ABCD Analysis of Human Eye

In the following passages, try to capture my journey to understand some of the basic optical properties of eyes. Started with some well known papers and books — listed in the Reference section. However, during the process the need for a more comprehensive, more accurate, modern and computer oriented method became apparent. Hence, adapted the commonly used ABCD matrix formalism of ray optics to examine a number of well known ophthalmological/optometrical methods. Namely — Red Reflex, Direct Ophthalmoscopy, and Indirect Ophthalmoscopy. The ABCD approach appears to work very well, and corroborates the numbers quoted in [kaschke] well.

Optical Components of Eye

Optically eye is composed of a number of refracting surfaces ( or lenses ) as shown in the following figures [aao, osa]. This particular model was derived by Gullstrand in early 20th century, he was awarded the Nobel Prize for his works in ophthalmology.

Figs 1: Gullstrand’s schematic eye model [aao].

The following figure shows a cross section of major parts of eye with labels. Why the structure is like that has probably has no concrete answer yet. Nevertheless, some parts have strong analogy with cameras. The iris is the shutter, retina is the sensor array (or negatives in old cameras). Like in a complex camera, there are many refractive surfaces — from left to right — cornea, aqueous humor, lens, vitreous body.

Fig 2: Cross section of the eye

The following table shows the refractive properties and dimensions of Gullstrand model. The powers are in diopters — which is the inverse of the focal length in meters.

Fig 3: Refractive properties and dimension of the Gullstrand’s schematic eye [aao].

Optical Issues of Eye

It is obvious that in such a complex system as the eye, many things can go wrong in every parts. Hence, there are many diseases of the eye. For example, the lens and cornea may not be uniform unlike a brand new camera lens and can be out of shape in many different ways. However, the system is fairly accommodating and can compensate for many of these imperfections. A common problem with any eye issue is that it is not possible to capture a picture of what the person is seeing — like pain — can only be verbally or pictorially described.

As is well known, the most common eye issue is refractive and is related to focusing. Some imperfections of focusing can be accommodated by conscious or sub-conscious flexing of the eye muscles. These muscles are essentially the auto-focus system of the eye — auto-focus is achieved by changing the shape of the lens. When it is no longer feasible for the muscle to compensate, eye glasses can provide additional compensation.

Another common eye issue is called cataract of the lens. Cataract is essentially opaque deposits in the eye and cause plethora of vision artifacts beside reducing eye sight. When the cataract is not so dense that effectively all lights are blocked, the vision artifacts seen by the effected can be due to diffraction, reflections, pin-hole etc optical phenomena caused by the opaque deposits. That may result in seeing multiple images is various combinations — similar to multiple images projected by tree leaves or other openings during a near total solar eclipse as seen in the following figure.

Fig 4 : Multiple images of the eclipsed sun through the leaves of a tree.

As the eye is a multi-element optical system — cornea, aqueous humor, crystalline lens, and vitreous humor, visual artifacts similar to multi-element camera system can also be perceived by the effected. Ghost images and flares [panasonic] can be observed if the the optical element axis is misaligned. Misalignment can occur for many reasons including cornea or lens shape deformation, eye muscle changes, cataract reflections, etc. Something like the following figure can the perceived. There are plethora of ghost image related eye diseases related to optics of the cornea and lens [aao-dip]. However, the exact optical mechanism of monocular diplopia is still not readily simulateable [springer]. Some works based of point spread function (PSF) has been done to show emergence of double images. PSF describes the intensity at the image as function of x,y and z co-ordinate. The spreading has been attributed to spherical aberration, coma aberration etc.

Fig 5: Ghosting ( to lower left of the bright sun ) and flaring in a multi lens point and shoot camera ( Canon PowerShot N2) .

There are 3 well known optics phenomenon that can cause multiple images — reflection, polarization and prism/refraction. Photographers sometimes use filters called multi-image/kaleidoscope filters that are essentially a number of prisms of various thickness. The following figure shows how adding a small prism/wedge, thus deforming the aggregate shape of the lens, results in 2 images. This type of phenomenon can occur in the eyes too, when the shapes and refractive indices change for some reason.

Fig 6: Formation of multiple images when a prism partially coves s lens.

