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1 Analysis of Refractive errors in the human eye using Shack Hartmann Aberrometry M. Jesson, P. Arulmozhivarman, and A.R. Ganesan* Department of Physics, National Institute of Technology, Tiruchirappalli 62, India. ABSTRACT The refractive errors of normal, healthy eyes were measured using an indigenously developed Shack Hartmann aberrometer. Measurement was made in both right and left eyes after dilation for a 6mm pupil size. Power vector method was used to represent the sphero cylindrical errors. Analysis was done for astigmatism with the rule, oblique astigmatism and defocus between the right and left eyes of the subjects, which showed a negative, positive and zero correlation respectively. No correlation could be detected for RMS values between right and left eyes, though Zernike between right and left showed bilateral symmetry in our subjects. It was found that with an increase of spherical aberration, defocus decreased slightly. The validity and repeatability of our Shack Hartmann aberrometry in measuring the refractive error was analyzed and repeatability coefficient was calculated. Optimal correction for greater retinal image quality has been discussed and far point vergence for detecting the point of maximum retinal image quality is suggested. Key words: Shack-Hartmann sensor, wavefront, aberration, refractive error, sphero-cylinder.. INTRODUCTION Aberration limits performance is a saying which goes well with Optics. In the Human eye, the visual performance is considerably limited by aberrations. To measure these aberrations and to correct them remains the challenging mission for the Optical Scientists for centuries. Major strides have been made over a century now in correcting the chief aberrations of human eye. Defocus and Astigmatism, which contributes up to 92 % of the total aberration, are corrected to a level of expedient viewing with spectacles or contact glasses. Nevertheless it has been proved beyond doubt from myriad studies that the visual performances are hindered by the presence of higher order aberrations. Hitherto, in the determination of refractive errors, higher order aberrations were considered clinically none effect. Conventional correction methods like refractive surgeries leave the subject with significant presence of higher order aberrations, which deleteriously mar the vision. Erroneousness in the conventional correcting methods of the refractive errors arises from the fact that spherical and cylindrical components are considered independent of each other. Instead it is found that the cylindrical lenses encompass spherical power in them, which is called as Mean Spherical Equivalent. The measurement of the spherocylindrical errors using power profile method involving Fourier analysis provides an accurate and comprehensive evaluation. This method makes available the numerical, graphical and statistical analysis of refractive errors in large population and lenses very simple. Fourier analysis has a constant and the harmonic terms, where the constant remains independent of the pure cosine terms. These two Fourier terms gives three components for the calculation of refractive errors. First, the constant term, which is the spherical equivalent corresponding to the exact physical lens. The second and the third component is the harmonic term corresponding to Jackson Cross Cylindrical lens (JCC) at θ = 4/3 or /8. The power profile of the sphero-cylinder lens is represented as M, J, J 4 and which could be plotted as a single point in a three dimensional space. The entire power profile of sphero-cylindrical lens combination is a vector sum expressed in a rectangular form as a single point in a three dimensional dioptric space, called Power Vector 2. The rationale of the present study is to analyze the refractive error in the light of higher order contribution measured using Shack Hartmann aberrometry and to explore ways to find out the optimum correction for sphero cylindrical lenses such that the optical quality of eye is greatest. Comprehensive analyses of the relation between refractive errors were studied meticulously and correlation between lower order and higher order aberrations have been plotted. *arg@nitt.edu, phone extn.3662, fax: , nitt.edu

2 2. METHODOLOGY 2. Shack Hartmann Aberrometry The Shack Hartmann aberrometer, designed and developed indigenously, has been used for determining the aberrations of the whole eye. Shack Hartmann is an objective technique, which is the most popular and robust method. This method has high accuracy, remarkable repeatability and incredible consistency compared to any other method involved in the measurement of aberrations of human eye. The principle of Shack Hartmann aberrometry is discussed elsewhere 3. The Experimental arrangement is in Figure. A collimated He-Ne Laser beam is launched into the eye with the power of 3 µw, which is far below the permissible exposure level prescribed by ANSI. The reflected light from the retina after having pass through the Lenslet array is imaged by the CCD camera. A Lenslet array is an assembly of small planovex lenses arranged regularly in a X square array format. The focal length of lenslet used was 4mm with an inter-lenslet distance of 47 µm. A 2/3 Sony CCD camera was used for imaging the array of focus spots. The CCD detector array was sub divided into sub apertures of 36x37 pixels, so that these spots are centered at each of these sub apertures. An aberrated wavefront passing through this lenslet would produce varying shifts of these spots over their sub apertures. The shift gradient of these spot positions and decomposition using Zernike polynomials would quantify the kind and magnitude of the aberrations inherent. Fig(): Shack Hartmann wavefront sensor experimental setup Subjects were university students between the age of 2 to 3 years. All of them have healthy eye and the subjects were dilated with % Tropicamide eye drops after getting their consent prior to the experiment. We determined the refractive errors in both the eyes at uniform pupil size of 6 mm in diameter. 2.2 Wavefront representation of Zernike polynomials Light through the sub apertures form a focused spot on the lenslet array in a regular arrangement along the optical axis. This is considered as a reference pattern. The position of the focus spot in each sub aperture is determined by the centroid of light distribution. Wavefront deviating from this reference pattern is an aberrated wavefront. If W(x,y) represents a wavefront, then partial derivative of it is given as W(x,y)/ x = x/f () W(x,y)/ y = y/f (2) Where f is the focal length of the lenslet array and x & y are the shift spots along x & y directions 4. 2

