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Abstract
This work presents the results of in vivo measurements of human corneal birefringence properties using the one-way reflective polariscope, in which the same module is used to generate and analyze the light polarization state. Data analysis was performed with the use of Mueller polarimetry techniques. The distribution of the azimuth angle and the phase retardation was determined on the paracentral and limbal areas. The results indicate growth of the phase retardation magnitude and the radial orientation of the azimuth angle in the cornea’s peripheral region, which confirms our assumptions and the results presented by other researchers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A human cornea consists of six layers: epithelium, Bowman’s layer, stroma, Descemet’s membrane, Dua’s layer, and endothelium. The layer in which birefringence properties mainly occur is stroma [1,2]. It is composed of 300-500 layers of collagen fibers (called lamellae), whose orientation changes from layer to layer in a more or less random way [3]. Fibers within one lamella (fibrils) are parallelly oriented. Born and Wolf stated that stromal birefringence results from the intrinsic birefringence of each fibril and the lamellae orientation [4]. Although the corneal birefringence phenomenon was revealed early, i.a. by Haindinger the models of corneal birefringence presented in the literature differ and are sometimes even contradictory [5–8]. Stanworth and Naylor were the first to introduce the anisotropic model of a cat’s cornea [8]. In their model, lamellas are randomly oriented and the cornea is described as a uniaxial crystal plate with its optical axis oriented perpendicularly to its surface. Bour and Lopes Cardozo conducted a subjective in vivo on a human cornea and reported that the ocular retardation changes from zero in the center of the pupil to reach the maximum value at the limbus [1]. Van Blokland and Verhelst measured human corneas in vivo using the Mueller-matrix ellipsometry [9]. They noted that a cornea behaves as a biaxial crystal with its fastest principal axis normal to its surface and its slowest one nasally downward. Some polarimetry-based experiments [10,11] and PS-OCT ones [12,13] showed that a cornea may be treated as a non-dichroic linear birefringent medium (ellipticity almost zero and dichroism negligible). They stated that the retardation’s behavior is closest to circularly symmetrical one increasing from the center of the cornea to its periphery. Other researchers using a polarizing microscope for in vitro measurements claimed that a cornea might be modelled as a biaxial material with low optical retardance [14,15]. Knighton et al. concluded that, in general, the cornea behaves as a biaxial material with its fastest axis perpendicular to its surface but in some corneas, some areas are better described as uniaxial material with the optical axis perpendicular to its surface [16].
There are several methods that allow determining birefringent properties of different media, for example, ellipsometric [9,17–22], polarimetric [10–16,23–26], and interferometric ones [27]. Divided accordingly to the way we obtain the optical phase difference. In direct methods, the optical retardation is obtained by way of compensating and introducing to the setup an element with well-known optical phase retardation. In indirect methods, the retardance is obtained by carrying out calculations calculation (using e.g. Stokes vector or Mueller matrix) after the measurement of the light polarization state changes [28–32]. There are many well-known methods that allow calculating the properties of a sample in transmitted light. However, when it comes to corneal birefringence in vivo measurements, the issue gets more complicated. It is necessary to build a setup in a reflective mode, where the generator and analyzer are placed in front of the eye. Separated lightning and observation systems and thereby generator and analyzer allow measuring the corneal birefringence properties. The analysis of the obtained results was impeded by multiplication of reflections from each of the optical elements and variable optical path of the incident beam components [1,33–35].
In Sobczak et al. [32] another setup was proposed in which the same module was used to generate different polarization states as well as to analyze the light polarization state after being reflected from the tested object. Crystalline birefringent plates were used to test the behavior of the setup. In the presented paper we used mentioned above setup to measure the birefringence properties of the cornea in vivo which is the main aim of this article. Thus, we claim that the information about the changes of light polarization states results mainly from the fact it passes through the cornea twice and gets reflected from the iris. Our measurement method is still based on transmission techniques (non ellipsometric ones) and it uses the simplest setup configuration encountered in the literature. The fact that light passes twice through the measured medium can increase the sensitivity of the method but narrows the range of measured values (see Discussion chapter).
2. Methods
To measure birefringent properties of a human cornea a measurement setup was designed based on Sobczak et al. [32]. The analysis of acquired data was performed using the Mueller polarimetry techniques.
