Optics Course (Phys 311)

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Geometrical Optics Refraction through Lenses

Lecturer: Dr Zeina Hashim Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 1

Objectives covered in this lesson :

1. The refracting power of a thin lens.

2. Thin lens combinations.

3. Refraction through thick lenses. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 2

The Refracting Power of a Thin Lens:

The refracting power of a thin lens is given by: 1 푃 = 푓

Vergence: is the convergence or divergence of rays: 1 1 푉 = and 푉′ = 푝 푖

∴ 푉 + 푉′ = 푃

A diopter (D): is a unit used to express the power of a spectacle lens, equal to the reciprocal of the focal length in meters. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 3

The Refracting Power of a Thin Lens: Individual Activity

What is the refracting power of a lens in diopters if the lens has a focal length = 20 cm ? Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 4

Thin Lens Combinations:

If the optical system is composed of more than one lens (or a combination of lenses and mirrors) which are located so that their optical axes coincide: the final image can be obtained by working in steps: 1. Consider the nearest lens only, find the image of the object through this lens. 2. The image in step 1 is the object for the second (adjacent) optical component: find the image of this object. This can be done both geometrically or numerically 3. Do the same for all optical components, nearest first. 4. The final image is the image of the original object through this optical system.

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 5

Thin Lens Combinations:

For a system composed of N lenses (and/or mirrors), the effective focal length of the system is given by: Study 1 1 1 1 = + + ⋯ + Sample Problem 34-5 푓 푓1 푓2 푓푁 in Halliday (8th ed.) The refractive power of the optical system is therefore: to get familiar with how 푃 = 푃1 + 푃2 + ⋯ + 푃푁 to solve problems of The lateral magnification of the optical system is: thin lens combinations.

푀 = 푚1 × 푚2 × ⋯ × 푚푁 Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 6

Refraction Through Thick Lens:

Description of a thick lens:

A thick lens is a refracting material which is bounded by two spherical refracting surfaces.

The image of a given object, formed by refraction at the first surface, becomes the object for refraction at the second surface.

The image formed by the second surface is then the final image.

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 7

Refraction Through Thick Lens: Radii of Curvature and refractive indices

Lens thickness (풅)

푭ퟏ 푽ퟏ 푯ퟏ 푵ퟏ 푵ퟐ 푯ퟐ 푽ퟐ 푭ퟐ

풏ퟏ 풏푳 풏ퟐ

푟1 is the radius of curvature of the first refracting surface that faces the object. 푟2 is the radius of curvature of the second refracting surface. 푛 is the refractive index of the medium at the side of the lens of where the object is. 1 푛2 is the refractive index of the medium at the other side. 푛퐿 is the refractive index of the lens.

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 8

Refraction Through Thick Lens: The 6 Cardinal Points

Cardinal Points: Lens thickness (풅) are points, on the optical axis of a thick lens, from 푭ퟏ 푽ퟏ 푯ퟏ 푵ퟏ 푵ퟐ 푯ퟐ 푽ퟐ 푭ퟐ which its imaging properties can be deduced. 풏ퟏ 풏푳 풏ퟐ

The 6 Cardinal Points:

푭 and 푭 : 1st and 2nd focal points of the thick lens. Cardinal Planes: ퟏ ퟐ 푯 and 푯 : 1st and 2nd principal points. are planes normal to the optical ퟏ ퟐ 푵 and 푵 : 1st and 2nd nodal points. axis at the cardinal points. ퟏ ퟐ

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 9

Refraction Through Thick Lens: Principal Planes

If a ray emerges the lens from 퐹1: . it is rendered parallel to the optical axis. 푭ퟏ 푭ퟐ 푯ퟏ . the extensions of the incident and resultant rays intersect in the 1st principal plane.

푭ퟏ 푭ퟐ If a ray emerges the lens parallel to the optical axis: 푯ퟐ

. it is refracted by the lens through 퐹2. . the extensions of the incident and resultant rays intersect in the 2nd principal plane.

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 10

Refraction Through Thick Lens: Principal Planes

If the thick lens were a single thin lens, the two principal planes would coincide at the vertical line drawn in the middle of the thin lens. Principal planes in thick lenses usually do not coincide, and they may be located outside the optical system. Once the locations of the principal planes are known, accurate ray- diagrams can be drawn.