3 A Wavefront error can be decomposed by Zernike polynomial and normalized. Zernike polynomials are product of three terms, a normalization constant and two functions, a radial coordinate and an angular variable defined in polar co ordinates. Individual Zernike modes are multiplied with a coefficient, giving a specific polynomial. The sum of the scaled wavefront is then fit into the real wavefront substituting with specific Zernike modes. W(ρ,ω) = Σ(C m n*z m n) (3) Shift in local values were calculated upto 4 th order mode (i.e., Z to Z ) and it was reconstructed using Modal wavefront estimation technique. 2.3 Refractive Error computations Refractive errors are characterized by sphero-cylinders 6. The second order mode representing the sphero cylindrical errors are assigned as Z 2, Z -2 2 and Z 2 2 respectively for defocus, astigmatism (4/3) and astigmatism (/8). The Zernike coefficients of the modes Z 2, Z -2 2 and Z 2 2 are C 2, C -2 2 and C 2 2 respectively. The Zernike components are converted to power vectors. The power vector components are M, J 4, and J 8. M is the spherical equivalent power and J 4/3 & J /8 are the powers of Jackson cross cylinder with axis at 4(or3) and (or8) degree. Power vectors are expressed in diopters. M = C 2/ r 2 (4) J o = C 2 2/ r 2 () J 4 = C -2 2/ r 2 (6) Where r is pupil radius and J is the total astigmatism, given as J = (J o +J 4 ) (7) The mean magnitude, which is the length of the power vector, also otherwise called as blur strength, is computed as m = (J M 2 + J 2 ) (8) The Square root of the area under the squared power profile gives the Root mean square value of the refractive error as RMS = (J 2 4/2 + M 2 + J 2 /2) The values M, J o, and J 4 are converted to Sphere (S), Cylinder (C) and axis (A) respectively as C = 2 (J J 2 ) (9) S = M C/2 () A = ½ tan - (J 4 /J ) () 3. RESULTS AND DICUSSION Fig. 2 shows the value of Zernike terms for all subjects measured for 6mm pupil size in diameter and Fig.3 shows the Zernike distribution across the eyes of the subjects of age group between 2 to 3 years. Apart from Spherical aberration, the means of all the aberrations were almost zero. It is found that 9% of the total aberrations were contributed by the second order aberration in our study group. Excepting defocus, the other significantly found aberration among our subjects was the Astigmatism with the rule. Aberrations decline with the increase of order considerably, except for Spherical aberration in the fourth order whose contribution is greater than any third order aberrations. As found in previous studies the inter-subject variability were prominent. The Intra subject study between the right and left eyes saw a near coherence in their aberration pattern as shown in fig.4. But the root mean square error for all subjects, as in fig.(), shows an apparent variability in RMS error between the right and the left eyes of the same subject. We could not find even one subject with same RMS error in both eyes. 3

4 Zernike modes ZERNIKE POLYNOMIALS 4 ZERNIKE DISTRIBUTION Fig.2: Zernike values upto fourth order mode Fig.3: Overall Zernike distribution for subjects ZER NIKE O RD ER ZER-L ZER-R RMS SUBJECT Fig.4: Plot between RMS error and Zernike order Fig.: RMS error for all subjects between Right & Left Eye The second order Zernike data were converted to Power vector description and then to Sphere, cylinder and axis. Defocus and Primary Spherical Aberration were plotted (fig.6), which establishes a slender negative correlation with the coefficient of correlation(r) of -. implying a slight decrease of defocus with the increase in spherical aberration. A scatter plot of Astigmatism with the rule between the right and the left eye show a downward trend suggesting a notable negative correlation (R=-.7) (fig.7). In the case of Oblique Astigmatism, the trend line is upward denoting a marginal increase, a positive correlation with R=.2 (fig.8). Defocus correlated between the right eye and the left eye shows a trend line horizontally straight across for all the subjects, suggesting a zero correlation. (fig.9). More near the value of R to, stronger is the correlation. C(2,) C(4,) Astigmatism Left Eye Z(2,2) Astigmatism Right Eye Z(2,2) Fig.6: Plot between Spherical and defocus Zernike terms Fig.7: Plot for Astigmatism with the rule between R & L eye 4

