Geometric Optics

Introduction

Geometric optics is the technology which underlies most of optical devices that are important to various aspects of our lives. These aspects range from simple things such as eyeglasses to complex instruments like Hubble Space Telescope. Basic physics is contained in the following laws; for all reflecting surfaces reflectance angle is equal to incidence and for refracting surfaces the angle of incidence and refraction are related by Snell’s law.

 

 

 

A set of curved surfaces that form a boundary between different optic media can be arrange as lenses in order to collect and focus light rays. In normal cases, these lens materials have higher refraction index than surrounding media. The focal length of a lens can be determined by finding the point where parallel rays of light that are incident on one side of the lens will focus. Ideally, all light rays should come to focus to a common local focal point independent of distance from the lens axis and wavelength of the light, λ.

The lens equation is given by 1/p + 1/q = 1/f. Where p is the object distance, q is the image distance and f is the focal length. Thin lenses have two focal points which are symmetrically located on either side of the lens because light rays can approach from either side. The figure below indicates roles two focal points play for both positive and negative lenses.

 

 

For a spherical convex mirror, the convex surface bulges out at the center and towards the object which is normally placed to the left of the mirror. Also the expression for the focal length ,f, does not involve the height of the object ,h, so any ray that is parallel to the axis is reflected to appear to originate at focal point, F.

A spherical concave mirror has its reflecting surface curving inwards at the center which is away from the extended object.

Referring to the figure above, we can find the image distance v= IC in terms of the object distance u = OC, where f = FC and h = OB. Thus in triangles OBC and IMC (treating h, u,v, f and IM as positive);

α = h/ u = IM/ v so that IM = hv/ u.

Also, in triangles PCF and IMF, 2β = h/ f = IM/ v−f giving M = h(v−f)/ f.

Equating expressions for IM:

hv/ u = h(v−f)/ f

Dividing by hv gives us 1/v + 1/u = 1/ f. Equations β=hr and 2β=hf can be used to show that the length f is half the radius of curvature. In the case of paraxial rays, since f is independent of h, it is known as the focal length and F as the focal point of the concave mirror.

PROCEDURE 1

I used an accompanied pdf which had four diagrams; one each for; a convex lens, a concave lens, a convex mirror, and a concave mirror. In each case, I found the focal length.

ANALYSIS

  Focal length, f (cm)
Convex lens 16.0
Convex mirror -1.6
Concave mirror 16.6
Concave lens -6.4

 

There was Parallax error which occurred when the measurement of the focal length is more or less than the true length because I positioned my eye at an angle to the measurement marking.

 

PROCEDURE 2

  • I considered an object of ho = 1cm.
  • Using the focal length, f,  I determined in procedure 1, I used the lens maker’s equation to determine;
    • the image distance, di, the height, hi, and the magnification, M, for an object distance, do = 1/2f,
    • di, hi, M, from do = 2f
    • di, hi, M, from do = 3f

ANALYSIS

  Convex lens Convex mirror Concave mirror Concave lens
Focal length, f (cm) 16.0 -1.6 16.6 -6.4
Object height, ho (cm) 1.0 1.0 1.0 1.0
Object distance, do (cm) 1.     ½ f = 8.0 0.8 8.3 -3.2
2.     2f = 32.0 3.2 33.2 -12.8
3.     3f = 48.0 4.8 49.8 -19.2
Image height, hi (cm) 1.     2 2 2 -2
2.     -1 -1 -1 -1
3.     0.5 -0.5 -0.5 0.5
Image distance, di (cm) 1.     -16.0 1.6 -16.6 6.4
2.     32.0 -3.2 33.2 -12.8
3.     24.0 -2.4 24.9 -9.6
Magnification, M 1.     -2 -2 -2 2
2.     1 1 1 1
3.     0.5 0.5 1.5 -0.5

 

Using lens maker’s equation; 1/object distance (do) + 1/image distance (di) = 1/ focal length (f) we do the following calculations:

  1. Convex lens
  2. f = 16 cm, do=8cm

1/ Image distance (di) = 1/16 – 1/8 = -1/16

1/ di = -1/16

Image distance = -16cm

Magnification = image distance / object distance

= -16/ 8 = -2

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (-2) * 1 = 2 cm.

