processing of large aperture ellipsoidal mirrors and high-precision surfaces. Therefore, low-cost high-precision large-aperture surface processing technology is worthy of further research.
7. Conclusions
A novel design of a multi-stage cascaded UWF SLO system is proposed. The system uses an ellipsoidal mirror, a strip-shaped ellipsoidal mirror and strip-shaped conicoid mirrors, which are cascaded by overlapping the focal positions to achieve high-resolution fundus imaging with 135° FOV. Meanwhile, the formulas of system vergence and angular magnification are obtained by recursive analysis. Optimal initial structure parameters of the system are dependent on the ratio of the semi-major axes, the eccentricities and the position parameters. The increase of the number of conicoid mirror can improve the system vergence uniformity, and the beam vergence distribution of three-stage CCM system at exit pupil of the receiving end is almost uniform. The optimal structure parameters of the system with the model eye are obtained by simulation and optimization. As compared to the two-stage CCM, the imaging quality of three-stage CCM system is improved significantly. The average RMS radius of spot diagram in the designed three-stage CCM system is 26.372 µm with 135° FOV, close to the diffractive limit resolution. Strehl ratio and MTF at 16.7 lp/mm of this system are 0.398 and 0.026, especially for the FOV of ±47.5° which are greater than 0.5 and 0.1, and the image resolution of the retina can be up to 30 µm for the three-stage CCM system. Besides, the tolerance analysis shows that system basically meets design requirements. The processing and assembly technology of the large aperture ellipsoidal mirrors will be further studied. It will lay the foundation for subsequent successfully building and clinical research of this system.
Funding
National Key Research and Development Program of China (2016YFC0100200).
Acknowledgments
The authors wish to thank Chaohong Li and Suzhou Microclear Medical Instrument Co., Ltd. for technical support and help.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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