5 Astigmatism Left Eye Z(2,-2) Astigmatism Right Eye Z(2,-2) Defocus Left Eye Z(2,) Defocus Right Eye Z(2,) Fig.8: Plot for oblique Astigmatism between R & L eye Fig.9: Plot for defocus between R & L eye Analysis to find any correlation between the refractive errors and the higher order aberration failed with out any significant relationship in both the eyes of all the subjects (fig). It only infers the uniqueness of variability of each eye. Few studies earlier have presented wavering results in regard to the correlation between the lower and higher order aberrations. However it was found that when defocus was corrected, spherical aberration reduced significantly 7. Also, to compensate for an aberrated wavefront in a highly spherical aberrated eye, defocus need to be increased considerably. An analysis of Total astigmatism, J, with the Mean Spherical Equivalent, M, gives a negative correction with the coefficient of correlation of.3(fig.), where as the plot between two astigmatic terms, with the rule and oblique (J4 & J8), projects a positive trend line indicating a not so significant correlation between the two astigmatism terms with R =.3. (fig.2). A perfect positive correlation (R =) was found between RMS error and Spherical Equivalent denoting a strong linear relationship as shown in figure 3. Higher Order Aberration Z(4,) Lower Order Aberrations (D) C(2,-2) R C(2,2) R C(2,) R C(2,-2) L C(2,2) L C(2,) L Fig.: Plot between Lower order and Higher order Aberration. J Right Left....2 M Fig.: Plot between Total Astigmatism (J) and Mean Spherical Equivalent (M) for both R & L eyes J J8 Fig2: Plot between J8 and J4 Right eye Left eye RMS Error (Mic) R 2 = Spherical Equivalent (D) Fig.3 shows a plot of Spherical equivalent with RMS error

6 The Shack Hartmann aberrometry used in the measurement of aberrations for the analysis of refractive errors were studied for its validity and repeatability 8. It is established that the instrument designed and development in-house in the Photonics laboratory of National Institute of Technology was consistent in the repeated measurements statistically. The mean magnitude of the vector deviation was found to be.3 D and its mean RMS.4. The repeatability coefficient computed for the aberrometry was.26, which corresponds well with the 9% limit accord 9. It is vital to evaluate the instrumental error as the error might sometimes under estimate or over estimate the refractive errors. Aberration measurement from such instruments would make the corrective procedure s success rate look abysmal. The ways to achieve an optimum correction from the refractive errors has been seriously analyzed of recently. Is there a gold standard for the optical quality?. So far no method could in its totality fit this standard, though Subjective refraction is considered to be a precise and accurate method. For achieving Optimum correction, if we could accurately locate the point of nearly zero error somewhere between the object or image plane that would solve the problem. In that case, Far Point is defined as a location in the object plane that is optically conjugate to the fovea. In other words, for an aberration free eye, far point is the center of curvature of the spherical wavefront reflected from fovea. At this Far Point, the quality of the retinal image would be greatest. But, in the case of aberrated eye, the far point cannot be located accurately but only approximately by shifting the Far Point axially to and fro. By the statistical method of least square fit to wavefront on the basis of Zernike approach, the result is a far point vergence. As all the subjects in our study group have positive spherical aberration, the Far point vergence computed (.4 D) which when compensated with a suitable spectacle would focus the rays to infinity, called the hyper focal point where object vergence is zero. 4. CONCLUSIONS Sphero cylindrical error has been analyzed by power vector method using Shack Hartmann aberrometry. Inter-subject variability and intra-subject variability was studied. Analysis of refractive error with higher order spherical term was performed and found that defocus slightly influences the spherical aberration. The increase of Mean spherical equivalent with RMS error was revealed. Our experimental device, the Shack Hartmann aberrometry, was evaluated for instrumental error in measuring the aberrations of human eye and the potion for finding optimum correction has been suggested. REFERENCES. J.Liang, B.Grimm, S.Goelz, and J.F.Bille, Objective measurement of wave aberrations of the human eye with the use of Hartmann-Shack wavefront sensor, Journal of Optical Society of America A,, 949 (994). 2. L.N.Thibos, W. Wheeler, and D.Horner, Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error, Optometry and vision Science, 74, 367, (997). 3. P.M.Prieta, F.Vargas Martin, S.Goelz, and P.Artal, Analysis of the performance of the Hartmann-Shack sensor in the human eye, 7, 388(2). 4. W.H.Southwell, Wavefront estimation from wave-front slope measurements, Journal of Optical Society of America A, 8, 998, (98).. J.Y.Wang, and D.E.Silva, Wavefront interpretation with Zernike polynomials, Applied Optics, 9, (98). 6. L.N.Thibos, X. Hong, A.Bradley, and X.Cheng, Statistical variation of aberration structure and image quality in a normal population of healthy eyes, 9, 2329(22). 7. X. Cheng, A. Bradley, X.Hong, L.N.Thibos, Relationship between Refractive error and Monochromatic Aberrations of the Eye, Optometry and Vision Science, 8. M.Rosenfield, and N.N.Chiu, Repeatability of Subjective and Objective refraction. Optometry and Vision science, 72, 77(99). 9. K.Naeser, and J.Hjortdal, Multivariate analysis of refractive data: mathematics and statistics of spherocylinders, J Cataract Refract Surg, 27, 8(2). 6

Pantoscopic tilt induced higher order aberrations characterization using Shack Hartmann wave front sensor and comparison with Martin s Rule.

Research Article http://www.alliedacademies.org/ophthalmic-and-eye-research/ Pantoscopic tilt induced higher order aberrations characterization using Shack Hartmann wave front sensor and comparison with