Structural issues of eye can become evident only in low light condition when the pupil is dilated and hence more of the lens etc are exposed.

Optical Techniques Used for Eye Examination

Given the importance of eye and its relative transparency, there is immense interest in trying to capture various types of optical images — plain photography, blue/IR photography, Optical Coherence Tomography (OCT), aberrometer to name a few.

On the surface, it should be simple to photograph parts of the eye all the way into the retina. However a few things make this difficult. The smallness of the pupil ( 2 mm to 5 mm diameter ) and lack of light inside the eye are probably the most difficult to overcome. The only way to get light into the eye is through the pupil, and the pupil shrinks in bright light making it even harder. A relatively close analogy is trying to capture a picture of the sensor array of a camera when the front aperture is small.

Lower order aberrations ( e.g. myopia, hyperopia, and astigmatism ) of eye optics are easily identified by tests such auto-refractometer or optometrist manually changing lenses and working co-cooperatively. Higher order aberrations ( e.g. halo, coma, trefoil) [sciencedirect] can have significant impact on vision quality and requires more specialized equipment to identify and correct. The following figure show some common aberration of the eye .

Fig 7: Visual perception of common aberrations of eye [rochester]

One commonly used technique to measure aberrations is called Hartmann-Shack (HS) sensor [rochester, aao-wave, um.es]. The following figure shows the major part of such a device — sensors, lenslets, light source [rochester]. There is also an inverse HS sensor, that replaces the sensor array with a display and asks the subject to align points.

Fig 8: Hartmann-Shack sensor [rochester].

A good reference of optical tools and instruments for ophthalmic/optometric use is [kaschke], however the math notation in the book is unusual and cryptic.

Optical Simulation of Eye

There have been some efforts to come up with accurate models of the eye to explain and treat many of the vision issues of optical origin. More complex models have been proposed [wiley, nih, nature].

The simplest tests done by ophthalmologists and optometrists to look into the internals of the eye are Direct Ophthalmology and Indirect Ophthalmology. Direct Ophthalmology requires the observer to be very close to the eye under test. Indirect Ophthalmology places a condensing lens between the eye under test and the observer. As an aside, it is not easy to perform Direct and Indirect Ophthalmology on oneself as the light focused on retina can be blinding.

To get a feel for what what type of lens and distances are needed for Indirect Ophthalmology, one can set up a ray tracing simulation. There are a number of software available for ray tracing simulation. The most well known is Zemax, which unfortunately is neither open source nor free . For rough and quick estimates, Python package RayTracing [raytracing] and Pyrate [pyrate] are available. Pyrate seems to be somewhat complicated — it has a purported integration with FreeCAD — which seems to not work. RayTracing although easier to use, appears to be very limited in functionality. Among free (not open source ) software there are WinLens [winlens] and Oslo.edu [oslo.edu], both of which are windows based. Oslo.edu’s software was found to be problematic by antivirus software hence was not able to use. WinLens is a fairly good starter software for simulations of simple situations. It is not able to do any image formation simulation.

Red Reflex

Red Reflex [utah.edu, eyeguru, wprr, wpre, aao-rr, nihdo] is the first landmark that is looked for in both Direct and Indirect Ophthalmology. It is similar to the Red Eye effect occasionally seen in flash photography. Red Eye used to be much more prevalent in old days when flashes did not correct by emitting at least 2 flashes. The first flash, called pre-flash, constricts the pupil and then when second flash flashes when the pupil is constricted and hence mostly prevents Red Eye. Both Red Reflex and Red Eye are apparently caused by the light reflected back from the back of the eye —retina and the choroid. From the point of view of photographing eyes, the modern camera flashlights are not well suited because of the pre-flash — there appears to be no easy way to disable the pre-flash without having to develop low level software which may or may not access to the underlying hardware.

Red Reflex is a consequence of optical conjugate ( e.g. image’s and object’s position can be swapped ). Light gets focused on to the retina and part of that light gets reflected back through the eye’s lens etc and focused back to the light source. Hence the light source and the viewer’s eyes should be as close as possible to see Red Reflex. The intensity of Red Reflex is determined by a number of factors — the intensity of source , the diameter of the pupil, the reflectivity of retina and other parts of the back of the eye, the transitivity of the parts of the eye. The most critical factors are, pupil diameter and closeness to the normal at the center of the pupil. Light transmitted increases linearly with area of the pupil or square of the diameter.

