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Abstract
Stokes lenses are variable power astigmatic lenses comprising of, in its standard version, two pure cylindrical lenses of equal but contrary power that rotate in opposite directions. Here, we present an optical device with variable and continuous astigmatic power which is based on a modified Stokes lens where two sphero-cylindrical lenses (in the form of pure astigmatic lenses) are combined in the classical way but merged with another fixed pure astigmatic lens for improving the capabilities of the resulting optical device concerning the expansion of the astigmatic range without worsening the dioptric power step resolution. The performance of this device is theoretically analyzed in virtue of the power vectors formalism including a three-dimensional (3-D) representation of the generated dioptric power as a function of both the meridian and the rotation angle between the cylinder’s axes. In addition, we have assembled a lab-made prototype of the proposed modified Stokes lens and validated its theoretical behavior by dioptric power measurements with an automatic focimeter. As conventional Stokes lenses, the applications of this new optical device range from astigmatism compensation in optical instruments to measurement of refractive error in subjective routines with the previously commented improved capabilities.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Initially discovered later than other visual problems and refractive errors, astigmatism is a nowadays well-known refractive error. The presence of astigmatism among the population was thought to be fairly uncommon but eventually it was found to be more usual than expected [1]. So, even though it was lately found, soon it became a productive research field of interest and its measurement/compensation took more time to be perfectioned up to its present status. From a historical perspective, the first description of astigmatism was made by Thomas Young 1800 [2] where he talked about the findings he made in his own eye by mean of an modified optometer [3]. Some years later, in 1825, George Bidell Airy was the first one to successfully correct this refractive error with a cylindrical lens [4,5], and he eventually named it as astigmatism by suggestion of his colleague William Whewell in 1846 [4]. Since then, the attention paid to astigmatism increased, specially from a clinical perspective, and methods to determine and compensate the value of ametropia in a more complete and proper way began to be developed [6]. In fact, George Gabriel Stokes introduced his astigmatic lens in 1849, one of the earliest devices specifically designed for astigmatism detection [7].
Stokes developed the idea of a combination of two cylindrical lenses of equal power but opposite sign, thus, one positive and the other negative, with their axes forming a right angle. The lenses were in a device with a rotatable mechanism allowing to change the angle between the cylinders axes from being parallel to forming a right angle. This rotation continuously varies the power from zero to twice the power of either lens [7]. Stokes also deduced that any sphero-cylindrical lens can be understood as a combination of a spherical lens, which power corresponds with the semi-sum of the power of each principal meridian, and an astigmatic lens, which power corresponds to the semi-difference in diopters between the principal meridians.
The instrument proposed by Stokes showed a different concept of astigmatism measurement compared with most of the methods described at that time, that were principally based on subjective observations with an optometer or similar instruments [5]. Stokes invention did not pass unnoticed but it was, in some way, misunderstood or underestimated at that moment [8]. It would be some years later when it recovered specific attention thanks to the work of Edward Jackson who revisited the Stokes lens concept including such type of lenses (although not variable in power) in his trial case as a useful tool in cylinder power [9] and axis [10] determination during patients’ subjective refraction [11]. These trial lenses are nowadays known as Jackson cross cylinders (JCC), as well as the technique to use them in world-wide accepted subjective refraction routines [12].
Nonetheless, the Stokes work [7] stated the first steps in two close areas of astigmatism study. On one hand, Stokes conceived his astigmatic lens as a tool to determine the amount of astigmatism during patient’s refraction. The procedure he proposed was very simple: the device is placed in front of the eye and the lenses are turned until the power is adjusted to compensate the eye’s refractive astigmatism. So, following a non-complex calculation, the astigmatic power of the lens is obtained from the angle between the cylinders axes [7]. And on the other hand, Stokes concept about refractive error as the combination of a spherical lens (including the astigmatism contribution) and a pure astigmatic lens (with a power of half the cylinder power) in addition with his calculations made with vectors could be pointed out as the beginning of the vector treatment of refractive error yielding in the vector formalisms appearing many years later [13–16]. But it was not envisioned in that sense at that moment because the general idea of astigmatism “as the difference between the two principal powers” prevailed [17,18].
Indeed, this conception of astigmatism generates the sphero-cylindrical classical notation of refractive error whose benefits and disadvantages have been described through the years [16,19–21]. Some inconveniences raise especially when mathematical treatment of the refraction is considered [16,19–21], and some of these drawbacks are even more relevant when paying attention on the obstacles that the Stokes lens found to become a more used method in clinical practice rather than a mere curiosity [18]. The first mathematical descriptions made by Stokes were difficult to understand or to apply on the clinical practice by his contemporaries. Some years later, Jackson published a graphic method with the aim of demonstrating the relations of equivalence between crossed cylinders and sphero-cylindrical lenses that facilitated the comprehension of Stokes concept [22]. Although there were discussions on this issue, the greatest advances in vector and matrix notations of dioptric power transcend long time after well-immersed into the 20th century [13–16,20]. Thus, alternative formalisms to the traditional sphero-cylindrical notation have been proposed through the years, but we will focus on power vector notation (or simply “power vectors”) which is a vector formalism easily and cleverly derived from Fourier decomposition and described by Thibos et al. in 1997 [16].
Power vectors describe the dioptric power in two principal forms: polar [M, J x α] and rectangular [M, J0, J45], where M represents a spherical power (or lens) in the form of a near equivalent spherical power, [J x α] represents an oriented Jackson cross-cylinder (JCC) power profile (or lens) with α orientation, and J0 and J45 represent both JCC power profiles with fixed orientations at 0-90° and 45-135°, respectively [16]. The transformation of sphero-cylindrical notation into these components, where spherical and pure astigmatic powers are independent each other, make easier statistical analysis and data treatment in refractive error management as well as allow a clear graphic representation of the dioptric power profile [15,19–21,23,24].
