The Gaussian formula and spherical aberrations of the static and

arXiv:0904.0291v4 [physics.class-ph] 12 Nov 2009

The Gaussian formula and spherical aberrations of the static and relativistic curved mirrors from Fermat’s principle Sylvia H. Sutanto1 and Paulus C. Tjiang2 Theoretical and Computational Physics Group, Department of Physics, Faculty of Information Technology and Sciences, Parahyangan Catholic University, Bandung 40141 INDONESIA E-mail: 1 [email protected], 2 [email protected] Abstract. The Gaussian formula and spherical aberrations of the static and relativistic curved mirrors are analyzed using the construction of optical path length (OPL) and the application of the Fermat’s principle to the OPL. The geometrical figures generated by the rotation of conic sections about their symmetry axes are considered for the shapes of the mirrors. By comparing the results in static and relativistic cases, it is shown that the focal lengths and the spherical aberration relations of the relativistic mirrors obey the Lorentz contraction with respect to the rest observer.

PACS numbers: 02.30.Xx; 42.15.-i; 42.15.Fr; 42.79.Bh

1. Introduction In every discussion of geometrical optics, the so-called Fermat’s principle hardly misses its part. Many textbooks on either elementary physics or optics discuss the application of the Fermat’s principle to derive the laws of reflection and refraction [1], but only few of them discuss the application of the principle on the curved mirrors and lenses. To the best of our knowledge, the first discussion of the application of Fermat’s principle on curved mirrors and lenses, particularly those of the spherical shapes, was given by Lemons with his proposal of the time delay function [2]. The more general treatment on the time delay function was given by Lakshminarayanan et. al. [3], and a non-technical discussion of curved mirrors and the Fermat’s principle was presented by Erb [4]. On the other hand, the discussion on the relativistic plane mirrors was recently brought by Gjurchinovski, both using Huygens construction [5] and Fermat’s principle [6]. The same discussion had been presented for the first time more than 100 years ago by Einstein through his famous work on the special theory of relativity [7]. However, to the best of our knowledge, no similar discussion on the relativistic curved mirrors has been found yet. The purpose of this work is to give an analysis on the relativistic curved mirrors and compare the results to the static ones. Although the common shapes of mirrors used in practical applications are the spherical and parabolic shapes, we shall consider the shapes of curved mirrors generated by the rotation of the two dimensional conic sections about their

The Gaussian formula and spherical aberrations from Fermat’s principle

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symmetry axes, inspired by Watson [8]. The calculations are performed straightforwardly without the introduction of time delay function, since it may provide clearer interpretations on the results. The paper is organized as follows : in Section 2, we shall review the general mathematical form of the conic sections, which shall be used to shape the curved mirrors. In this case, we shall follow the discussions of conic sections provided by Watson [8] and Baker [9]. In Section 3 we shall discuss the application of Fermat’s principle to obtain the Gaussian formula and the spherical aberration relations of the static curved mirrors, with some analysis on the focal length of the mirror and the physical behaviour of the spherical aberrations. The similar discussion for the relativistic curved mirrors will be given in Section 4, with some analysis on the results compared to the static cases. The discussion concludes in Section 5, with some remarks on the future work for the relativistic curved lenses. 2. General curved mirror : the conic sections Let us consider the curved mirrors whose shapes are the geometrical figures generated from the rotation of two dimensional conic sections about their symmetry axes. These shapes are well known in the constructions of astronomical instruments [8]. It is worth to note that the discussion of conic sections is also useful for construction of mathematical models to predict the theoretical change of corneal asphericity after the eye surgery [10], although it is beyond our scope of discussion. The general mathematical description of the two dimensional conic sections [8, 9, 10] is given by (1 − e2 )l2 + h2 − 2Rl = 0,

(1)

where l is the horizontal distance from the optical center of the mirror system, h is the vertical distance from the principal axis of the mirror system, e is the eccentricity of the conic section, and R is the apical radius of curvature (apex), given by

R=

  2  32        1 + dl   dh d2 l     dh2    

(2) h=0

The apex region of a given conic section is illustrated by Figure 1. Eq. (1) is known as Baker’s equation [9, 10]. Solving Eq. (1) for the horizontal distance l, we get 

l± (h) = ± 

R−

q

R2 − (1 − e2 )h2 1 − e2



.