Fig 9: Red Reflex with glass ball

For Red Reflex practice, an eye model can be used [nih-em]. However just to get a feel, it is possible to construct a rather simple setup with just a glass sphere ( e.g. marble or ball lens) of about 15 mm or larger diameter, square red paper, and pieces of square paper with a hole anywhere form 2 mm to 8 mm diameter corresponding to pupil diameter. The hole ideally should be circular, but any shape works. The pieces of papers should be larger than the diameter of the sphere, so as to cover it completely. Place the glass ball at the center of the red paper. The sphere could be raised by half of the radius, so that focal point is at the red paper. Place the paper with the hole on the top of the sphere on the other side from the red paper, see above figure. Now hold a small torch near eye, and look into sphere through the hole. The appropriate distance from the glass sphere depends on the size of the sphere and can be found easily by starting from up close to the sphere and moving away. Bright red in the hole should be visible after a certain distance. If too close, then there is not going to be any reflex as light gets de-focused over a larger area of the red paper, hence less intensity. From distance of about 10 times the diameter of the ball, the light will essentially be focused into the red paper.

The focal length from center of sphere is given by the following formula.

f = 2nR/4(n-1)

For glass with refractive index 1.5, f becomes 1.5R . The Lens Formula gives the image distance (v) in terms of object distance (u) and focal length (f)

v = fu/(u-f) = f*mf/f(m-1) = mf/(m-1) ~ f , when m is large for u = mf

As object and its image are conjugate, the image of the illuminated area of the red paper forms at around the torch lamp. If the cone of light coming back from the other end of the sphere is bright enough and wide enough to cover the eye also, Red Reflex becomes visible to the observing eye also. The cone of reflected light is determined by the pupil size (area).

For Red Reflex to be visible a number of conditions have to be satisfied. The following figure shows a simplified model of Red Reflex. The light source has to be far enough to be focused on the retina — assuming a fully corrected lens or emmetropic eye. The pupil diameter has to be wide enough to allow a cone of light that covers both eye and the light source. The rendition on the right of the following figure tries to capture what actually gets viewed as Red Reflex.

Figure 10: Simplified optics of Red Reflex.

A simple formula for the distance between the light source and the observer lens and pupil diameter is a triangle ratio as follows. The formula ignores bending of rays at exit and entrance of the glass sphere due to refraction, and hence the collimation effect. The cone of light in reality will have an intensity distribution being highest at the center then tapering off very quickly. Hence the following formula is likely a significant overestimation.

<< (u+2.5r)p/(5r)

In the simple emulation setup above, a 60 diopter convex lens in place of sphere can also be used. However, in that case, the lens has to be separated from the paper by about 16 mm. Also, it appears that glass spheres are lot cheaper than 60 D lenses.

Direct Ophthalmoscopy

Direct Ophthalmoscopy (DO) was invented by Helmholz in 1851 [nih-do]. In DO, only the lens of the eye of the observed is used and the observer uses just their eyes — if we ignore any correction needed for both the observed and observer eyes’ refractive errors (i.e both are emmetropic) . The following figure shows the construction of a modern Direct Ophthalmoscope [timberlake]

Fig 11: Consctruction of a Direct Ophthalmoscope [timberlake]

The magnification in DO can be easily obtained by the lens equation as D/4 — where D is diopter of the observed’s eyes — for a perfect/emmetropic eye, it is 60 [24]. The 4 in denominator comes from dividing 100 cm by the near point distance, 25 cm, used for reference as being the minimum distance where human eyes can form an image.