In addition, the concept behind the Stokes lens instrument has been applied to some other devices based on the rotation of different optical elements for providing a variable and continuous generation of different optical effects such as variable aberration generation based on two rotating phase elements [25,26], variable prism compensator (Risley prism mount) [27], tunable zoom system based on toroidal lens rotation [28,29], and, of course, variable dioptric power using both refractive and diffractive optical elements [30–34]. Focusing on the Stokes lens itself, some versions have been proposed with different features and uses. It is of special interest to remember Foley and Campbell’s variable power astigmatic lens which was made from the combination of two identical cylinders instead of being equal but opposite powers [32], and Arines and Acosta’s one which combined two identical cross-cylinder lenses [33,34]. Regarding the different uses that have been given to this type of lenses of variable astigmatic power, we found the enhancement of peripheral fundus images [35] and eye fundus camera [34], user’s astigmatism compensation in device eyepieces [33] or in a Hartmann-Shack sensor [36], residual astigmatism compensation in digital microscopy systems [37] and dioptric power measurement in focimetry applications [38,39].
In a recent paper [38], Ferrer and Micó assembled and characterized a low-cost Stokes lens including two cylinder lenses of equal but opposite dioptric powers in a Risley prism mount extracted from an old phoropter. This Stokes lens generates a variable astigmatism in the range of |3.0|D and was experimentally validated for astigmatism compensation on a digital microscope and a manual focimeter. These results led to the development of a vectofocimeter [39], which is a manual focimeter modified to measure the dioptric power of ophthalmic lenses based on power vectors using polar notation where the M component is measured as usual with the focimeter’s wheel while the [J x α] is measured with a Stokes lens. Our aim in this manuscript is to propose and validate a variation of the Stokes lens original design together with an intuitive mathematical and graphical interpretation/visualization that would help to understand as well as would solve some of the disadvantages found on previous designs [38,39]. Essentially, we have noticed two main drawbacks while reporting previous works. First, there is a limited maximum astigmatic range that can be generated (up to |3|D in [38] and to |3.5|D in cylindrical power [39]). The reason for those limits is that the minimum readable increment in the angular dial (specified in prismatic diopters – PD) when rotating the cylinders in the Stokes lens mount corresponds to |0.25|D steps in dioptric power (the regular dioptric power step when classify ophthalmic lenses). Thus, considering higher cylindrical powers will result in unacceptable dioptric power steps for astigmatism determination. Nonetheless, these astigmatic powers cover most of the cases since astigmatisms above |3.5|D are infrequent in population [40,41]. But it is interesting to expand up the astigmatic generated power to cover a higher range, thus including a higher population collective. And second, it is quite convenient to have the maximum sensitivity for astigmatism generation in the Stokes lens device for low astigmatisms since, again, the higher astigmatism prevalence in population is done for low values (67% below |1|D and 86.5% below |1.5|D according to [41]).
To expand up the astigmatic range, we add a fixed JCC lens on top of a standard Stokes lens. Assuming that we are using cylinders of generic power ±C (equivalent to pure astigmatic JCC lenses of power J = ${\mp} $C/2), the standard Stokes lens will produce a continuous harmonic astigmatic variation ranging from -C to + C or, equivalently, from 2J to −2J in terms of JCC lenses. After combination with a fixed JCC lens of power 2J = -C, it will provide a final continuous astigmatic variation from 0 to −2C (or from 0 to 4J). Thus, the generated astigmatic range is not only increased but also shifted because, instead of oscillating around 0D (as in a classical Stokes lens arrangement), the proposed device ranges from 0 to −2C. This result doubles the generated amount of astigmatism while avoiding redundancy in the generated cylindrical power with different sign (from -C to + C) by virtue of the transposition operation of the sphero-cylindrical notation. Obviously, one can start with cylinders having higher powers, let’s say for instance ±2C, and assemble a standard Stokes lens arriving to a variable astigmatic power ranging from −2C to +2C. And one can be tempted to affirm that both types of Stokes lenses (standard lens with ±2C cylinders and our modified one including ±C cylinders with a fixed JCC lens) are equivalent but there are two important differences. First, the harmonic power profile generated in the classical design oscillates around 0D but goes from 0D to |2C| in our modified design thus producing a range of Jacksonian powers all of the same sign (positive or negative). Second, the minimum step in power introduced by the classical lens will be higher than 0.25D because the internal cylinders have double power whereas our modified design maintains the 0.25D resolution while doubling the astigmatic range available.
Another advantage of our design is that our modified Stokes lens provides lower variation in astigmatism for values close to 0D than the classical Stokes one, thus increasing the resolution in astigmatism generation for low astigmatisms. In a standard Stokes lens, the generated harmonic power profile ramps up from 0D (parallel cylinder’s axes) to its maximum/minimum value (orthogonal cylinder’s axes) depending on the rotation direction [38]. And the initial power variation (the one corresponding with lower astigmatism values) is stronger than the one happening around the maximum/minimum values because of the harmonic power generated profile. With our proposed modified Stokes lens, it is possible to interchange this situation thus having softer power variation at the beginning and thus increasing the resolution for astigmatism determination at low values (most probable values when dealing with refractive errors).
