Chapter 7

Geometrical Optics (general theory)

(April 27 , 2005)

A brief summary to the last lecture

• Capillarity and surface absorption ： Capillary tube in liquid, capillary phenomena, stemmed from ( 起源于 ) additional pressure under curved surface & surface tension.

• Geometrical optics: light rays, dielectrics, light speed, refractive index, laws of reflection and refraction (three angles), total internal reflection (from optically denser medium to optically thinner medium), and refraction of a spherical surface.

Some questions for you to answer

1. What is capillary tube? ( 刘潇强 )

2. What is capillarity? ( 周婉明＊ )

3. What is air embolism? （吴丽玲 * ）4. What causes the capillary phenomena? (In other

words, how is it formed?) 陈文杰

5. What is the surface-active agent? What is the non-surface-active agent? Which one could change the surface tension greatly? （陈娜平 * ）

1. Does light travel at the same speed in different medium? 　 Give your definition on the refractive index? ( 陈志勇 )

2. What are reflective and refractive laws? 麦小婷 *

3. What is the equation of single spherical surface? 卢 锋

4. How to determine the signs of the object distance, image distances and curvature radius? 谭晓君 *

5. What is the first focal distance and what is the second focal distance? 　陈 琪

1. Converge ( 会聚 , 收敛 ), diverge ( 发散 )

2. Lens ( 透镜 ) ， circumference ( 圆周 ) ， virtual ( 虚的 )

3. Negligible ( 可忽略的 ), unimportant

4. Graphical ( 绘画的，图示的 ) ， dioptric ( 屈光的，折射的 , 视力矫正用的 ) ， coaxial （共轴的）

5. un-deviated （无偏离的）， un-displaced

6. Diopter ( 屈光度 ) ， Substitute ( 代替 ( 品 ))

7. Reversed ( 颠倒的 ) ， principal points ( 主点 )

8. `Cardinal points ( 基点 ): focus ( 焦点 , focal point), in focus ( 清晰 ); out of focus ( 模糊 ) (phase, order, question, shape, touch with …)

7.2 (b) Focus and coaxial spherical system

3. Coaxial spherical systemUsually there are several refractive surfaces in an optical system. For example, we could make a system of a few lenses and each lens has two surfaces. If all the centers of the spherical surfaces are in a line, the system is called coaxial spherical system and the line is called light axis.

• If we want to calculate the image of the system, we can calculate the image of spherical surface one after another.

• The image of first surface is regarded as the object of the second surface and so on.• Especially, pay attention to the signs of object distance, image distance and curvature radius of each spherical surface.

Example 7-2: A glass ball has a radius of 10cm and refractive index ( 折射率 ) of 1.5. Use paraxial rays ( 近轴光线 ) formula to calculate the image of an object that is 40cm from the ball.

Solution: for the first surface, we know

,10,40,5.1,1 121 cmrcmunn

·O

u1

v1

u220cmv2

n1 (air)

n2 (glass)

n1 (air)I1I2p1 p2

This is the first step. I1 is the virtual ( 虚的 ) image for the first lens and it is regarded as the object of the second spherical surface. So for the second surface, we have:

cmrcm

vunn

10,40

)20(,1,5.1 1221

cmv

vr

nn

v

n

u

n

60

10

15.15.1

40

1

1

1

12

1

2

1

1

Substituting the above data to the paraxial ray formula, we have:

cmv

v

r

nn

v

n

u

n

4.11

10

5.111

40

5.1

2

2

12

2

2

2

1

7.3 Thin lenses• A lens is a simple coaxial system and it is an optical system including two refracting surfaces.

• If the thickness of a lens is much smaller than curvature radius, the object and image distances, the thickness of the lens can be negligible in comparison with them. Such a lens is called thin lens.

• Lens can converge and diverge light.

7.3.1 Some special rays for converging and diverging lenses

(1). Light rays parallel to the axis of a converging lens are refracted through the focal point on the opposite side of the lens.

First and second focal point of a lens.

Planes through the focal points of a lens are called focal planes.

(2). Light rays parallel to the axis of a diverging lens are refracted so that their backward continuations pass through the focal point on the same side of the lens.

(3) Definition of converging and diverging lenses

A lens is converging if the glass is thinner around the circumference than at the center and diverging if the situation is reversed.

(4). Graphical method

A light ray through the center of a thin lens continues un-deviated and un-displaced. A light ray parallel to the axis passes through the second focal point of a converging lens

F Real image

Virtual image

7.3.2 The lens equation

u v1

v

n0

nn0

According to the formula of single spherical surface, we can use it one by one and finally calculate the lens equation.

u1 = u, v1 = -u2,

v2 = v, r

nn

v

n

u

n 1221

For the first surface, n1= n0, n2 = n

1

0

1

0

r

nn

v

n

u

n

For the second surface, n1= n, n2 = n0

2

00

1 r

nn

v

n

v

n

The above two equations can be added together, then we have

210

0 1111

rrn

nn

vu

If lens is in the air, n0 = 1,

21

11)1(

11

rrn

vu

1

0

1

0

r

nn

v

n

u

n

2

00

1 r

nn

v

n

v

n

first focal distance for thin lens

210

0

1

1111

rrn

nn

f

1

210

01

11

rrn

nnf

second focal distance for thin lens:

210

0

2

1111

rrn

nn

f

1

210

02

11

rrn

nnf

Therefore we have f1 = f2 . we can suppose that they are all equal to f and for air medium, we have

21210

0 11)1(

111

rrn

rrn

nn

f

Finally we obtain the lens equation,

fvu

111

Pay attention to the symbol of curvature radius (r2 < 0 generally).

