南开大学光学工程内部课件Sep_16th

Lens
第一章第6节
Lens—What is a lens
What is a lens In traditional sense, a lens is an optical system consisting of two or more refractive interfaces where at least one of these is curved.
n1 n2 n2 s' s s s' n1
| s ' | tan 1 L, | s | tan 2 L s ' tan 1 s tan 2 tan sin for 1 s' sin 1 s sin 2 n1 s' n2 s ( n1 sin 1 n2 sin 2 )

Refraction at curved surface
n1
Yo
i
V
MБайду номын сангаас
n2
S
•
u
S0

R
i
C Si
n
u
Yi
•
S’
The upper figure depicts a wave from the point source S impinging on a spherical interface of radius R centered at C. Here we assume n2>n1.
s’
s
Refraction at curved surface
Refraction at a spherical surface Image we have a point source S (in other words, we have a point object S) whose spherical waves arrive at a spherical boundary of two transparent media. We prefer that the wave traveling in the second medium converges to a point S’. In fact, it is also the basic requirement of optical instruments.
d {n2 [ R ( S i R ) 2 R( S i R ) cos ] 2 } d
2 2
1
0
Refraction at curved surface
So we have:
n1 R( S0 R) sin n2 R( Si R) sin 0 2 SM 2 MS '
Refraction at curved surface
Similar, the second or image focus is the axial point Fi where the image is formed when S0= . And the second or image focal length fi as equal to Si in the special case, we have
Lens—Type of lenses
3. Convex (converging or positive) lens and concave (diverging or negative) lens: With the assumption that the refractive index of the lens is larger than that of the environment of the lens, convex lens is thicker at the center and so tends to decrease the radius of wavefronts. On the other hand, concave lens is thinner at the center and tends to cause the wavefronts to be more diverging than it was upon entry.
Why are focusing instruments necessary?
Refraction at curved surface
Imaging In order to image S at location P, the time it takes for each and every portion of a wavefront leaving S to converge at point P must be identical. So:
So, fo Xo Si, fi
Xi R Yo, Yi
+ left of V + left of V + right of V
+ right of V + if C is right of V + above optical axis
Refraction at curved surface

First order, paraxial or Gaussian optics With the paraxial condition (small values of ), we have sin= and cos=1. In this case, we have an approximation:
n1 n2 n2 n1 P S0 S i R
(1)
Refraction at curved surface
P is called optical power. We could have this formula with Snell’s law rather than Fermat’s Principle.

Lens—Type of lenses

Type of lenses
We limit ourselves centered systems of spherical surfaces (for which all the nonplanar surfaces are centered on a common axis, or in other words, all surfaces are rotationally symmetric about a common axis)
which followed with
n1 n2 1 n2 S i n1 S0 ( ) SM 2 MS ' R MS ' SM
Refraction at curved surface
Discussion Sign convention for spherical refraction surfaces and thin lenses
n2 fi R n2 n1
Refraction at curved surface
Refraction at curved surface
•
Refraction at aspherical surfaces
Refraction at spherical surfaces is limited by paraxial condition. Refraction at aspherical surfaces will help us to perform many types of really “perfect” reshaping operation of light.
Imaging by a Flat refracting surface

The image formed by a flat refracting surface is on the same side of the surface as the object The image is virtual The image forms between the object and the surface The rays bend away from the normal since n1 > n2 L

SA n1 AP n2 const . S0 n1 Si n2
Refraction at curved surface
where S0 and Si are the object and image distance measured from the vertex or pole V respectively. Obviously, this is the equation of a Cartesian Oval. S will be perfect imaged at P with the help of Cartesian Oval.
Refraction at curved surface
Perfect reshaping between converging (diverging) wave and flat wave

With the help of ellipsoidal or hyperboloidal surface, we can reshape precisely converging (diverging) waves into plane wave or vice versa
Lens—Thin-lens equations
The simplest case —— thin-lens equations Let’s now locate the conjugate points for a lens of index nl surrounded by a medium of index nm. It is the simplest case of a lens.
Refraction at curved surface
Fermat’s Principle maintains that the optical path length (OPLSS’) will be stationary （实际上，物与像之间根据费马原 理具有等光程性）, i.e.:
1 d (OPLSS ' ) 2 2 d {n1 [ R ( S 0 R ) 2 R( S 0 R ) cos ] 2 } d d
Refraction at curved surface

Focus and focal length If an point is located at F0, where
n1 f0 R n2 n1
according to equation (1), it is imaged at infinity (Si=). The location F0 is called first or object focus. And the special object distance is defined as the first or object focal length.
Lens—Type of lenses
1. Simple lens and compound lens: Simple lens —— consist of one element, i.e., it only has two refracting surfaces Compound lens —— more than one elements. 2. Thin and thick lens: —— if its thickness is effectively negligible or not
The emerging wavefront segment corresponding to paraxial rays from a point source S is essentially spherical and will form a “perfect” image at its center point located at S’.