Fig 12: Magnification of Direct Ophthalmoscope [libretexts]

The derivation of the magnification is as follows.

linear magnification = 𝑚=−𝑑𝑖/𝑑𝑜=-ℎ𝑖/ℎ𝑜

angular magnification = 𝑀 = θ𝑖𝑚𝑎𝑔𝑒/θ𝑜𝑏𝑗𝑒𝑐𝑡 = ℎ𝑖(25𝑐𝑚)/𝐿ℎ𝑜

= (−𝑑𝑖/𝑑𝑜)(25𝑐𝑚/𝐿) = (1−𝑑𝑖/𝑓)(25𝑐𝑚/𝐿) = (25𝑐𝑚/𝐿)(1- (𝐿−ℓ)/𝑓)

at L = ∞ , M = -25cm/f = -(100/f)/(100/25) = - D/4

A more modern way is to use spreadsheet and ABCD formulation [mit.edu, utk.edu, cern ] as shown in the following figure. It is pretty easy to see that the linear magnification remains constant at -1 irrespective of the distance between the observed and the observer. Hence the image size seen by the observer also remains constant irrespective of the distance. This makes DO much easier to perform than IO ( described later). Of course, as distance increases intensity/brightness falls off inversely to the square of distance — hence very rapidly.

Fig 13: Using spreadsheet and ABCD formulation for DO (angle and height are transposed, hence DCBA).

DO suffers from a number of problems — closeness of the observed and observer, small field of view ~ 5 degrees due to high magnification and small aperture (pupil) of the observer eye lens. The field of view for DO purpose is quantified as how much of the back of the eye can be viewed by changing relative positions of the observer and the observed.

For analysis, the observed eyes can be considered a magnifying lens or loupe. With that approximation, it is possible to get an expression for linear field of view (LFOV). The top part of the following figure pictorially shows what is the LFOV. The bottom figure shows the situation with a thin lens.

Fig 14: Linear filed of view of a single lens and DO.

The LFOV for a single aperture without any lens can be expressed as follows.

lfov = da/l * (do+l) = da * (1+do/l)

Introduction of the lens in the aperture changes the path of rays. In this case, to the observer eye, the rays appear to emanate from a point in the image plane. Hence, we can just use the image distance, di, in the expression for LFOV for a simple stop aperture above. This provides the LFOV in the image plane. To get the object plane LFOV, we apply the equation for linear magnification from before.

lfov’ = da* (1-di/l) , positive/negative distance for real/virtual image

lfov’/lfov = di/do => lfov = -lfov’ * do/di => lfov = -da*do/di * (1-di/l)

lfov =-da*( do/di -do/l) = -da/l * do * (l/di-1) =da/l * (1-l/di)/(1/f — 1/di)

The last expression is same as given by [kaschke] for loupe. If the linear magnification of the lens can considered high compared to do/l , then the expression simplifies to following.

lfov ≈da/m

Let us try to see if ABCD matrix can also provide us with the LFOV. The ABCD matrices for the above is space of length do, followed by lens of 1/f , followed by space of length l. The following figure shows each of the 3 matrices and the final matrix.

Fig 15: ABCD matrices for DO

LFOV can be expressed as follows also. The last part in parenthesis is exactly same as the B element of the ABCD matrix.

lfov = da/l * (l+do — do*l/f) ~ da*f/l , when do ≈ f

lfov ≈da*4/D, when l ~ 25 cm .

The B element is the contribution to x (height) of the ray at the observer eye due to angle of the beam. Plugging in the numbers for B and l from the spreadsheet, lfov comes out to be = da*16/250 =da/15.625.

In [Kaschke] the DO FOV is given as follows. Not exactly sure how it was derived and what assumptions were made. Specially the numerator, addition of pupil diameters, is hard to comprehend!

angular field of view = αfov = (dpupil +dphys)/L ,

diameter of the retina visible = dfov = αfov/Deye

Another way to look at the field of view is how the cornea+lens and pupil interacts. Assuming there is no cornea+lens , the field of view becomes the hole of the pupil — measured by its diameter [Timberlake]. When viewed through the eye’s refractive system, the retina is enlarged by the magnification of the eye. Hence approximate diameter of the retina visible through the pupil is as the following equation.

fov = dpupil/M = 4*dpupil/D

Which for a 2 mm pupil of an emmetropic eye becomes = 2000/15 ~ 133 micro-meter in diameter.

Red Eye is DO at distance, where the observer eye is replaced by a camera. The following figure shows the ABCD formulation in spreadsheet for Red Eye. The approximate camera specs are taken from [arizona.edu] — 5 mm focal length and 5 mm distance between images and lens.

Fig 16: Red Eye at a smartphone camera with ABCD formulation (angle and height are transposed in the above sheet, hence DCBA).