2.1 Measurement setup
The following is the scheme of the measurement setup used in our experiment: monochromatic light leaves the light source S and passes through to the collimating lens CL. The collimated light is partially reflected on the beam splitter BS and enters the polarization state generator PSG (consisting of a linear polarizer P and two liquid crystal variable retarders LCVR). The light passing through the PSG gets polarized in a various manners dependent on the LCVR setting [30]. The polarized light goes through the cornea, bounces off of the iris E, and once again passes back through the PSG (which now plays the role of an analyzer), the beam splitter and the imaging lens IL to the camera C (see Fig. 1).
Fig. 1. Setup scheme: S – light source, CL – collimating lens, BS – beam splitter, PSG – polarization state generator (consisting of polarizer P and two Liquid Crystal Variable Retarders LCVR1 and LCVR2), E – eye, IL – imaging lens, C – camera.
The light source is a monochromatic high-power LED with an average wavelength of λ=660 nm and a full width at half maximum FWHM=25 nm. A Carl Zeiss Jena DDR Tessar 4.5/300 photographic lens stands for the collimating system. The PSG is made up of one Melles-Griot linear polarizer and two Meadowlark Liquid Crystal Variable Retarders. A CMOS camera (Basler aCA1440-220um) and imaging photographic lens PZO Poland AMAR/S 4,5/105 form the data acquisition setup. The distances between the elements are respectively: S-CL is 300 mm, CL-BS is 50 mm, BS-PSG is 70 mm, PSG-E is about 250 mm, BS-IL is 200 mm, and IL-C is adjusted to receive sharp cornea image and is about 150 mm.
2.2 Theoretical background
During each measurement six input lights of different polarization states (with the same intensity I0) were induced and six output light intensities (I1, I2, I3, I4, I5, I6) were recorded. The PSG allows producing four linear light polarization states (horizontal and vertical polarizations as well as linear ones with azimuth angles +45° and -45°) and two circular ones (the right-handed and left-handed). The PSG also acts as the analyzer and forms six polariscope systems in which light intensities were measured.
We assumed that our measurement sample – a human cornea – is a non-dichroic, linearly birefringent medium. The polariscopic equations for this kind of medium, from which one can obtain the azimuth angle α and the phase retardance γ, are following:
In the absence of an eye in the setup, the six intensity distributions Ioj of the backgrounds were recorded. The subtraction of the intensity distributions from the intensity distributions of the output light enables us to obtain the light intensity distributions deprived of extra, unnecessary information (Eq. (4)).
It should be emphasized that the light passed through the cornea twice. That is why the value of the measured phase retardance was doubled in comparison to the equations mentioned above (Eqs. (1)–(4)). Combing Eqs. (1)–(3) and taking into account Eq. (4) one can calculate wanted azimuth angle α and optical phase retardance γ distributions from the following equations:
One can notice the formulas for calculating the azimuth angle α (Eq. (5)) and the phase difference γ (Eq. (6)) result in limited range solutions. The possible range obtained for the azimuth angle is [0°, 45°) and for the phase retardance [0°, 90°).
2.3 Image processing
Each measurement resulted in six eye images for the respective six polarization states, and six background images (illustrating the level of illumination when the subject is absent with some residual reflections). The background image was sequentially subtracted for every eye image to receive six corrected images for subsequent polarization states. Because an eye is in constant movement, the next step was, first, to localize the center of the pupil and then centralize all the other images in reference to the first one. The final step of the image processing part of the analysis was to assign a zero value to the light intensity outside the area of interest (zero intensity values in the pupil and extra-limbus areas). Next, the birefringence parameters were calculated from the processed images: the azimuth angle α and the phase retardance γ. The process was presented in Fig. 2.
3. Results
The numerical simulations were performed using Matlab software to confirm the assumption that the human cornea is a non-dichroic, linearly birefringent medium (Fig. 3). Figure 3(a) presents an axisymmetric distribution of the azimuth angle for the assumed medium. Figure 3(b) shows the medium’s simulated phase retardation. The distribution of the medium’s parameters adapted in the simulations result from our experiments as well as the ones carried out by other authors.
The solution of polariscopic equations provided by the formulas (Eq. (5) and Eq. (6)) allows calculating the azimuth angle and the phase retardance in a limited range only. Double-crossing the medium increases the sensitivity of the measurement but reduces the measuring range. Finally the azimuth angle is specified in the range [0, 45°) and the phase difference [0, 90°). That is why our distribution simulations accounted for these limitations, which was presented in Fig. 4.