In summary, we want the highest possible astigmatic value to expand the usability of the Stokes lens while having the best possible step resolution for low astigmatisms. Thus, in this manuscript we propose a novel Stokes lens design made from the combination of a standard Stokes lens in addition with a fixed JCC lens capable of satisfying both previous targets. The initial point (initial value) comes from the addition of the maximum power generated by the standard Stokes lens (orthogonal cylinder’s axes) with the power of the fixed JCC lens. By setting these two values equal with opposite signs, the initial value ideally becomes 0D. And the continuous variation of the power profile for the modified design provides better resolution near 0D and increased range of astigmatic powers achievable, all without affecting the mean spherical equivalent power (M) of the lens. In this paper, Section 2 introduces the theoretical calculations behind the proposed design as well as the graphical visualization of the generated power profiles to conclude with a lab-made prototype of the proposed device. Section 3 presents the step-by-step experimental characterization of the modified Stokes lens prototype. And Section 4 analyzes and discusses the main outcomes as well as concludes the paper.
2. Methodology
2.1 Theoretical framework
This subsection step by step reviews how to solve the drawbacks previously mentioned, that is, increasing/shifting the generated astigmatic range without worsen the dioptric step accuracy. We will introduce a second variable in the mathematical framework: the relative angle (δ) between the lenses integrating the standard Stokes lens. Thus, the power profile of modified Stokes lens will depend on the relative angle as well as on the lens meridian (θ). For the shake of convenience, Fourier polar notation [16] is used in the theoretical analysis since the math treatment from this notation is clearer than using the sphero-cylindrical formalism. Equation (1) shows the power profile of a sphero-cylinder lens (S, C x β) which is decomposed into the sum of a spherical lens M = S + C/2 and a JCC lens of power J = -C/2 with axis α = β:
For the sake of generality, let us start describing the addition of two arbitrary sphero-cylinders rotating in opposite directions (±δ). Thus, the resulting power profile (PSUM) can be calculated as the addition of the two individual ones (P1, P2) coming from Eq. (1) in the form of:
Naming MSUM to the addition of the M component of each lens (MSUM = M1 + M2) and using the trigonometrical identity for the angle difference, Eq. (2) can be rewritten as:
Equation (3) describes a combination of any pair of sphero-cylindrical lenses rotating each other in opposite directions being δ the individual rotation angle of each cylinder axis, so the relative angle between them is 2δ. In order to define a Stokes lens, some conditions must be imposed to that combination of lenses, that is, MSUM = MSL = 0D, being MSL the nearest equivalent sphere of the Stokes lens, and |J1| = |J2| = J. On its classical implementation, two pure cylinders with equal but opposite powers are combined, that is, (|C| x α) and (-|C| x α), thus verifying both previous conditions. But as previously introduced, they can also be sphero-cylinders in the form of JCC lenses, that is, (-|C|/2, |C| x α) and (|C|/2, -|C| x α). Note that we have used the modulus of C just to emphasize the different sign of the cylinders. However, it is quite normal in optometry to always use the negative sign convention for the cylinders; so, the previous lens combination is transformed into (|C|/2, -|C| x (α + 90°)) and (|C|/2, -|C| x α). In this case, α1 = α + 90° and α2 = α. But no matter the initial values α1 and α2 of the axes since they do not change the behavior of the resultant Stokes lens: they just indicate the starting orientation of astigmatic component generation. In previous works [38,39], the relation between the lenses axes has been set to α1 = α2 + 90°, as in the above case (parallel axes when considering positive and negative cylinders), meaning that the initial position provides a null resulting total astigmatic power. But here, we start from the maximum achievable astigmatic power (orthogonal axes when considering positive and negative cylinders). That case verifies that α1 = α2 = α, that is, (|C|/2, -|C| x α) and (|C|/2, -|C| x α). From these premises and considering some trigonometrical manipulation, Eq. (3) turns into:
where J is defined as the modulus of the astigmatic component of one of the lenses included in the Stokes lens, that is, |J1| = |J2| = J = -C/2.Looking to Eq. (4), the dioptric power profile describes a harmonic variation, regarding the meridian θ, for the astigmatic component J which is centered around zero because MSL equals to 0D (at least theoretically) and where the maximum and minimum values are +2J and −2J (or -C and C as stated at the intro section), respectively. This behavior is like the one reported in Eq. (1) but now, the generated power profile also includes the dependence with the angle δ that each cylinder is rotated. This new variable δ defines the astigmatic power generated in the Stokes lens for every relative position of the individual lenses which is independent of the lens meridian (θ) and/or global orientation of the Stokes lens (A). For instance, the maximum/minimum powers are reached for δ = 0° and δ = 90°, respectively, that is, our initial position for the proposed Stokes lens and the position where each cylinder is rotated 90° from such initial position. Also, the generated power becomes 0D when δ = 45° meaning that both axes are parallel. Figure 1(a-d) represents some of these cases regarding the disposition of the lenses combination as well as defines the concept of rotation angle between cylinders (δ) and power meridian (θ). Also, representation of power vectors in the astigmatic plane (double-angle space) is included (from e to h) for all the analyzed cases allowing to easily visualize the resultant vector (black arrow) coming from the addition of the individual ones (green for positive cylinders, red for negative ones, and orange for the fixed JCC lens).