  1. f = 16 cm, do=32cm

1/ Image distance (di) = 1/16 – 1/32 = 1/32

1/ di = 1/32

Image distance = 32cm

Magnification = image distance / object distance

= 32/ 32 = 1

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (1) * 1 = -1cm

  • f = 16 cm, do=48cm

1/ Image distance (di) = 1/16 – 1/48 = 1/24

1/ di = 1/24

Image distance = 24cm

Magnification = image distance / object distance

= 24/ 48 = -0.5

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (-0.5) * 1 = 0.5cm

  1. Convex mirror
  2. f = -1.6 cm, do=-0.8cm

1/ Image distance (di) = -0.625 + 1.25 = 0.625

1/ di = 0.625

Image distance = 1.6cm

Magnification = image distance / object distance

= 1.6/ -0. 8 = -2

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (-2) * 1 = 2cm

 

  1. f = -1.6 cm, do=-3.2cm

1/ Image distance (di) = -0.625 + 0.3125 = -0.3125

1/ di = -0.3125

Image distance = -3.2cm

Magnification = image distance / object distance

= -3.2/ -3.2 = 1

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (1) * 1 = -1cm

  • f = -1.6 cm, do=-4.8cm

1/ Image distance (di) = -0.625 + 0.208 = -0.417

1/ di = -0.417

Image distance = -2.4cm

Magnification = image distance / object distance

= -2.4/ -4. 8 = 0.5

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (0.5) * 1 = -0.5cm

  1. Concave mirror
  2. f = 16.6 cm, do=8.3cm

1/ Image distance (di) = 1/16.6 – 1/ 8.3 = -5/ 83

1/ di = -5/ 83

Image distance = -16.6cm

Magnification = image distance / object distance

= -16.6/ 8.3 = -2

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (-2) * 1 = 2 cm

  1. f = 16.6 cm, do=33.2cm

1/ Image distance (di) = 1/16.6 – 1/ 33.2 = 5/ 166

1/ di = 5/ 166

Image distance = 33.2cm

Magnification = image distance / object distance

= 33.2/ 33.2 = 1

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (1) * 1 = -1 cm

  • f = 16.6 cm, do= 49.8cm

1/ Image distance (di) = 1/16.6 – 1/ 49.8 = 10/ 249

1/ di = 10/ 249

Image distance = 24.9cm

Magnification = image distance / object distance

= 24.9/ 49.8 = 0.5

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (0.5) * 1 = -0.5cm

  1. Concave lens
  2. f = -6.4 cm, do= -3.2cm

1/ Image distance (di) = -0.15625 + 0.3125 = 0.15625

1/ di = 0.15625

Image distance = 6.4cm

Magnification = image distance / object distance

= 6.4/ 3.2 = 2

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (2) * 1 =-2 cm.

  1. f = -6.4 cm, do= -12.8cm

1/ Image distance (di) = -0.15625 + 0.078125 = -0.078125

1/ di = -0.078125

Image distance = -12.8cm

Magnification = image distance / object distance

= -12.8/ -12.8 = 1

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (1) * 1 =-1 cm.

  • f = -6.4 cm, do= -19.2cm

1/ Image distance (di) = -0.15625 + 0.05208 = -0.1042

1/ di = -0.1042

Image distance = -9.6cm

Magnification = image distance / object distance

= -9/6/ -19.2 = 0.5

Height of image/ height of object = – image distance/ object distance

Height of image = – (di/do) * height of object

= – (0.5) * 1 = 0.5 cm.

PROCEDURE 3

  • Created an Excel table with columns for M and do
  • From the sim I entered the default M and do into my table.
  • Dragged the object, (the point on top of the arrow), to do=9.5cm and recorded my data.
  • Repeated this process for do = 8.5cm, 8.0cm, 7.5cm, 7.0cm, 6.5cm, 6.0cm, 5.5cm, 5.0cm.
  • My plot required more data as I approached f, so between 5.0cm and 4.0cm I collected at least 5 points, with the last as close to, but not at 4.0, as I managed.
  • Added a trend line, thinking carefully about what the fit should be, and showed your equation and r value.

CONCLUSION

From the data obtain in procedure 1 we observe that convex lens and concave mirror have a positive focal length while convex mirror and concave lens have a negative focal length which is true from our introduction. From procedure 2 we observe that when focal length is halved the magnification doubles, when focal length is doubled the magnification becomes one and when it is tripled the magnification becomes 0.5. This is applicable to all lenses and mirrors. Lastly, from the graph in procedure 3 we observe that magnification decreases with increase in object distance.

A thin lens is a transparent material having two spherical refracting surfaces that have a thin thickness compared to radii of curvature of the refracting surfaces. They are concave and convex lens. Chromatic dispersion is the bulk material dispersion, which is, change in refractive index with optical frequency. In spherical aberration, parallel light rays passing through the central region of the lens get focused farther away than light rays that are passing through the edges of the lens.

The magnification when the object distance is the focal length is infinity. In order to get a linear graph a graph of image distance against object distance should be plotted where the slope is the magnification.

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