Fig. 4. The simulation of the azimuth (a) and phase retardation distributions (b) taking into account the formulas’ limitations.
Knowing what the distributions of the azimuth angle and the phase difference should be like, we performed a simulation of the light intensity distributions for different polarization states (Fig. 5(a)-(c)). It is worth remembering that for a non-dichroic medium the simulated distributions look the same for subsequent pairs of polarization states: linear horizontal and vertical polarization state (Fig. 5(a)), linear one with azimuth angle +45° and -45° (Fig. 5(b)) and right-handed or left-handed circular one (Fig. 5(c)). That is only one figure for each pair was shown. The remaining part of Fig. 5 presents the light intensity distributions obtained in the experiments (Fig. 5(d)-(i)). Figures 5(d) and 5(g) show in sequence the light intensity distributions, which can be associated with the horizontal and the vertical polarization states in PSG. Similarly, Figs. 5(e) and 5(h) present the light intensity of the polarization with azimuth angles, respectively. +45° and -45° while Figs. 5(f) and 5(i) the ones with right-handed and left-handed polarization. It is worth noticing that the simulation assumption can also be applied in corneal intensity light distributions. That is why the mean light intensity distributions were used to calculate the azimuth angle and the phase retardance. Comparing the simulated distributions and measured ones one can observe major similarities.
Fig. 5. The distributions of the light intensity for different polarization states: (upper row) the simulated ones, (medium and lower row) measured ones for the human cornea. L0°, L90°, L+45°, L-45° - the light intensity distribution of linear polarization with the azimuth angle subsequently 0°, 90°, +45°, -45° and CR, CL - right- and left-handed circular polarized light intensities.
Using imaging processing and Eq. (5) and Eq. (6) we were able to obtain the results of the azimuth angle α and the phase retardance γ of the human corneas (Fig. 6). The results for the right eye (R) and left one (L) were presented in Fig. 6. It is to be noted that the results for the right eye are a mirror-like reflection of the results for the left eye when it comes to their distribution with a midsagittal axis in a nasal plane. The measured distributions (azimuth and phase difference) are similar to the simulated one from Fig. 4. When the limitations from the Eq. (5) and Eq. (6) are taken into account one can conclude that a human cornea in the paracentral area may be described as a non-dichroic linear birefringent medium.
Fig. 6. The results of experimental distribution of the azimuth angle α (a, b) and the phase retardation γ (c, d). Left column (a, c) stands for the right eye and right column (b, d) for the left eye.
4. Discussion
The issue of birefringence has been investigated by many researchers over the years. Brewster was the first to report about the cornea’s anisotropic properties [36]. However, for many years researchers used animal corneas in in vitro experiments to describe its birefringent properties. In 1856 His analyzed corneas of vertebrates. He found great similarities between corneas of humans, cows, sheep, pigs, rabbits, guinea pigs, doves and crows [6]. Other researchers (with scattering methods) discovered that there are regions where corneal collagen fibers are essentially parallel to each other [8,37]. Wang and Bettelheim illustrated in 1975 the maps of birefringent interference patterns (isochromes) in the corneas of various species (lamb, pig, rabbit, carp, sea bass, human) [38]. They showed that corneal birefringence may vary in detail, but generally shows similar behavior. The lowest birefringence was noticed at the center of the cornea and its value increased towards the limbus. What is more, human cornea showed the smallest birefringence among other vertebrates. The shape of the isochromes are more or less mirror images of the right and left eye with the nose being the midsagittal axis. Moreover, in most mammals the isochromes are densely packed in the nasal part of the cornea. Later experiments carried out on rabbit, bovine, porcine, and canine corneas confirmed their discovery [11,39–41].
The in vitro measurements using human corneas were also performed. Jaroński and Kasprzak described a cornea as a uniaxial-like structure [42]. They noticed the retardation greater than zero in the cornea’s center and showed a constant growth in the limbus’ direction. The azimuth angle in the center of the cornea shows the inclination in one direction although the linear ordering of the radial orientation subsides towards its periphery. On the other hand, Bour and Lopes Cardozo stated that the retardation in the center was equal to zero and the azimuth angle was radially oriented [1]. Based on mineralogy knowledge and using a polarizing microscope Bone and Draper measured the cornea’s central part birefringence [14]. They concluded that this area may be characterized as a negative biaxial material with the optic axial plane oriented nasally-downward. Fanjul-Velez et al. used the PS-OCT to test human corneas in both in vivo and in vitro conditions [43]. They presented a polarimetric model with the biaxial-like birefringence in the center and a quasi-radially symmetric high-birefringence area at the corneal periphery.