Fig. 1. Working scheme for the proposed modified Stokes lens (from a to d) and the corresponding power vectors representation in the astigmatic plane (from e to h). In detail, (a) depicts the standard Stokes lens considering axes cylinders (pos/neg in green/red, respect.) at right angle as starting point, that is, α1 = 90° and α2 = 180°. Note that this case is equivalent to the previously described example when α1 = α2 = α considering negative sign convention in the sphero-cylindrical notation. Also, a generic meridian (θ) is included to clearly show its independence from the axes initial position (α) and from the cylinder’s rotation angle (δ). From this initial position, the cylinders can be rotated ±δ to continuously vary the astigmatic power from 0D to ±2J. (b) shows the inclusion of the fixed JCC lens (pos/neg in clear green/orange, respect.) with a power of ${\mp} $2J resulting in a null total astigmatic power when combining (a) for δ = 0°. (c) presents the case when δ = ±45°, thus providing 0D of astigmatic power for the standard Stokes lens so the total one comes from the fixed JCC lens (${\mp} $2J). (d) depicts the case where δ = ±90° meaning that the total astigmatic power becomes doubled, that is, ±4J. From (e) to (h), the resulting astigmatic power for the 4 previous situations is depicted in the astigmatic plane (J0, J45) where the red/green/orange/black arrows colors represent, respectively, the neg/pos cylinders of the conventional Stokes lens, the fixed JCC lens and the modified Stokes lens (final generated power). In addition, the whole frame can be rotated as a whole (angle A) to any other initial orientation to provide any oriented astigmatic component.
Our next step is the combination of the power profile coming from Eq. (4) with a fixed JCC lens of the same power to the maximum one generated by the conventional Stokes lens in such a way that, on this initial position, the total generated power will be 0D (Fig. 1(b) and 1(f)). This can be easily achieved by adding a JCC lens that in Fourier polar notation takes the form of [MJCC, 2J x (α+90°)], being MJCC the near equivalent sphere of the JCC lens which we want to set to 0D. Note that this fixed JCC lens comes from the addition of, for instance, two static crossed pure cylinders of powers (|C| x α) and (-|C| x (α + 90°)) or as a fixed sphero-cylindrical lens of power (|C|/2, -|C| x (α + 90°)), and setting that |C| = |2J| for generating the same maximum power than the classical inner Stokes lens. Equation (5) describes the resulting power PT as a combination of the standard Stokes lens (Eq. (4)) with the fixed JCC lens (Eq. (1) for [MJCC, 2J x (α + 90°)]):
Summarizing, the generation of the astigmatic power provided by the proposed modified Stokes lens starts from 0D and increases as δ is varied. But since we are starting from the maximum astigmatic power in the standard Stokes lens (orthogonal axes of the cylinders), the rate of change in the generated dioptric power is lower than in previous reported cases [38,39], thus verifying our first target. Moreover, the generated astigmatic power range ramps up from 0D to a maximum value of 4J meaning that the astigmatic power profile has been shifted from oscillating around 0D (as in conventional Stokes lenses) to 2J, which is the power of the fixed JCC lens. So, we have also increased the range of measurable astigmatic power without penalizing the sensitivity of the device (our second target).
Let us exemplify it with a theoretical example (close to the one that is going to be experimentally presented later). Let’s suppose that the standard Stokes lens is built from two pure cylinders of power ±1.50D with orthogonal axes, that is, C1 = (+1.5 × 0°) and C2 = (−1.5 × 90°). This cylinder combination yields in SL = (+1.5, −3 × 90°) in sphero-cylindrical form and the Fourier polar notation becomes [JSL x α] = [+1.5 × 90°]. And this astigmatic power is added to the one coming from the fixed JCC lens that, in this case, is [JJCC x α] = [−1.5 × 90°] to allow a null total power at the starting point. The JCC lens can be generated from the combination of two pure cylinders of powers (−1.5 × 0°) and (+1.5 × 90°) or as a sphero-cylinder of power (+1.5, −3 × 0°). The near equivalent spheres (MSL, MJCC and MT) are all null and Eqs. (4) and (5) result in PSL(θ, δ) = −1.5 cos(2δ) cos(2θ) and PT(θ, δ) = −3 (cos2δ - 1) cos(2θ), respectively. Thus, as the δ value changes continuously, the astigmatic generated power also varies continuously from 0D (initial value for δ = 0°) to a maximum value of 4J = 3D for δ = 90°.
2.2 Graphical visualization mode
Equations (4) and (5) are 3-D plotted to easily see the dependence with the two variables, the lens meridian (θ) and the turning angle (δ) of the cylinder’s axes in the standard Stokes lens. Since it is a theoretical example, we are considering the combination of the two following cylinders (|C| x α) and (-|C| x (α + 90)) yielding in a sphero-cylindrical notation of (|C|, -|2C| x (α + 90)), or [2J x (α + 90)] being J = -C/2, for the classical Stokes lens part, and (|C|, -|2C| x α), or [2J x α], for the fixed JCC lens where, without loss of generality, α is set to 90°. First, the classical representation of the power profile of a Stokes lens coming from Eq. (4) in the form of PSL(θ, δ) = −2J cos(2δ) cos(2θ) is presented at Fig. 2 where some illustrative sections are included corresponding to 2-D representations for different values of δ and θ.
Fig. 2. 3-D Representation of astigmatic power profile of a classical Stokes lens (top-center), where some interesting sections have been displayed corresponding to δ = 0° (top-left), δ= 90° (bottom-left), δ = 45° (bottom-center), θ= 0° (top-right) and θ = 90° (bottom-right).
As it was expected, Fig. 2 shows the same power variation for θ and δ because Eq. (4) is identical (double angle cos function) for both variables. Thus, the astigmatic power profiles for θ = 0°, 90° coincide with the ones for δ = 0°, 90° and they show a maximum astigmatic value of ±2J which is centered around 0D. In addition, δ = 45° means that both cylinders are with parallel axes and no resulting power is generated.