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 11

Refraction Through Thick Lens: Nodal Points

A ray directed towards the first nodal point (푁1): . emerges from the optical system parallel to the incident ray,

. but displaced, so that it appears to come from 푁2.

푭 푵 푵 푭 ퟏ ퟏ ퟐ ퟐ

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 12

Refraction Through Thick Lens: Positions and signs

Lens thickness (풅)

Distances shown in the figure (and their values) 풇ퟏ 풇ퟐ

are: 푭ퟏ 푽ퟏ 푯ퟏ 푵ퟏ 푵ퟐ 푯ퟐ 푽ퟐ 푭ퟐ . Positive: if arrow is to the right 풏ퟏ 풏푳 풏ퟐ (i.e. from object side to the other side). 풔 . Negative: if arrow is to the left (i.e. to object) 풔ퟏ ퟐ

풗ퟏ 풗ퟐ

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 13

Refraction Through Thick Lens: Positions and signs

Lens thickness (풅) 푓1 is the distance from 퐻1 to 퐹1 풇ퟏ 풇ퟐ 푓2 is the distance from 퐻2 to 퐹2 푭 푽 푯 푵 푵 푯 푽 푭 푠1 is the distance from 푉1 to 퐻1 ퟏ ퟏ ퟏ ퟏ ퟐ ퟐ ퟐ ퟐ 풏 풏 풏 푠2 is the distance from 푉2 to 퐻2 ퟏ 푳 ퟐ

푣1 is the distance from 푉1 to 푁1 풔ퟏ 풔ퟐ

푣2 is the distance from 푉2 to 푁2 풗ퟏ 풗ퟐ

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 14

Refraction Through Thick Lens: Positions and signs

Lens thickness (풅) 푓 (-)  퐹 is to the left of 퐻 풇ퟏ 풇ퟐ 푓 (+)  퐹 is to the right of 퐻

푠 (-)  퐻 is to the left of 푉 푭ퟏ 푽ퟏ 푯ퟏ 푵ퟏ 푵ퟐ 푯ퟐ 푽ퟐ 푭ퟐ

푠 (+)  퐻 is to the right of 푉 풏ퟏ 풏푳 풏ퟐ

푣 (-)  푁 is to the left of 푉 풔ퟏ 풔ퟐ 푣 (+)  푁 is to the right of 푉 풗ퟏ 풗ퟐ

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 15

Thick Lens Equations:

Focal Lengths: ퟏ 풏푳 − 풏ퟐ 풏푳 − 풏ퟏ (풏푳 − 풏ퟏ)(풏푳 − 풏ퟐ) 풅 = − − 풇ퟏ 풏ퟏ 풓ퟐ 풏ퟏ 풓ퟏ 풏ퟏ 풏푳 풓ퟏ 풓ퟐ

These equations are 풏ퟐ 풇ퟐ = − 풇ퟏ NOT to memorize ! 풏ퟏ

푓2 = 푓1 if the lens is surrounded by a single medium (at which 푛2 = 푛1) Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 16

Thick Lens Equations:

Principal Planes Positions from the Vertices: These equations are 풏 − 풏 풏 − 풏 풔 = 푳 ퟐ 풇 풅 풔 = − 푳 ퟏ 풇 풅 NOT to ퟏ 풏 풓 ퟏ ퟐ 풏 풓 ퟐ 푳 ퟐ 푳 ퟏ memorize !

Positions of Nodal Points from the Vertices:

풏ퟐ 풏푳 − 풏ퟐ 풏ퟏ 풏푳 − 풏ퟏ 풗ퟏ = ퟏ − + 풅 풇ퟏ 풗ퟐ = ퟏ − + 풅 풇ퟐ 풏ퟏ 풏푳 풓ퟐ 풏ퟐ 풏푳 풓ퟏ Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 17

Thick Lens Equations: 풑 > ퟎ 풑 < ퟎ Image and Object Distance:

This equation 풇 풇 − ퟏ + ퟐ = ퟏ is NOT to 풑 풊 memorize ! 풊 > ퟎ 풊 < ퟎ

The distances p , i , and f are measured relative to their corresponding principal planes. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 18

Thick Lens Equations:

Image and Object Distance: Is the image (and object) real or virtual ?? In thin lenses: Objects are always real. Image is real : if it is on the opposite side of the lens of where the object is. Image is virtual: if it is on the same side of the lens where the object is. In thick lenses: It is more complicated  we will not study it in this course. The object can be virtual with thick lenses ! Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 19

Thick Lens Equations:

Lateral Magnification:

This equation 풏 풊 풎 = − ퟏ is NOT to 풏ퟐ 풑 memorize !

