Lecture # 7 Geometrical Optics. Lenses. Optical system of human eye

1. Lecture # 7Geometrical Optics. Lenses. Optical system of human eye

2. Plan of the lecture • Some concepts and practical applications of geometrical optics • Thin lenses. Lens aberrations 3. Light conducting and light perceiving systems of human eye 4. Sensitivity of eye to colour and light 5. Human eye as an optical system 6. Eye defects

3. SOME CONCEPTS AND PRACTICAL APPLICATIONS OF GEOMETRICAL OPTICS • The operation of many optical devices applied in clinics, and medical and biological laboratories, is based on the laws of geometrical optics. • Geometrical, or ray, optics is the branch of optics that uses the concepts of light rays. • Light rayis an imaginary line, along which the luminous energy propagates. • In geometrical optics the wave nature of light is not taken into account, i.e. one may neglect the effects of light interference and diffraction. It is possible when the light wavelength tends to zero ( 0). Thus, geometrical (ray) optics is the ultimate case of wave optics on the assumption of a very small wavelength.

4. 1 2   n1 n2 Geometrical optics is based on the following laws: • The law of rectilinear propagationof light in an optically homogeneous and isotropic medium. • The law of independence of light rays: when light rays intersect, they do not perturb each other, but continue their propagation in the former direction. • The law of light reflection: the incident ray, the reflected ray, and the perpendicular (normal) to the interface of two media, which is at the point of ray incidence, are in one plane; the angle of reflection is equal to the angle of incidence . α‛

5. The law of light refraction (Snell’s law): the incident ray, the refracted ray, and the perpendicular to the interface, which is at the point of ray incidence, lie in one plane; the ratio of the sine of angle of incidence  and the sine of angle of refraction  is a constant value for these two media; this constant is called the index of refraction of the second medium with respect to the first one (the relative index of refraction ): α‛ β

6. The relative index of refraction n21is equalto the ratio of the absolute indices of refraction of the second medium (n2) and the first medium (n1), i.e. The absolute index of refraction of the medium indicates how many times the velocity of light in vacuum c is greater than that in the given medium v , i.e. Hence, the relative index of refraction shows how many times the velocity of light v1 in the first medium is greater than that of light v2 in the second medium:

7. As light passes from one medium to another one, its velocity of propagation and wavelength change, though its frequency does not. • If light is incident on the interface from the side of the optically denser medium (the medium with a higher absolute index of refraction)(n2<n1), the phenomenon of total internal reflectioncan be observed: light does not pass into the second medium, but is totally reflected from the interface. air water

8. Total reflection occurs when angle of light incidence is greater than the critical angle of total internal reflection: where n2 is the index of refraction of the medium, in which light passes with refraction; n1 is the index of refraction of the medium, from which light falls on the interface. • When light ray falls on the interface of a medium with air (n2≈1): where n is the absolute index of refraction of the given medium.

10. Functioning of optical fibresis based on the effect of total internal reflection. The branch of optics dealing with light and image transmission by optical fibres is called fibre optics. Optical fibres are transparent fibres encased in a substance whose index of refraction is less than that of the fibre. When light enters such a fibre, it reflects repetitively and propagates along the fibre.

11. Optical fibres are the key components of endoscopes and laparoscopes (special instruments for examining internal cavities - the stomach, bronchi, rectum, and others). Medicine harnesses laser radiation by transmitting it over optical fibres into the internal organs for healing tumours.

12. 2 n1 3 cr n2 1’ 2’ 3’ • When light passes from a medium with a smaller index of refraction (less denser medium) to a medium with a greater one (denser medium), the angle of refraction (β) is smaller than the angle of incidence (α). Since the angle of incidence is limited by 90°, the angle of refraction cannot exceed a critical value known as the critical angle of refraction(βcr). 1

13. The value of the critical angle of refraction is determined by formula where n1 and n2 are indices of refraction of the first and second media respectively. If the first medium is air, then n1≈1, and where n is the absolute index of refraction of the second medium.

14. Refractometeris an instrument for the direct determination of the refractive index of different liquids. It enables the identification of organic liquids to be made using the refractive index. • In medicine, refractometersare used for determining of the concentration of a substance in a solution (for example, the content of a protein in blood serum etc.).

15. THIN LENSES LENS ABERRATIONS • A key component of many optical instruments, devices is the lens. • The lens is a transparent body bounded by two spherical surfaces, which has an index of refraction differing from that of the environment. • The straight line passing through the centres of these spherical surfaces is called the principal optical axis of the lens. • If the lens thickness is far less than the radii of its bounding surfaces, the lens is called a thin lens. Further, we will deal only with thin lenses.

16. We can neglect the thickness of the thin lens and consider the principal opticalaxis intersecting the lens in point O. This point is called the lens optical centre. In passing throughthe optical centre of the thin lens O, no ray refracts. • All straight lines that pass through the optical centre of the lens at an angle to the principal opticalaxis are called auxiliary axes. Every lens has one principal optical axis and an infinitely large number of auxiliary axes.