Now, the addition of a fixed JCC lens is considered, so Eq. (5) results in PT(θ, δ) = −4J (cos2δ−1) cos(2θ) and produces the astigmatic power profiles presented at Fig. 3 in a similar style than for Fig. 2. Once again, α is set to 90°. First of all, we can observe the notorious change on the 3-D representation since the astigmatic power profile is not symmetrical and it is no longer centered around 0 when visualizing from the δ perspective: it ranges from 0D to 4J and from 0D to −4J depending on the meridian θ. And what does the meridian mean? The angle θ is related with different angular directions when looking the device from a frontal point of view and, in that sense, a positive power for θ = 0° implies a negative power for θ = 90° as usually happens when a J0 component is transformed to a J90 one, that is, [J x 0°] = [-J x 90°]. Thus, every 90° in θ, the astigmatic power profile changes its sign. For intermediate values, the sign is maintained but the power profile does not reach the maximum/minimum astigmatic values of ±4J. Note that, in essence, a specific change of θ degrees in the meridian is like to globally rotate the whole device the same value but on opposite direction (-θ degrees), thus justifying the inversion in the sign of the power every 90°.
Fig. 3. 3-D Representation of astigmatic dioptric power profile of the modified Stokes lens (top-center), where some sections have been displayed which correspond to δ = 0° (top-left), δ = 90° (bottom-left), δ = 45° (bottom-center), θ = 0° (top-right), and θ = 90° (bottom-right).
For the meridian perspective, no power variation is found when δ = 0° because this is our starting point, and the astigmatic power starts to increase until its maximum and minimum values of 4J D and −4J D when δ = 90° and 270°. The case for δ = 45° is the case of null power for the standard Stokes lens and it only remains the astigmatic power introduced by the fixed JCC ranging from −2J to 2J as we have previously commented (Fig. 1(g)).
Just as a final clarification, δ represents the turning angle of the lenses inside the Stokes lens, but 2δ does not correspond to the relative angle between the lenses as it was in previous references [38,39] because now we are not starting with the cylinder axes in parallel configuration but orthogonal. Thus, the relative angle γ between the cylinders can be calculated as γ = 2|δ| - 90, and the comparison of the astigmatic power profiles in a standard Stokes lens and the modified one as a function of γ and considering θ = 90° is shown in Fig. 4. It is now evident how the addition of the fixed JCC lens shifts the generated astigmatic power profile so the J component obtained would be only negative (or positive for other θ values) but oscillating around the value of the fixed JCC lens (±2J) and not around 0.
Fig. 4. 2-D representation of the astigmatic power profile generated by the conventional Stokes lens (a) and the modified one (b) with respect to the relative angle γ and considering θ = 90°.
2.3 Construction and characterization
Once settled the theoretical frame, a modified Stokes lens according with the previous features has been built in our lab. On one hand, a standard Stokes lens has been assembled following likewise steps as in previous publications [38,39] by using a Risley prism mount from a manual focimeter Topcon LM-8 where the prisms have been replaced by ophthalmic sphero-cylindrical lenses of theoretical cylindrical powers +1.5D and −1.5D. Before being assembled, the dioptric power of the lenses has been measured with an automatic focimeter Topcon CL-300 resulting in (S1 = 0.00 ± 0.01 D, C1 = −1.51 ± 0.01 D) and (S2 = +1.45 ± 0.01 D, C2 = −1.46 ± 0.01 D). The lenses were manually cut down in a diameter of 19 mm to replace the prisms in the mount while trying to keep the same optical center.
As it was discussed in previous papers [38,39], the utilization of a Risley prism mount with the scale in prismatic diopters adds some limitations because of the reading accuracy as well as it forces the mathematical transformation between prismatic diopters and degrees. To avoid this disadvantage, an angular scale was printed according to the Risley mount diameter (see Fig. 5(a)). Thus, the lenses were introduced in the Risley prism mount with their axes orthogonal one each other and the resulting lens power was (SSL = +1.45 ± 0.01 D, CSL = −2.97 ± 0.01 D), being this position our starting point and the initial value at the angular scale from which the turning angle δ will be measured. And before adding the fixed JCC, the standard Stokes lens was characterized using the automatic focimeter and following a similar procedure as in Ref. [38] but in this case each measurement was made in steps of approximately 2 degrees (our maximum sensitivity in the printed angular scale). Thus, 3 measurements were annotated for every angular position from which mean value and errors were computed.
Fig. 5. (a) Risley prims mount included angular scale and (b) standard Stokes lens (left) compared to the modified one after adding the fixed JCC (right).
After standard Stokes lens characterization, a sphero-cylinder lens with the following power (SJCC = 1.45 ± 0.01 D, CJCC = −2.97 ± 0.01 D) is selected to be included as fixed JCC lens. This lens has been cut down to a diameter of approximately 28 mm to be placed in the back part of the Risley prism mount as it can be seen from Fig. 5(b) (right). It was attached as close as possible to the standard Stokes lens avoiding contact for not interfere with the rotation mechanism. Also, the fixed JCC lens is not affected by the rotation of the lenses inside the prism mount.
Finally, a similar dioptric power characterization over the modified Stokes lens has been made. The results obtained from every characterization either with or without the fixed JCC have been compared with those expected from the theoretical calculations.
3. Experimental results
3.1 Standard Stokes lens characterization
The results concerning the dioptric power measurement of the standard Stokes lens are shown in Fig. 6 and compared with the predicted results coming from the particularization of Eq. (4) according to the used sphero-cylinders (as we have stated on the numerical example included in the end of the section 2.1). Here, the lens mount is fixed and only the turning angle δ is modified. All the comparisons have been made using Fourier polar notation.
Fig. 6. Dioptric power characterization for the standard Stokes lens (SL) considering power vectors (M, J) notation and θ = 90°. (a) M component variation (solid blue line), measurement errors (blue shadowed zone) and averaged M value (black dotted line). (b) J component profile (black solid line), theoretical prediction from Eq. (4) (red dashed line) and measurement errors (blue shadowed zone). (c) is a zoom of the ROI marked in (b) to clearly show the 3 profiles.