(3)

It is clear from Eq. (3) that l(h) is an even functions of h, with l(0) = 0 due to the fact that the l(h) vanishes at the optical center. The ± signs refer to concave / convex mirror systems. There are four types of mirrors based on the variation of eccentricity e : • Spherical mirror (e = 0).

The Gaussian formula and spherical aberrations from Fermat’s principle

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Figure 1. The schematic diagram of a conic section. The area inside the dashed circle is the apex region of a conic section.

Figure 2. The schematic diagram of the static curved mirror system. The (S0 ,h0 ) and (Si ,hi ) are sets of coordinates related to the object and its associated image, respectively. The solid line A-C-B is the optical path length (OPL) of the light.

• Elliptic mirror (0 < e < 1). • Parabolic mirror (e = 1). • Hyperbolic mirror (e > 1). 3. Applications of the Fermat’s principle on the static curved mirrors Figure 2 shows the propagation of light from point A to point B, with reflection of light at point C on the surface of the mirror. The optical path length (OPL) of the light from A to B is OPL(h) = S0

v u" u t

l(h) 1− S0

#2

"

h − h0 + S0

#2

+Si

v u" u t

l(h) 1− Si

#2

"

#2

hi − h + .(4) Si

The Gaussian formula and spherical aberrations from Fermat’s principle

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The traveling time of light can be obtain by dividing the OPL (4) with the speed of light c of the corresponding medium : v u"

l(h) S0 u t(h) = t 1 − c S0

#2

"

h − h0 + S0

#2

v u"

l(h) Si u + t 1− c Si

#2

"

#2

hi − h + . (5) Si

The application of Fermat’s principle can be accomplished by vanishing the first derivative of Eq. (5). The result is h−h0 S0



− 1−

l(h) S0

i l(h) 2 S0

h

dt (h) = rh dh 1− = 0.

+



dl dh

h−h0 S0

i2

hi −h Si



+ 1−

l(h) Si

i l(h) 2 Si

h

− rh 1−

+



dl dh

hi −h Si

i2

(6)

Since it is very difficult to solve Eq. (6) for h, we shall analyze it by taking the Taylor expansion of Eq. (6) near h = 0 : ∞ X 1 dn t dt (h) = (0)hn−1 = 0, (7) n dh (n − 1)! dh n=1 leading to

dn t (0) = 0, dhn

n = 1, 2, 3, ...

(8)

3.1. The Gaussian formula of the static mirrors The first two terms of Eq. (8) after the insertion of Eq. (5) lead to     l(0) ˙ ˙ + h0 l(0) l(0) + Shii 1 − 1 − l(0) S0 S0 Si r r + = 0,    h2i h20 l(0) 2 l(0) 2 1 − S0 + S 2 1 − Si + S 2 0

(9)

i

and

1 S0

+

1 Si

1 + l˙2 (0) − 1 −



r



1−



 l(0) 2 S0

+

1 + l˙2 (0) − 1 −



= 0,

r



1−



 l(0) 2 Si

+

l(0) S0 h20 S02 l(0) Si h2i Si2



¨l(0)

−

1 S0



1−



1−



¨l(0)

−

1 Si





1−

1−

l(0) S0



˙ + l(0)

 l(0) 2 S0

l(0) Si



+

h20 S02

˙ + l(0)

 l(0) 2 Si

+

h2i Si2

h0 S0

3

2

2

 hi 2 Si

3 2

(10)

˙ where l(h) and ¨l(h) are the first and the second derivatives of l(h) with respect to h, respectively. In order to get a clear interpretation on these terms, we consider two usual assumptions in geometrical optics : • It is common to use the so-called paraxial ray approximation, i.e. when the rays intersect the principal axis at small angles. The consequences of the approximation are h 0 S0

≪ 1,

h i Si

≪ 1,

(11)

The Gaussian formula and spherical aberrations from Fermat’s principle i.e. we may neglect the higher order contributions of

h0 S0

and

hi Si

5

on Eq. (8).

