THE TIME DELAY FUNCTION IN THE
DERIVATION OF THIN LENS AND MIRROR FORMULA VIA FERMAT’S PRINCIPLE
Paulus Cahyono Tjiang & Sylvia Hastuti Sutanto Department of Physics, Parahyangan Catholic University
Bandung 40142
Abstract
The time delay function occurring in the derivation of thin lens and mirror formula via Fermat’s principle is derived explicitly for the case of the thin spherical lens and mirror, and the associated focal length is obtained from the second derivative of the function. The derivation procedure can be used to obtain any time delay function and focal length associated with a thin lens/mirror with a well-defined surface function.
I. INTRODUCTION
The thin lens and mirror formula,
1 1 1
so + si = f (I.1)
known as the Gaussian formula, is usually derived from the law of reflection and refraction using a geometrical approach. There is an alternative way in deriving the formula, i.e. using Fermat’s principle, suggested by Don S. Lemons1. In Lemons’
derivation, there is a function which depends on the curvature of the lens/mirror surfaces, called the time delay function, whose functional form is not explicitly expressed in his paper. The time delay function has an important role in determining the focal length f of the thin lens/mirror.
In this paper, we will discuss the contribution of the time delay function in the derivation of thin lens/mirror formula from Fermat’s principle and derive its functional form explicitly. The paper will be organized as follows : in Section II, we will take a brief review of Lemons’ derivation of thin lens/mirror formula, and the role of the time delay function in the derivation. In Section III and Section IV, we will focus our attention to the time delay function, derive its functional forms for the case of thin spherical lens and mirror, and obtain the associated focal lengths.
We will summarize and conclude our discussion in Section V, with a note on the contribution of time delay function in understanding optical aberrations.
II. FERMAT’S PRINCIPLE AND DERIVATION OF THE GAUSSIAN FORMULA
Fermat’s Principle, discovered by Pierre de Fermat, states that
Out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.2
One of the applications of Fermat’s principle is the derivation of the law of reflection and refraction, known as Snell’s Law. In the derivation of the law, one determines the optical path length, i.e. the length of the light path from one point to another via an optical element (such as mirror surface or boundary of optical media), determines the time spent by light to travel along the path, and finds its minimum by equating its first derivative to zero. We shall follow the same procedure in deriving the Gaussian formula.
Figure 1 shows the system of a single lens [Fig. 1 (a)] and the system of a single mirror [Fig. 1 (b)]. The arrows in the figures represent the direction of light propagation from point A to point B.
2 Feynman, R. P., Lectures on Physics, Vol. I, Addison-Wesley, Massachusetts, 1963, pp. 26-3
optical axis
A B
y y
ho hi
O principal
so si axis
(a) The Lens System
optical axis A
ho
B
hi y y
si O principal axis
so
(b) The Mirror System
Figure 1.
The optical path length l from A to B for both figures is
( ) ( ) ( )
l= so2 + −y ho 2 + si2 + y−hi 2 +cD y (II.1) and the time spent by light to travel from A to B is
( ) ( ) ( )
t l c
s y h
c
s y h
c D y
o o i i
= = + −
+ + −
+
2 2 2 2
(II.2)
where c is the speed of light in vacuo. The function D(y) is the time delay function, which is a function of y only. The physical meaning of D(y) is the time spent by light between reaching and leaving the optical element.
As a special case, let us consider only the rays near the principal axis, called the paraxial rays. In this case, the rays intersect the principal axis at very small angles.
The consequences of the case are
a. y−ho <<so ; y−hi <<si
b. y≈ y
Then we can rewrite equation (II.2) as follows :
( ) ( ) ( )
t s c
y h s
s c
y h
s D y
o o
o
i i
i
= + −
+ + −
+
1 1
2
2
2
2 (II.3)
Expanding the equation (II.3) using the Taylor expansion, we get
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
t s c
y h s
y h s
s c
y h s
y h s
D D y D
y D
y
o o
o
o o
i i
i
i i
= + −
− −
+
+ + −
− −
+
+ + + + +
1 2 24 1
2 24
0 0 0
2
0 6
2 2
4 4
2 2
4 4
2 3
... ...
' ...
" '''
(II.4)
Retaining the second order terms, we get
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
t s c
y h s
s c
y h
s D D y D
y s
c s
c D h
s c h
s c h s c
h
s c D y
s c s c D y
o o
o
i i
i
o i o
o i
i
o o
i
i o i
= + −
+ + −
+ + +
= + + + +
− + −
+ + +
1 1 0 0 0
2
0 2 2 0 1 1
0 2
2
2
2
2
2
2 2 2
'
''
' ''
(II.5)
The shortest time, i.e. the minimum of t can be obtained by equating to zero its first derivative with respect to h :
( ) ( )
dt dh
h cs
h
cs D
cs cs D y
o o
i
i o i
= − + + −
+ + +
=
' ''
0 1 1
0 0 (II.6)
which solutions are
( ) ( )
h s
h
s cD
s s cD
f
o o
i i
o i
+ =
+ = − ≡
'
''
0
1 1
0 1 (II.7)
The second equation in (II.7) is indeed the Gaussian formula. The right-side term -cD`` (0) is interpreted as 1
f , where f is the focal length of the optical system. In the next section we shall see that the time delay function D(y) is an axially symmetric function, i.e. D(y) = D(-y). It means that D`(0) = 0, and the first equation in (II.7) will describe the well-known lateral magnification formula.
III. THE TIME DELAY FUNCTIONS OF SPHERICAL MIRRORS
Figure 2 shows a concave mirror [Fig. 2 (a)] and a convex mirror [Fig. 2 (b)]
with surface radius R. The arrowed lines show the propagation of light from point A to point B.