As Fig. 6 shows, the power profiles exhibit a behavior very close to the theoretical prediction. Thus, the M spherical component is close to zero (averaged M value equals to −0.038 ± 0.015 D) and the J astigmatic component varies following a harmonic profile. In our initial position δ = 0°, where the lenses are forming a relative angle of approximately γ = 90°, the measured values are [M, J] = [−0.027 ± 0.015, 1.506 ± 0.005] D which reveal a M close to zero and a J component close to 1.50D. For δ = 270° (see Fig. 6(c)), we found [M, J] = [−0.041 ± 0.015, −1.505 ± 0.005] D again similar to theoretical values.
And Fig. 7 includes the 3-D representation of the astigmatic power variation coming from Eq. (4) for the standard Stokes lens in the form of PSL(θ, δ) = −0.035 + 1.485 cos(2δ) cos(2θ). The values of MSL and JSL can easily calculated from the two lenses used in its assembly as follows. The sphero-cylindrical values (S1, C1 x α) = (0, −1.51 x α) and (S2, C2 x α) = (+1.45, -1.46 x α) are transformed into Fourier polar notation [M1, J1 x α] = [−0.755, +0.755 x α] and [M2, J2 x α] = [+0.72, +0.73 x α], and the result added to provide [MSL, JSL x α] = [−0.035, +1.485 x α]. Note that this PSL(θ, δ) equation is really close to the theoretical one (see last paragraph of section 2.1). And as in that case, the dioptric power variation is symmetrical whether θ or δ change, and when δ = 45° the dioptric power obtained is almost zero, with an average value of −0.011 ± 0.005 D.
Fig. 7. 3-D view of the astigmatic dioptric power profile generated by the lab-made assembled standard Stokes lens as a function of θ and δ (top-center) and including some interesting 2-D sections corresponding with δ = 0° (top-left), δ = 90° (bottom-left), δ = 45° (bottom-center), θ = 0° (top-right), and θ = 90° (bottom-right).
3.2 Modified Stokes lens characterization
After adding the fixed JCC lens on the back part of the prism mount, the modified Stokes lens has been characterized following a similar procedure. The obtained results are shown in Fig. 8 as 2-D representations of the power profile of M and J components and in Fig. 9 as 3-D plot of the astigmatic power profile as a function of both θ and δ.
Fig. 8. Dioptric power characterization for the modified Stokes lens considering power vectors (M, J) notation and θ = 0°. (a) M component variation (blue line), measurement errors (blue shadowed zone) and averaged M value (black dotted line). (b) J component profile (black solid line), theoretical prediction from Eq. (5) (red dashed line) and measurement errors (blue shadowed zone). (c) is a zoom of the ROI marked in (b) to clearly show the 3 profiles.
Fig. 9. 3-D view of the astigmatic dioptric power profile generated by the lab-made assembled modified Stokes lens as a function of θ and δ (top-center) and including some interesting sections corresponding with δ = 0° (top-left), δ = 90° (bottom-left), δ = 45° (bottom-center), θ = 0° (top-right), and θ = 90° (bottom-right).
Figure 8 shows the generated power profile for M and J components measured over the modified Stokes lens. First, in Fig. 8(a) the M component present an average value of −0.036 ± 0.015 D which is similar than for the standard Stokes lens case but showing higher oscillation around the mean value. Figure 8(b) presents the comparison for the J component and for the case of θ = 0° between the theoretical curve obtained from Eq. (5) when considering the theoretical values for the lenses used in the experimental implementation (see last paragraph of section 2.1) and the measured ones. One can see again a harmonic variation but this time the oscillation is not centered around zero but shifted up thus making a variation power range from 0D to around 3D in astigmatic component. The blue shadowed zone referring to the J component error is also regular for all the range long, being 0.005 D in amplitude. The measured values at δ = 0° are [M, J] = [−0.038 ± 0.015, −0.041 ± 0.005] D, and they are in good agreement with the expected ones.
Figure 9 includes the 3-D plot of the astigmatic power profile as a function of θ and δ coming from Eq. (5) in the form of PT(θ, δ) = −0.07 - 2.97 (cos2δ - 1) cos(2θ). To obtain this equation, we can first compute the Fourier polar notation of the fixed JCC lens which is experimentally assembled, that is: (SJCC, CJCC x (α + 90°)) = (+1.45, −2.97 x (α + 90°)) so [MJCC, JJCC x (α + 90°) = [−0.035, +1.485 x (α + 90°)] = [−0.035, −1.485 x α]. And then add it with the power of the standard Stokes lens [MSL, JSL x α] = [−0.035, +1.485 x α]. Thus, the resulting power in the initial position becomes [MT, JT x α] = [−0.07, +0.00 x α] which is close to 0D as expected. And the power when the standard Stokes lens is rotated 90° is [MT, JT x α] = [−0.07, −2.97 x α]. From these [MT, JT] values, the above equation is obtained and, again, it is really close to the theoretical one (see last paragraph of section 2.1). Figure 9 also shows some useful sections where we can see that the measured values in Fig. 8(b) perfectly fit with the plot when θ = 0° (Fig. 9 top-right) showing a dioptric power variation being positive all the δ range. On the contrary, the dioptric power obtained when θ = 90° (Fig. 9 bottom-right) only reach negative values. In this case, when δ = 45° (Fig. 9 bottom-center) the result is a harmonic profile centered in zero with maximum and minimum values of 1.40 ± 0.02 D at θ = 0° and, −1.56 ± 0.02 D at θ = 0° coming from the fixed JCC lens. For the value δ = 0° (Fig. 9 top-left), the contribution due to the harmonic oscillation is null and it only remains the one coming from the residual near equivalent sphere value (MT = −0.070 ± 0.015 D). For the value δ = 90° (Fig. 9 bottom-left), we found the dioptric power profile of a JCC type lens considering J component that reaches values of +2.90 ± 0.02 D and −3.04 ± 0.02 D.