17. A lens is called converging (convex), orpositive, if a pencil of parallel monochromatic rays passing along the principal optical axis after refraction in the lens intersect at the point lying on the principal axis at the other side of the lens. This point is called principal focus (or principal focal point) of a converging lens. There are two of these for a lens. • If parallel rays, after being refracted in the lens, diverge, the lens is called a diverging(concave), or negative, lens. In this case, the refracted rays do not intersect in the focus, but their virtual prolongations do. In so doing, the focus is at the same side of the lens as the pencil of rays incident on the lens. In this case, the focuses are called virtual.

18. Let n2 be an absolute refractive index for material of lens, and n1 be an absolute refractive index for the environment. Then for n2>n1 converging lenses may be double convex or plane-convex. They are thicker in the centre than at the edges. Diverging lenses may be double concave or plane-concave. They are thinner in the centre. For n2<n1 , the lens classification is opposite to that of n2>n1 .

19. Focallength (F) of a lens is the distance between the optical centre of the lens and the principal focus. • Focal power of the lens(D) is reciprocal to the lens focal length : • Unit of focal power is dioptres (dptr). • As evident, the shorter the focal length, the stronger the lens refracts, and value D being greater. Thus, the lens power characterises the lens refractivity.

20. Lens focal power (D): • where n is the relative index of refraction of the lens material with respect to the surrounding medium; • R1 and R2 are the radii of the spherical surfaces bounding the lens. • If the lens surface is convex, the corresponding R is taken with the sign "+"; if it is concave, it is taken with the sign "-". • If values D and F turn out positive, then the lens is converging; if they are negative, the lens is diverging. • The power of a system of tightly packed lenses (which can be considered as a unit lens), is equal to the algebraic sum of powers of all the component lenses, i.e.

21. Planes passing through the principal focuses and perpendicular to the principal optical axis are called focal planes of the lens. • If a beam of parallel monochromatic light rays falls on a converging lens, then after refraction in the lens all the rays will intersect at the point belonging to the focal plane. • If a beam of parallel monochromatic light rays falls on a diverging lens, then after refraction in the lens diverging beam of rays forms, and their virtual prolongations will intersect at the point belonging to the focal plane.

22. Image of Object in Lens • The image of an object in a lens is the geometrical locus of all the points being the images of each point of the object in the lens. To build the image of an object in the lens, one usually builds the images of its extreme points. • The image of an object formed by a lens is formed by means of two rays from each point of the object. • The image of the point is at the intersection of these rays after their passing through the lens. In case of a virtual image, the image of the point is at the point of intersection of the extensions of the rays passing through the lens.

23. As a rule, any two of the following three typical rays are used for this purpose The object image in the lens can be real or virtual, erect or inverted, magnified, diminished, or the same size.

24. The image character is determined by the mutual positions of the object and the lens. The distance from the object to the lens is denoted by d, and that from the lens to the image is denoted by f . • If the object is between the converging lens and its focus, the image is virtual, erect, and magnified. • If the object is far from the lens, at a distance meeting the condition F < d < 2F then the image is real, magnified, and inverted.

25. If for a converging lens, d = 2F the image is real, inverted and equal to the object in size. • If d > 2F the image is real, inverted, and diminished.

26. Diverging lens always give a virtual, erect, and diminished image of the object.

27. The formula relating the values and is often called the thin lens formula or simply the lens formula. It is written as follows: If the image is virtual, value f is taken with the sign "-". • The magnification of the lens (k) is the ratio of the image dimensions (H) and the object dimensions (h), i.e.

28. Lens Aberrations • All the above formulae are strictly valid only for paraxial rays, i.e. rays, which are close to the principal optical axis at small angles thereto. • When using non-paraxial rays, certain errors or defects, which are called lens aberrations, are observed in the image. We will briefly consider the following kinds of aberrations: spherical aberration, chromatic aberration, astigmatism, and distortion.

29. Spherical aberration is caused by the lens edges refracting the rays more than the central part of the lens. Hence, the principal focuses for lateral and central rays do not coincide. The object image on the screen is fuzzy (washed) to some extent. The use of wide beams in optical systems leads to spherical aberration of the image obtained.

30. Chromatic aberrationis caused by dispersion, i.e. the dependence of the index of refraction (n) of a substance on the light wavelength (λ). • Light of different colours (and different wavelengths) has different speeds, or different indices of refraction in the glass and therefore the focal length of a given lens is different for different colours. So if we take a white spot, the image will have colours. When we focus for the violet, the red is out of focus. This makes the edges of the object image on the screen coloured.

31. Considering astigmatism, let us note that we can only observe the so-called astigmatism of oblique rays in strictly spherical lenses. Such astigmatism is observed in the case if rays are incident on the lens at big angles to the principal optical axis. At this, unequal refraction of rays, which pass through the lens in different meridional planes, occurs. In this case, a point image is not a point, but two mutually perpendicular lines in different planes. Optical axis

32. Distortion is deformation of a large object image. This occurs because the lens linear magnification for object points, which are at different distances from the principal optical axis, is different. Due to this, the rectilinear contours of an object, which are in a plane perpendicular to the principal optical axis, take the form of an arc in the image. • In optical instruments, a specially selected system of lens is used for removing aberrations.