The consequence of the use of paraxial rays is that the length of A`C and CB` are almost paralel to the principal axis, so the length of A`-C-B` ≈ 2 A`C, which depends on y only. The function of concave mirror curvature is
( )
xconcave y = − +R R2 −y2 (III.1)
and the length of path A`-C-B` ≈ 2 A`C can be written as
( ) ( )
dconcave y = −2 R−∆l +2 R2 − y2 (III.2)
The time delay function of the mirror is
( ) ( ) ( )
D y d y
c c R l
c R y
concave
concave
= = −2 −∆ +2 2 − 2 (III.3)
A optical axis
A` C B B` y
O principal
∆l axis
R
(a) The Concave Mirror.
B optical axis
B`
C
A` R
A y
O principal
axis
∆l
(b) The Convex Mirror.
Figure 2.
It is obvious that D(y) has an axially symmetric property. The focal length of the concave mirror is
f
( )
cD
R
concave
concave
= − 1 =
0 2
'' (III.4)
which agrees with the result from the geometrical approach.
It can be shown using the same procedure that the time delay function of the convex mirror is
( ) ( )
D y
c R l
c R y
convex = 2 −∆ −2 2 − 2 (III.5)
and its focal length is f
( )
cD
R
convex
convex
= − 1 = −
0 2
'' (III.6)
IV. THE TIME DELAY FUNCTIONS OF THIN SPHERICAL LENSES.
Now we shall focus our attention on the thin spherical lens. Figure 3 shows a thin biconvex lens [Fig. 3 (a)] and a thin biconcave lens [Fig. 3 (b)] with the surface radii R1 and R2. The arrowed lines show the propagation of light from point A to point B.
The thinness of the lens means that ∆l1, ∆l2, ∆l3, and ∆l4 are small quantities and the path A`-D`-E`-B` can be considered as a straight line across the lens, which is almost paralel to the principal axis by the consequence of using the paraxial rays.
Then it is clear that the length of A`-D`-E`-B` depends on y only. The travelling time of light along the path depend on the speed of light in the optical media. The speed of light is c outside the lens and v c
= n inside the lens, where n is the refractive index of the lens.
The functions of the biconvex lens’ curvature is
Left-hand side : xbiconvexL
( )
y =(
R1 −∆l1)
− R12 − y2 (IV.1)Right-hand side : xbiconvexR
( )
y = −(
R2 −∆l2)
+ R22 −y2 (IV.2)The length of path A`-D`-E`-B` is
( ) [ ] [ ( ) ]
[ ]
d y R R y l l R R R y R y
R R y
biconvex = − − + + − − + − + −
+ − + −
1 1
2 2
1 2 1 2 1
2 2
2
2 2
2 2
2 2
∆ ∆
(IV.3) and the time delay function of the lens is
( ) [ ] [ ( ) ]
[ ]
D y
R R y
c
n l l R R R y R y
c
R R y
c
biconvex = − −
+ + − − + − + −
+ − + −
1 1
2 2
1 2 1 2 1
2 2
2
2 2
2 2
2 2
∆ ∆
(IV.4) Once again we see that the time delay function is an axially symmetric function.
optical axis A` D` E`
B`
y
O
principal
∆l1 ∆l2 axis
R2 R1
(a) The Thin Biconvex Lens.
optical axis A` D` E` B`
principal ∆l3 ∆l1 ∆l2 ∆l4 axis
R1 R2
(b) The Thin Biconcave Lens.
Figure 3.
The focal length of the biconvex lens is
( ) ( )
f f cD n
R R
biconvex
biconvex
biconvex
⇒ = − = − +
1 0 1 1 1
1 2
'' (IV.5)
which once again agrees with the result from the geometrical approach.
It can be shown using the same procedure that the time delay function of the thin biconcave lens is
( ) [ ( ) ] [ ( ) ]
( )
[ ]
D y
l R R y
c
n R R l l R y R y
c
l R R y
c
biconcave = − + −
+ + + + − − − −
+ − + −
∆ ∆ ∆
∆
3 1 1
2 2
1 2 1 2 1
2 2
2
2 2
4 2 2
2 2
(IV.6) and its focal length is
( ) ( )
f f cD n
R R
biconcave
biconcave
biconcave
⇒ = − = − − +
1 0 1 1 1
1 2
'' (IV.7)
V. SUMMARY AND CONCLUSIONS
The time delay function appearing in the derivation of Gaussian formula via Fermat’s principle has played an important role in determining the focal length of the thin spherical lens and mirror. The functional form of the function depends on the curvature of the lens/mirror sufaces. Once we know the function of the lens’ or mirror’s surface curvature, we can easily derive the associated time delay function.
Some approximations has been made, such as the use of paraxial rays and the
thinness property of the lens, to get a simple time delay function which depends on a single variable only.
The time delay function can be used to obtain the focal lengths of other thin lenses/mirrors. For example, for the paraboloid mirror with the curvature function x(y) = y2 , the associated time delay function is
( ) ( )
D y
c l y
paraboloid = 2 − 2
∆ and the associated focal length is
f ( )
= cD1 = 0 0 25
" .
. The time delay function also gives an important contribution in the abberations phenomena, such as astigmatism and spherical abberation, by retaining the higher order contribution in equation (II.4). This will be described in a following paper.
Although the time delay function has been derived with some approximations, we expect that the concept of time delay function will be of great use in understanding many optical phenomena, especially understanding via Fermat’s principle.
V. ACKNOWLEDGEMENTS
We would like to thank Dr. Aloysius Rusli and the staff members of the Department of Physics, Parahyangan Catholic University, for their valuable comments and corrections.