Finally, to have a better view of what we referred as lower variation in generated astigmatic power for values close to 0D between both Stokes lens modalities, Fig. 10 includes together the dioptric power profiles coming from individual cylinders of ±1.5D when considering a standard Stokes lens (blue line) starting with parallel axes [38] and the proposed modified Stokes lens (pink line). Note that this latter case is the same one included at Fig. 9 top-right. Figure 10(a) highlights that the power profile of the modified Stokes lens comes from the classical Stokes lens one but it is upshifted to oscillate around 1.5D (the J value of the fixed JCC lens) and moved 90° to the right. The first issue allows to double the astigmatic range generation from [−1.5, +1.5] D to [0, +3] D interval while the second one provides a lower astigmatic variation for the minimum readable angle in the scale (2°). Figure 10(b) provides a magnification of the astigmatic generation power starting from 0D where it is easy to see how the standard Stokes lens ramps up to J = 0.125D (C = −0.25D) approximately for the first measurement point (δ = 2°) while the modified Stokes lens takes the same value for the third measure point (δ = 6°). So the generation of astigmatic power is slower in the proposed lens modification. Consequently, the dioptric power rises more abrupt in the standard Stokes lens than in the modified Stokes lens where it takes more control (measurement) points to reach the same value of dioptric power step. Figure 10(b) also includes the dioptric power values coming from the readings at δ = 2, 4, 6 and 8° just to reinforce the slower variation around 0D in generated dioptric power with the modified Stokes lens design.
Fig. 10. Dioptric power variation of the conventional Stokes lens (blue plot) and the modified Stokes lens (pink plot) with the increment of δ for θ = 0°. (a) shows the range between 0° and 360°, while (b) depicts a zoom of the initial ramping up of the dioptric power. The J = 0.125D is marked with a dashed black line, and the point where each power profile cuts this value is marked with a vertical line and a circular marker. The values for the two closest measurement points are included and marked with a point of the same color for each function.
4. Discussion and conclusions
As we have presented in previous sections the experimental behavior of the modified Stokes lens closely follows the theoretical predictions. This can be easily observed through comparing the symmetry and the position of maximum and minimums of the generated astigmatic profiles at Fig. 2 vs Fig. 7 and Fig. 3 vs Fig. 9. While Figs. 2 and 3 are viewed in a theoretical fashion with no specific value for the J component, Figs. 7 and 9 are the 3-D representations considering the measurements performed with real lenses. Through these 3-D views, not only the dioptric power profile of the lens is obtained (classical representation as a function of the meridian θ being δ constant) but also the dioptric power variation as a function of the turning angle δ of the lenses integrating the standard Stokes lens (being θ constant). However, since the lenses used in the experimental implementation do not have exactly their corresponding theoretical dioptric powers, small differences arise between theoretical and experimental results as Figs. 6 and 8 have shown. Because of this, there is a small but not null remaining averaged near equivalent sphere for both the standard Stokes lens (MSL = −0.038D) and the modified one (MT = −0.036D) as well as the maximum/minimum astigmatic powers (JSL = 1.505D and JT = 3.038D) are also slightly different from the theoretical predictions (1.5D and 3D) for δ = 0°. Furthermore, the experimental results shown in Fig. 6(b) fit with those shown in Fig. 7 (bottom-left), as well as the Fig. 8(b) is comparable to Fig. 9 (top-right), where the maximum/minimum astigmatic powers used to plot the curves are JSL = 1.485D and JT = 2.97D, respectively. Nevertheless, M values are close to 0D, so they can be considered as pure astigmatic power lenses at least from an ophthalmic/optometric point of view.
Analyzing in depth the proposed modified Stokes lens, the dioptric power profile when δ = 0°, which is the starting position, should give a null contribution to either M or J component, but it is not exactly the case. These differences can be seen with more detail at Fig. 8 for both components. The M component of the modified Stokes lens has an average power of −0.036D (Fig. 8(a)) while the theoretical prediction resulting from the addition of the near equivalent spheres of the 3 measured lenses is −0.070D, shown in Fig. 9 (top-left). In addition, Fig. 8(a) also exhibits a variation in the dioptric power which seems to follow a similar periodicity to that of the J component, having minimum values or approximately −0.05D around δ = 0°, 180° and 360° and maximum values close to 0D for δ = 90° and 270°. And regarding the J component, the measurements closely follow the theoretical predictions (Figs. 8(b) and 8(c)). For example, there is a difference between the measured profile and the theoretical prediction of 0.042D (assuming cylinders of ± 1.5D) at Fig. 8(c), being the measurement more powerful than the prediction. Although these differences, the modified Stokes lens shows the expected behavior reaching a J value of approximately |3|D with the turning angle, with a negligible introduction of remaining M component.