When m is positive  object and image have the same orientation.

When m > 1  image > object. When m < 1  image < object. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 20

Refraction through thin lenses: Group Work

Work in pairs: (a) What happens to the positions of the nodal points and positions of the principal points if the lens was in air? Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 21

Thick Lens Equations:

Final notes:

When the lens is in air:  the focal lengths will have the same magnitudes, and the focal length equation will become: ퟏ ퟏ ퟏ ퟏ 풏푳 − ퟏ 풅 = = (풏푳 − ퟏ)( − − ) 풇 풇ퟐ 풓ퟐ 풓ퟏ 풏푳 풓ퟏ 풓ퟐ  the usual thin-lens equations will be valid;

1 1 1 푖 + = , 푚 = − where 푓 = 푓 = −푓 푝 푖 푓 푝 2 1 Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 22

ퟏ 풏푳 − 풏ퟐ 풏푳 − 풏ퟏ (풏푳 − 풏ퟏ)(풏푳 − 풏ퟐ) 풅 풏ퟐ Exercise: = − − 풇ퟐ = − 풇ퟏ 풇ퟏ 풏ퟏ 풓ퟐ 풏ퟏ 풓ퟏ 풏ퟏ 풏푳 풓ퟏ 풓ퟐ 풏ퟏ

Determine the focal lengths and the principal points for a 4-cm thick, biconvex, lens with refractive index of 1.52 and radii of curvature of 25 cm, when the lens caps the end of a long cylinder filled with water ( n= 1.33) ? Using the thick lens equations:

1 1.52 − 1.33 1.52 − 1 (1.52 − 1)(1.52 − 1.33) 4 = − − 푓1 1(−25) 1 +25 1(1.52) (+25)(−25) 1.33 ⇒ 푓 = −35.74 푐푚 푓 = − −35.74 = 47.53 푐푚 1 2 1 to the left of the first principal plane to the right of the second principal plane Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 23

풏푳 − 풏ퟐ 풏푳 − 풏ퟏ Exercise: 풔ퟏ = 풇ퟏ풅 풔ퟐ = − 풇ퟐ풅 풏푳 풓ퟐ 풏푳 풓ퟏ

Using the thick lens equations:

1.52−1.33 1.52−1 푠 = −35.74 4 = 0.715 푐푚 , 푠 = 47.53 4 = −2.60 푐푚 1 1.52 −25 2 1.52 +25

Thus, the principal point 퐻1 is situated 0.715 cm to the right of the left vertex of the lens, and 퐻2 is situated 2.60 cm to the left of the right vertex. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 24

How to draw Rays with Thick Lenses:

Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 25

Homework :

Q1: Two-lens systems: In Figure, stick figure O (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions.

Lens 1 is mounted within the boxed region closer to O, which is at object distance 푝1. Lens 2 is mounted within the farther boxed region, at distance d. Each problem in the table (labeled 80 – 87) refers to a different combination of lenses and different values of distances, which are given in centimeters. The type of lens is indicated by C for converging and D for diverging; the number after C or D is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).

Find (a) the image distance 푖2 for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification M for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object O or noninverted (NI), and (e) on the same side of lens 2 as object O or on the opposite side. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 26

Homework :

Q1: Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 27

Homework :

Q2: Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 28

Homework :

Q3: A small object is placed 20 cm from the first of a train of three lenses with focal lengths, in order, of 10, 15, and 20 cm. The first two lenses are separated by 30 cm and the last two lenses are separated by 20 cm. Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive, (b) the middle lens is negative, (c) the first and last lenses are negative. Provide ray diagrams for each. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 29

Homework :

Q4: Go back to the thick-lens example given in the previous slides and study it well. Phys Geometrical Optics: Refraction (Lenses) Lesson 2 of 2 311 Slide 30 (last)

Summary:

Next lesson will cover:

1. The refracting power of a thin lens. Field of View

2. Thin lens combinations. + Aberrations 3. Refraction through thick lenses.

Any Questions?