33. Lecture # 8 27.02.2018 Human eye structure Light Polarization

34. Human eye structure • A human eye has spherical form with diameter d 2425 mm. • The eyeball has three coats (or layers): • outer – fibrous coat (sclera and cornea) • middle – choroid, vascular coat (choroidea, ciliale body, iris) • internal - sensory nervous coat (retina)

35. The contents of the eyeball: • Humor of two chambers (anterior and posterior) divided by the iris • Crystalline lens • Vitreous humor (vitreous body)

36. Eye’s Light Conducting System: • Cornea • Aqueous humour of Anterior chamber • Iris • Humour of Posterior chamber • Crystalline lens • Vitreous humour (vitreous body) 1. Corneais the front part of the sclera. Sclera (or fibrous coat) protects the eyeball and assists in maintaining its shape. • Light enters the eye through the cornea. The cornea - is transparent - has the highest curvature, optical strength and refraction: n=1.38; D(cornea) = 40 dptr.

37. 2. Aqueous humor is a transparent watery fluid, which fills the anterior and posterior chambers of the eyeball; n=1.33

38. Iris is the front part of the choroid; it determines the colour of the eyes. Pupil is a small aperture in the centre of the iris. • Quantity of light coming into the eye is regulated with diameter of the pupil (it is changed from 2-3 mm at bright light to 7-8 mm at weak light).

39. 4. Humour of Posterior chamber Posterior chamberis a space between the iris and the crystalline lens. Anterior and Posterior chambers are connected with each other through the pupil.

40. 5.Crystalline lens • It is situated immediately behind the posterior chamber. • It is a transparent elastic structure, consisted of large number of transparent layers just like an onion. • Ithas the shape of a biconvex (converging) lens. It refracts light onto the retina. • D (at rest) = 20 dptr; D (at accommodation) = 30 dptr • n = 1.44

41. 6.Vitreous humour (Vitreous body) is a transparent jelly-like substance, which helps maintain the shape of eyeball and assists in the refraction of light n = 1.33, D=1 dptr Total focal power of the eye in rest (when eye looks into infiniteness) is Deye= 63-65 dptr.

42. Retina is the innermost layer lining the inside of the eyeball, and it does not cover the front region of the eyeball. Retina is composed of nervous tissue. It contains photoreceptors (receptor cells sensitive to light) - rodsandcones. The number of rods is about 130 million, number of cones is about 7 million. Eye’s Light Perceiving System is retina

44. The rods and cones are irregularly distributed over the retina: • Cones are mainly located in the macula lutea (or yellow spot) – an oval area in the centre of the posterior part of the retina; rods – in the periphery of retina. • The fovea centralis is situated at the centre of the macula lutea and contains numerous cones, but no rods. It is the region where light is mainly focused and where there is the greatest acuity of vision. • Blind spot (or optic disk) is a place where the opticnerve enters the eyeball; where photoreceptors are only absent, so light is not perceived.

45. All contents of the eye is under the pressure exceeding the ambient pressure by 18-26 mm of mercury column. This pressure is called an intra-eye pressure.

46. SENSITIVITY OF EYE to LIGHT and COLOUR • Dark adaptationof the eye is the ability of the eye to adapt itself to different brightness due to following mechanisms: 1) changing the pupil diameter; within 2 to 8 mm; 2) changing of concentration of the photosensitive substancecontained in the photoreceptors, its decomposition causing receptor excitation; 3) screening of cones and rods with a dark pigmentcontained in the choroid, which is able to move toward the vitreous humor in the process of adaptation. • By means of adaptation the eye perceives brightness (luminance) in the range from 10-7 to 105cd/m2 (candela per square metre), 1 cd/m2 = 1 nit.

47. Cones create the colour sensation; they make the human eye able to distinguish colours. But the cones lack very high photosensitivity and require sufficiently bright light for functioning. So, the cones form the apparatus of central coloured daylight (bright) vision. • Rods cannot differentiate colours. But the rods possess very high photosensitivity and function sufficiently well at twilight when lighting conditions are low, and the cone apparatus does not function. So, the rods form the apparatus of peripheral achromatic twilight vision.

48. kλ • Brightness curve is the spectral sensitivity of the eye. • Quantity K is called the relative spectral sensitivity of the eye (or relative visibility). 1 2 relative visibility Rod (1) and cone (2) brightness curves • Two curves: (1) sensitivity of the eye in the dark (rods brightness curve); (2) for the eye in the light (cones brightness curve).

49. For normal eye K=1 at =555 nm in the light, and K=1 at =510 nm in the dark. This means: • the peak sensitivity of cones is  = 555 nm (yellow-green colour) • the peak sensitivity of rods is  = 510 nm (grey-blue colour).

50. The simplest theory of colour visionin man is proposed by Young and Helmholtz and widely accepted. It supposes that in the eye there are three different pigments, which receive the light. These pigments have different absorption spectra. One pigment absorbs strongly in the red, another absorbs strongly in the blue, the other absorb in the green.