Apart from Fanjul-Velez et al., other researchers also conducted many in vivo experiments. Van Blokland and Verhelst measured the central part of corneas (about 6 mm diameter in the pupil plane) with Mueller matrix ellipsometry [9]. They concluded that a cornea may be modeled by the biaxial crystal with its fastest principal axis normal to its surface and its slowest, nasally downward. They noted that ellipticities of the eigenstates are small in the central part but grow bigger near the pupil boundary and near the temporal minimum in the retardation (which is statistically relevant). Knighton et al. measured the central and paracentral parts of a cornea (about 8 mm diameter) and noticed that most of them could be described by the biaxial model. However, they also noticed that some corneas behave more like a uniaxial crystal with the optical axis perpendicular to the surface. They emphasized that the corneal birefringence varies strongly among humans and depends on its position during measurements. Bueno and Vargas-Martin tested the paracentral and peripheral areas of cornea and stated that, in most cases, its slow axis is oriented nasally downward [35]. The magnitude of the phase retardation increases in limbus direction, however, its increment is minimal in the paracentral and peripheral areas of the cornea. Mastropasqua et al. divided corneas into two models of birefringence in paracentral and peripheral areas using the polarimetric interferometry [44]. Their first model assumes that a cornea is a linear uniaxial medium while the second one describes the cornea as a biaxial material with an angle between the binormal axes variable for each subject. However, the method does not allow measuring the central part of the cornea. Hence, there is a possibility that the separation between axes exists with the angle being extremely small. They also noticed the enantiomorphism between the subject’s two eyes. Utilizing the PS-OCT Beer et al. presented the retardation and the axis orientation maps obtained with the standard scanning pattern [45]. They proved that the low retardation in the paracentral area strongly increases towards the cornea’s periphery. Its orientation was directly associated with the lamellar orientation in the stroma. The axis orientation shows mirror symmetry between the left and right eye in each subject. Hitzenberger et al. and Gotzinger et al. based their research on the use of the PS-OCT [12,13]. Their results indicate that the phase retardation is the lowest at the corneal vertex and increases in a radial direction and also with stroma’s depth. The optical axis orientation was approximately proportional to the azimuth angle of the medium, although, a preferential optical axis orientation was observed.
5. Conclusions
In vivo measurements of the corneal birefringent properties were conducted using the reflective mode polariscopic method based on the Mueller polarimetry. The research focused on the paracentral and limbal areas of a human cornea. Our results show constant growth of the phase retardation magnitude and the radial orientation of the first eigenvector’s azimuth angle in the peripheral region of the cornea. From this, it follows that the paracentral cornea structure behaves more like a biaxial linear birefringent medium rather than a uniaxial one. The binormal axes are lying in cornea’s plane in its peripheral part as opposed to central part where they are perpendicular to corneal plane (observed by i.a. Misson [15]). Moreover, from the similarity of the intensity distributions in the respective polariscope pairs (Fig. 5(g)-(i)), both linear (L0° and L90°, L+45° and L-45°) and circular (CR and CL), it can be deduced that the peripheral cornea (at least) can be treated as a linear birefringent medium since the ellipticity of the medium would differentiate the respective intensity distributions.
Obtained outcomes are in good accordance with the simulations on this kind of material and the experiments run by other researchers that measured paracentral and peripheral cornea [11–13,16,35,42–46]. Our results do not coincide with other studies probably because of the different area of interest [9,14,43]. We focused on the paracentral and peripheral corneal regions, not on central ones. The results are in agreement with the data obtained by other researchers using the in vitro and in vivo methods [11,42,46] acquired with other techniques such as PS-OCT [12,13,43,45] or scanning laser polarimetry [16].
To conclude, our measurement setup allows measuring the azimuth angle α and the phase retardation γ of a human cornea. The results of our measurements have been confirmed by the presented simulations. Based on the obtained data and simulations it is possible to create a new model of corneal birefringence. Determining the direction of the growth of the azimuth angle α and the phase retardation γ values may be possible by conducting an experiment using, e.g. different birefringent wave plates, wedges and a circular polarizer.
Disclosures
The authors declare that there are no conflicts of interest.
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