Modeling of systems that change the polarization state of light

Polarized light is widely used in optical systems for various purposes. Our retinal birefringence scanning systems measure the property of parts of the retina to change the polarization state of light, which is called “birefringence”.

 

 

 

 

 

 

 

The principle of Retinal Birefringence Scanning for detecting central fixation and eye alignment.

Polarized near-infrared light is reflected from the foveal area in a bow-tie pattern of polarization states; the pattern is similar to the faint pattern observed surrounding the point of fixation when a subject views a clear background through a polarized filter (the Haidinger brush phenomenon 6). With our previous instruments, the foveal area was probed with a circular scan of frequency  f .  When the eye fixated on a point at the center of the circular scan, the double-pass polarization state of the light changed at twice the frequency of the scan (2f).  With paracentral fixation, the change in the polarization state was only at the frequency of the scan  (f).  

 

 

An early version of the RBS scanner – basic principle

 

 

The model developed in our lab describes mathematically retinal birefringence scanning (RBS) in a double-pass system using Stokes vector analysis and Mueller matrix multiplication. The cornea was modeled as a linear retarder, whereas the foveal area was modeled as a radially symmetric birefringent medium. The model has been shown to accurately predict the frequency and phase of RBS signals obtained with our eye fixation monitor during central and paracentral fixation. The method is based on analysis of polarization changes induced by the retina and cornea, and by reflection from the fundus:

                  

where  S  is the 4-element Stokes vector, and  M  is the 4x4 Mueller matrix, whose values are functions of the azimuth  θ  and the retardance  δ  of the corresponding retarder. 12   Since we use vertically polarized light, the incident Stokes vector is Sin = {1,-1,0,0} (unit intensity beam used). In the above equation, the subscript  r  stands for retinal, while the subscript  c  stands for corneal.  Thus, the polarization state of light is altered when light passes  through the cornea,  then through the retina (the birefringent radially symmetric Henle fibers surrounding the fovea), and is then reflected by the deeper layers of the retina, and returns through the retina and cornea. The typical values for  θr ,  δr , θc  and  δc  are given in publications by our laboratory.   During a scan, the values of  θr   and  δr   at each moment depend on the position of the scanned spot of light on the nerve fiber layer surrounding the fovea.  The azimuth  θr  of the retinal retardance depends on the orientation of the fibers at the point being scanned, while the retinal retardance  δr  depends on the distance from the foveal center (we used the radial function from Eq.1 for this purpose).  The ocular fundus at our wavelength (780 nm) exhibits a high degree of polarization preservation 2 and can be modeled with a good approximation as a reflective surface. The  Mrefl is

 

 

which simply changes the sign of Stoke vector elements S2 and S3.  The 180º phase change at reflection thus reverses the sign of the azimuth and the handedness of the reflected polarized light. In the Mueller matrices for the retina and cornea for the return pass, (θ)  has been replaced by   (-θ).  The model provides all four elements of the output  Sout , thus enabling comparison among the utilities of the four Stokes components for retinal birefringence scanning.  All calculations in this model are performed in MATLAB.

The full model, as presented in the illustration below, includes also retarders (WP=wave plate; HWP=Half-Wave Plate, etc, some of which may be rotating).

 

 

 

Stokes representation of the polarization state of light (Please see also the above illustration).

 

 

Signal strength variations for variations in corneal and retinal azimuth and retardance

 

 

The overall bow-tie light intensity function can be approximated with the following formula:

                                                                 

 

where q  is the azimuth relative to the fast axis of the Henle fiber birefringence, q=atan(yb/xb),  and   r = (xb2+yb2)½   is the distance from the origin of the bow-tie distribution in millimeters, on the same scale as the bow-tie image on the surface of the photodetector.  The exponents that give the closest match to the profile shown in Fig. 6 are as follows:   τ1 =  3.7,  τ2 = 50.0,  τ3 =  0.6, τ4 =  5.0,  and   τ5 =  0.8.  The model uses millimeters in the detector plane (1.96 mm/degree of visual angle).

 

The computer model can be used for optimizing the parameters of optical systems that use polarized light. An example of employing the computer model for maximizing the amplitude of the RBS signal as a function of corneal retardance and corneal azimuth is given below, where Panels A and B represent two different designs.

 

References:

1.      Gramatikov B.I.,  Zalloum O.H.Y., Wu Y.K.,  Hunter D.G.,  Guyton D.L.      A Directional Eye Fixation Sensor Using Birefringence-Based Foveal Detection.   Applied Optics, Vol. 46, Issue 10, pp. 1809-1818,  April 2007,

https://www.osapublishing.org/vjbo/fulltext.cfm?uri=ao-46-10-1809&id=130944

2.      Gramatikov B.I.,  Zalloum O.H.Y., Wu Y.K.,  Hunter D.G.,  Guyton D.L.      Birefringence-based eye fixation monitor with no moving parts.  Journal of Biomedical Optics, 2006, 11(3), May-June, pp. 034025-1 - 034025-11,

 http://www.ncbi.nlm.nih.gov/pubmed/16822074

  1. Irsch, K, Gramatikov, B.I., Wu, Y-K, Guyton, D.L.  Improved eye-fixation detection using polarization-modulated retinal birefringence scanning, immune to corneal birefringence. Optics Express,  22(7):7972-7988, published on 18 March 2014, DOI:10.1364/OE.22.007972, http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-22-7-7972

http://www.opticsinfobase.org/oe/fulltext.cfm?uri=oe-22-7-7972&id=282349

4.       Irsch,K., Gramatikov,B.I., Wu, Y. K., Guyton,D. L. A new pediatric vision screener employing polarization-modulated, retinal-birefringence-scanning-based strabismus detection and bull’s eye focus detection with an improved target system: Opto-mechanical design and operation.  Journal of Biomedical Optics, 2014, Jun 1;19(6):67004. doi: 10.1117/1.JBO.19.6.067004.

http://biomedicaloptics.spiedigitallibrary.org/article.aspx?articleid=1881172