The source of those differences can be related with the construction itself of the modified Stokes lens and the measurement procedure. The readings provided by focimeters (manual or automatic ones) are quite dependent on the position where the measured lens is placed in the apparatus. In our case, the standard Stokes lens does not show a wrong position but when adding the fixed JCC lens in the modified implementation, there are 3 lenses making a bulky device where some lenses are not in proper measurement position: we have estimated that the standard Stokes lens is 4 mm away from the position where the same lens was measured alone. Due to this, the modified Stokes lens cannot be understood as a simple addition of the three lenses and this fact could probably be the source of the error behind the observed dioptric power differences. In our previous work [39], something similar happened when placing the Stokes lens in the manual focimeter 7 cm away from the lens to be measured. It was observed that the higher the generated astigmatic power the greater the effect of the distance over the spherical component meaning that the correction is not constant for all the measurement range. Here, the influence of the J component over the distance correction could be the cause of the variation of the M component in the modified Stokes lens following a similar shape than the J component. As previously stated, the addition of the three individual lenses gives a resulting near equivalent sphere of −0.070 D. But using the distance correction proposed in Eqs. (6-7) of Ref. [39], the result is modified to −0.048D which is really closer to the obtained value (−0.036D). Some slight differences are also observed in the J component (Fig. 6(b), 6(c), 8(b) and 8(c)) that can be also related to the experimental procedure of construction and measurement, including displacements of the lenses inside the device that are not predictable from the calculations. In any case, the differences are below 0.12D (the typical tolerance value in clinical procedures).
All in all, we can say that the lab-made assembled modified Stokes lens has met the proposed targets, that is: i) to increase the astigmatic range in with previous works [38,39] for covering a higher astigmatism range (up to |6|D of cylinder, |3|D in terms of J component) and in order to expand the usability of the Stokes lenses in general, and ii) having the highest possible sensitivity in the readings for low astigmatisms to better quantify those cases. This is achieved by starting with orthogonal cylinders axes in the standard Stokes lens and adding a fixed JCC lens to neutralize the power generated by the Stokes lens. The result shows a gentle variation in the astigmatic power generation at the beginning, that is, for low astigmatism values (Fig. 8(b) and Fig. 10(b)). But what is also relevant is the shift of the dioptric power profile which is generated as the rotation angle changes. In a conventional Stokes lens, the astigmatic power profile oscillates around 0D, thus being able to generate both positive and negative J components with the same lens. This is supported by Figs. 2 and 7 (right graphs with δ variable while θ is constant) and Fig. 6(b). Obviously, this is an interesting feature since there are astigmatisms with positive J value while others have negative ones. In a standard Stokes lens, one can change the sign of the J component using a double strategy. On one hand, the cylinders can be rotated clockwise and anticlockwise, that is, ±δ or ${\mp} $δ, thus generating J component with a different sign. And on the other hand, the cylinders axes can be rotated always in the same direction but the whole Stokes lens rotated, that is, changing the global angle A. In this latter case, imaging that a positive J0 component is generated in the form of [|J0| x 0°] and the Stokes lens mount is rotated 90° thus defining an astigmatic component equal to [|J0| x 90°] which is equivalent to [-|J0| x 0°]. So somehow, the capability of a classical Stokes lens for generating positive/negative J components using two different manners is redundant. And the proposed modified Stokes lens sacrifices the capability of generating J with different signs by rotating the cylinders into different directions in return of arriving to a higher J absolute value. Thus, to change the J sign, the whole device must be rotated 90° as it can be seen through Figs. 3 and 9 (right graphs).
Finally, the proposed modified Stokes lens is potentially suitable for being used as the conventional one, that is, astigmatism compensation in optical instruments or astigmatism determination in clinical optometry practice. Because of all previous discussions, we conceive the proposed Stokes lens modification as an improvement of previous designs where the astigmatic range has been matched to other instruments used in clinical practice for astigmatism determination in patients, such as trial cases or phoropters, where astigmatism ranges up to a maximum of 6D in cylinder power at 0.25D steps. However, higher or lower values can be achieved by customizing the lens according to the needs of the target application. For instance, we envision different strategies to further increase the astigmatic dioptric power. Probably the simple option is to include lenses having higher dioptric power, thus arriving to higher astigmatic range generation. This case will need the appropriate angular scale (or automatized rotation mechanism) having finer resolution steps to ensure that the astigmatism generation between consecutive readings is 0.25D as maximum. Second, the inclusion of additional fixed astigmatic lenses changes the shift in the power profile, allowing to cover higher astigmatic range. But, in this case, the minimum value will not be zero, thus introducing astigmatism in all the positions. And finally, the merge of several Stokes lenses would increase the dioptric power achieved while keep a null dioptric power variation in a certain position. But in this last option, the device will be less compact and will have other problems coming from presence of high order aberrations and additional corrections because the lenses will not be in proximity (thin lens approximation). The inclusion of other type of lenses (for instance, doublets) could be also an option, but since the meniscus design in ophthalmic lenses is very well optimized in terms of both primary aberrations and thickness reduction, they have been shown an excellent choice in the construction of the classical Stokes lens. Further studies should be done to find out what option is more efficient regarding the final target in future applications.
In summary, since George Stokes first described his namesake lens as a tool for the determination of the astigmatism of the eye, different designs of the device for different uses have been described, being the Jackson cross-cylinder one of the most known and especially relevant in optometry. But other applications involving astigmatism measurement and/or compensation have been developed with the Stokes lens as accessory or as the main component for different optical instruments such as digital microscopes, eyepieces or focimeters. The modification of the Stokes lens proposed in this manuscript supposes an improvement over previous designs mainly for the area of focimetry where some drawbacks were found on previous works. Moreover, an intuitive theoretical framework (including equations and 3-D plots) has been introduced considering the turning angle δ as a variable for astigmatic power generation and a modified Stokes lens has been assembled and characterized at the lab fulfilling the theoretical predictions. The result exhibits nice capabilities for its use in optometric clinical practice as well as for astigmatism measurement or compensation, in general. Future steps can be directed to a more complete experimental characterization of the proposed lens by measuring the high order aberrations.
Funding
Ministerio de Cicencia e Innovaciones, Grant (PID2020-120056GB-C21); Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana (ACIF/2020/037).
Acknowledgments
S. Ferrer-Altabás is recipient of a contract from the Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital de la Generalitat Valenciana and European Social Fund.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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