The linear magnification from the above figure is -0.3125. So a 2 mm section on the observed retina casts a 0.625 mm image on the camera sensors. A typical smartphone sensor is about 1 cm and has 1080+ pixels. Given that Red Eyes should cover more than half of the sensor or should appear over half of the picture. The reason that is not the case is the aperture of the pupil. The pupil aperture restricts visible area and the amount of light that can come out of eye. Hence further away the camera is, the less pixels the eye covers in the camera sensors. The lfov is inversely proportional to the distance as was shown previously for a single magnifier. Red Eye from up close should reveal structures within the eye. However. the separation of the flash LED from the camera lenses makes very close distance to the eye not work well.

Indirect Ophthalmoscopy

Indirect Ophthalmoscopy (IO) is considered an improvement upon DO. IO offers wider field of view as the condenser lens can be moved very close to the observed eye, more comfortable distance between observed and observer, depth perception with stereoscopic ophthalmoscopes, ability to project more light into the eye, ability to collect more reflected light. The following figure shows the important parts of IO.

Fig 17: Indirect Ophthalmoscopy.

In IO, a 20 to 30 D lens, condenser lens, is placed close to the observed eye. The combination of the 2 lenses produces a real inverted image of the retina in front of the observed eye, which then is viewed by the observer.

To model IO for spreadsheet computation, consider the following ABCD matrices — S4, L3, S3, L2, S2, L1, S1. Where Lx and Sx respectively represent lens (thin) and space.

Fig 18: ABCD matrices for Indirect Ophthalmoscopy

In the IO example in [kaschke] — distance between L2 and L3 is 56 mm , distance between L1 and L2–450 mm is used. The following figure shows the ABCD spreadsheet after plugging in the numbers. L1 and L3 are assumed to be emmetropic (i.e. 60D lens), and S1 and S4 are assumed to be air of 16 mm, L2 is assumed to be 20D thin lens.

Fig 19: ABCD matrix calculation of Indirect Ophthalmology (angle and height are transposed, hence DCBA).

Playing around with the number in the spreadsheet it was found that image formation in the retina of the observer is more sensitive to eye focal length and distance to retina than other parameters.

The linear magnification at the retina of the observer is 0.124 vs -1 in case of DO , about 8x less.

The lfov, can be similarly estimated from the B element of the ABCD matrix upto the eye. Plugging in the numbers for B and l from the spreadsheet, lfov comes out to be = da*133.33/450 =da/3.37 . About 4.6x higher than DO.

The following figure shows the ABCD matrix for IO with smartphone in place of eye. The major difference here is the distance between the condenser lens and the camera is almost half — 194 mm vs 450 mm.

Fig 20: ABCD calculation of IO with smartphone camera.

The ABCD formulation can be a easy and powerful tool to analyze and understand aspects of optics involving eyes. It is probably possible to simulate some form of image formation at retina with existing software and measurement techniques. However it does not look like any work flow currently exists for such work. In not too distant future, it should be possible to accurately predict the images formed on retina, and hence provide insights into what some one is or should be seeing.

References:

1. [aao] 020–2021 BCSC Basic and Clinical Science Course, https://www.aao.org/bcscsnippetdetail.aspx?id=30bf26c8-801d-40bb-8bc6-9dea73fd1adf
2. [osa]Optics of the eye and its impact in vision: a tutorial, https://www.osapublishing.org/aop/fulltext.cfm?uri=aop-6-3-340&id=301593 .
3. [panasonic] Lens Characteristics: Flare, Ghosting and Aberration, https://av.jpn.support.panasonic.com/support/global/cs/dsc/knowhow/knowhow15.html
4. [aao-dip] Diplopia , etc , https://www.aao.org/focalpointssnippetdetail.aspx?id=8eb21322-ab3c-49f9-a772-b9ba8bd9783f
5. [springer] Monocular Diplopia, https://link.springer.com/referenceworkentry/10.1007/978-3-540-69000-9_645
6. [sciencedirect] Effect of third-order aberrations on dynamic accommodation, https://www.sciencedirect.com/science/article/pii/S0042698906003488
7. [rochester] Ocular Wavefront Sensing, https://www.cvs.rochester.edu/yoonlab/research/ows.html
8. [aao-wave] https://eyewiki.aao.org/Wavefront_Testing
9. [um.es] Assessment and Hartmann-Shack Aberrometry, https://lo.um.es/lo.um/wp-content/uploads/2010/01/2010_JRS_Cause-of-monocular-diplopia_GM-Perez.pdf
10. [kaschke] Optical Devices in Ophthalmology and Optometry, https://www.google.com/books/edition/Optical_Devices_in_Ophthalmology_and_Opt/UKQTAgAAQBAJ
11. [timberlake] The Direct Ophthalmoscope How it Works and How to Use It, https://web.media.mit.edu/~raskar/Eye/TheDirectOphthalmoscope.pdf
12. [wiley] Optical models of the human eye, https://onlinelibrary.wiley.com/doi/full/10.1111/cxo.12352
13. [nih] The Optical Design of the Human Eye: a Critical Review, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3972707/
14. [nature] Simulation of visual acuity by personalizable neuro-physiological model of the human eye, https://www.nature.com/articles/s41598-019-44160-z
15. [raytracing], https://github.com/DCC-Lab/RayTracing
16. [pyrate], https://salsa.debian.org/mess42/pyrate/
17. [winlens], https://www.qioptiq-shop.com/en/Optics-Software/Winlens-Optical-Design-Software/Free-Winlens-Basic/Free-software-Winlens-Basic.html
18. [oslo.edu] , https://www.lambdares.com/edu/
19. [utah.edu] HOW TO USE THE DIRECT OPHTHALMOSCOPE, http://morancore.utah.edu/basic-ophthalmology-review/how-to-use-the-direct-ophthalmoscope/
20. [eyeguru] How to use the indirect: Key tips and tricks, https://eyeguru.org/essentials/indirect-ophthalmoscope-tips/
21. [wprr] Red reflex, https://en.wikipedia.org/wiki/Red_reflex
22. [wpre] Red eye effect, https://en.wikipedia.org/wiki/Red-eye_effect
23. [aao-re] Photos Can Help Diagnose Children’s Eye Problems and Save Sight, https://www.aao.org/eye-health/tips-prevention/diagnosing-children-from-photographs
24. [nih-do] The demise of direct ophthalmoscopy, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4404284/
25. [libretexts] The Simple Magnifier, https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.08%3A_The_Simple_Magnifier#:~:text=M(L%3D%E2%88%9E)%3D,length%20gives%20a%20stronger%20magnification .
26. [osa-uwf] Understanding the relationship between visual-angle and eye-angle for reliable determination of the field-of-view in ultra-wide field fundus photography, https://www.osapublishing.org/boe/fulltext.cfm?uri=boe-12-10-6651&id=460079
27. [nih-em] A simple eye model for practicing indirect ophthalmoscopy and retinal laser photocoagulation, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6488265/
28. [AAOOI] 2020–2021 BCSC Basic and Clinical Science Course, https://www.aao.org/bcscsnippetdetail.aspx?id=ab5ac44c-47e7-4540-a80e-5b894a1063c4
29. [ncsu.edu] https://cvm.ncsu.edu/wp-content/uploads/2016/05/Heinrich_ophthalmoscopy-16-notes.pdf
30. [oculist] Chapter 63, Principles of Ophthalmoscopy, AUGUST COLENBRANDER, http://www.oculist.net/downaton502/prof/ebook/duanes/pages/v1/v1c063.html
31. [reviewofop] Condensing Lenses: Sharpen Your Skills in Choosing and Using, https://www.reviewofoptometry.com/article/condensing-lenses-sharpen-your-skills-in-choosing-and-using
32. aao-bio] Binocular Indirect Ophthalmoscopy, https://eyewiki.aao.org/Binocular_Indirect_Ophthalmoscopy
33. [utk.edu] Modern Optics, http://electron9.phys.utk.edu/optics421/
34. [mit.edu] http://web.mit.edu/6.161/www/Geometric-Optics-9-07.pdf
35. [arizona.edu] https://wp.optics.arizona.edu/jsasian/wp-content/uploads/sites/33/2018/11/Mobile-phone-lenses-JS.pdf
36. [cern] https://indico.cern.ch/event/266133/attachments/474621/656940/Gillespie_Aachen_CP_vs_P_optics.pdf