1/f is called lens power (or focal power), also denoted by D, with unit of diopter (1/m).

7.3.3 Systes of two lenses

Many optical instruments include combinations of two or more lenses. In systems of multiple lenses, the image formed by one lens becomes the object for the next lens (see the following figure).

u1 v1 u2 f2

f1

However, same as before, sometimes the second lens is placed between the first lens and the image. The original image disappears, and the second lens may or may not form a new image, but the image that the first lens would have formed still serves as the object for the second lens. Because the first image is not actually formed but still functions as the object for the second lens, we call it a virtual object. The object distance u of a virtual object is negative.

11

111

fvu

21

111

fvv

u1= uvv1= -u2

For the first lens, u1 = u

For the second lens, u2 = -v1, v2 = v,

21

1111

ffvu

The case of two connected thin lenses:

21

111

fff

Suppose that the focal distance for the group of thin lenses is f, we have

For the more lens systems, we could use this method one by one to solve them. Written as dioptric strength,

21 DDD

Example 7.3: A converging lens of focal length 12 cm is placed 52 cm from another converging lens of focal length 8 cm. Calculate the image position of an object that is 16 cm in front of the first lens.

52cm

16cm

f1=12cm f2=8cm

·· ·

v1

u2

v2

u1

air air air

Solution: the image formed by the first lens can be obtained by the given data and the thin length equation. Substituting the first object distance of 16 cm and the focal distance of first lens into thin lens equation, we have

111

111

fvu

cmvv

48 12

11

16

11

1

As the distance between the two lenses is 52 cm, the object distance for the second lens is 52 – 48 = 4 cm. Using the focal distance for the second lens of 8 cm, the final image can be obtained as

cmvvfvu

88

11

4

11112

2222

The image is virtual.

7.4 Thick lens*

Thick lenses are similar to the thin lenses and they also contain two systems of coaxial spherical surfaces. The difference between them is that thickness of the thick lenses cannot be negligible while the thickness of thin lenses can be ignored. As before, such a system can be solved by spherical surface, but it contain a lot of trivial details especially for coaxial optical system of more spherical surfaces.

Cardinal points ( 基点 ): 1. Two focal points (F

1, F2), 2. Two principal points ( H1, H2, first principal plane B1H1A1 and second principal plane B2H2A2), 3. Two nodes (N1, N2).

Cardinal points ( 基点 , 最重要的点 ):

1. Two focal points (F1, F2) ： they have the same definition as above.

2. Two principal points ( H1, H2, first principal plane B1H1A1 and second principal plane B2H2

A2): the extension of incident ray and backward extension of refracted ray (note that the refracted line is parallel to the light axis) meet at point A1. First principal plane…

3. Two nodes (N1, N2), through these points, the incident line and refractive line are parallel.

Positions of the three pairs of cardinal points are based on the specific conditions of the refractive system. When the refractive system is put in one medium, in the air for example, the two focal lengths can be proved to be equal to each other. f1 = f2 = f, N1 and H1 are at the same position, N2 and H2 are at the same position. The same equation for the thin lens can be obtained.

fvu

111

Note that the object distance is from the first principal plane and the image distance is from the second principal plane, not from the surface of lens.

• Cornea ( 角膜 ), membrane ( 膜 , 隔膜 ).

• Aqueous humor ( 水状液 , n = 1.336, 眼房水 )

• Crystalline lens ( 水晶体 , n =1.437)

• ligament ( 韧带 )

• Ciliary( 睫毛的 ) muscles ( 睫状肌 ) .

• Vitreous humor ( 玻璃状液，透明的液体 )

• Retina ( 视网膜 )

• Yellow spot ( 黄斑 )

• Ciliary muscle ( 睫状肌 ) can change the curvature of the lens in order to focus objects at different distances in front of the eye.

• Vitreous humor ( 玻璃状液 ) with refractive index 1.336 is a thin jelly which is behind the crystalline lens.

• Retina ( 视网膜 ) is a delicate film of nerve fibers which is a large part of the inner surface of the eye.

• Yellow spot ( 黄斑 ) is a slight depression in the retina. Its center is a minute region, about 0.25mm in diameter and it is a most sensitive parts in the retina as it contains cones excessively.

• myopia ( 近视 ) ， myopic, nearsighted, shortsighted,

• Hyperopic eye ( 远视 ), farsighted ( 远视的，有远见的 )

• Astigmatism ( 散光 ) ， astigmatical ( 散光的 )

• retinal image ( 视网膜图像，眼膜图像 )

• Glasses ， spectacles ( 眼镜 ) ，• prescribe ( 规定，开处方 ) ， • magnifying glass ( 放大镜 )

• fibrescope ( 纤镜 ) ， microscope ( 显微镜）• unaided eye ( 肉眼 ) ，