• It is customary to consider the mirrors whose curvatures are symmetric with respect to their associated principal axes. Consequently, the function l(h) must be an even function : l(−h) = l(h), and all odd derivatives of l(h) vanish at h = 0 [2]. The shapes of mirrors discussed in Section 2 satisfy this condition. Based on the assumptions above, Eqs. (9) and (10) lead to hi Si h0 hi + = 0 ⇐⇒ =− , (12) S0 Si h0 S0 1 1 + = 2¨l(0). (13) S0 Si Eq. (12) expresses the usual lateral magnification formula, and Eq. (13) is the Gaussian formula relating the object distance S0 and the image distance Si , with the focal length f defined as 1 . (14) f≡ 2¨l(0) Using the horizontal distance given by Eq. (3), it is interesting to note that the second derivative ¨l(0) = ± R1 regardless of the eccentricity e. Hence the focal length for any shape of mirror generated by a conic section is R (15) f± = ± . 2 If the conic section is a circle, Eq. (15) gives the usual focal length of a spherical mirror, since the apex of a circle is equal to its radius. In general, Eq. (15) is equally true for any shape of mirror generated by a conic section, where R is generalized to the apex of the associated conic section. 3.2. Spherical aberrations in the static mirrors Spherical aberration occurs in the curved mirror system when the reflected light rays coming from the outer region of the mirror do not arrive to the same focal point as those from paraxial region [11]. The schematic diagram of the phenomenon is shown in Figure 3. At the paraxial region, the Gaussian formula (13) holds. We may obtain the focal length of the curved mirror by setting S0 → ∞, i.e. when light source is relatively far away from the mirror, and the distance of the associated image Si is equal to the focal length. By Eq. (12), the image height hi = 0 regardless of the object height h0 . As we consider light rays coming from the outer region, we have to start from Eq. (6), take S0 → ∞ and hi = 0, and solve the resulting equation for Si . The result is   h 1 − l˙2 (h) . (16) Si (h) = f (h) = l(h) + ˙ 2l(h) Inserting Eq. (3) into Eq. (16), we obtain 

f (h, e) = ± 

R−

q

R2 − (1 − e2 )h2 1 − e2



R2 − (2 − e2 )h2  + q 2 R2 − (1 − e2 )h2

(17)

The Gaussian formula and spherical aberrations from Fermat’s principle

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Figure 3. The schematic diagram of the spherical abberation.

Figure 4. The spherical aberration of concave mirrors of various conic sections. The apical radius used for the plot is R = 5 unit length.

It is clear from Eq. (17) that in general, the intersection of the reflected rays to the principal axis depends on the vertical distance h and the shape of mirror. Now we may analyze the behaviour of the spherical aberration based on Eq. (17) :

The Gaussian formula and spherical aberrations from Fermat’s principle

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(i) The expansion of Eq. (17) in terms of Taylor series at paraxial region gives R (1 − e2 )h2 (1 − e2 )(3 − e2 )h4 f (h, e) = ± − − 2 4R 16R3 ! (5 − e2 )(1 − e2 )2 h6 + O(7) . (18) − 32R5 The expansion (18) gives the same results as obtained by Watson using different approach [8]. Eq. (18) indicates that the intersection point of the reflected ray (or the extension of the reflected ray in the case of convex mirrors) to the principal axis is coming closer to the optical centre for mirror shapes with eccentricities 0 ≤ e < 1 as the vertical distance h increases, and is moving away from the optical centre for hyperbolic mirror (e > 1). For the parabolic mirror (e = 1), all the reflected rays (or the extension of the reflected ray in the case of convex mirrors) come to the single point whose horizontal distance is ± R2 from the optical centre. This means that the parabolic mirror does not suffer the spherical aberration, as expected. (ii) Figure 4 shows the intersection of the reflected ray f (h, e) vs. vertical distance h graph for selected conic sections. All mirrors generated by conic sections (except the parabolic mirror) suffer the spherical aberrations. At a sufficiently long vertical distance h, the spherical aberration suffered by hyperbolic mirror is worse than by either the spherical or elliptic mirrors. However, it is interesting to note from Figure 4 and Eq. (18) that the spherical aberration may be reduced by constructing the mirrors generated by the conic sections with the eccentricity closed to 1. It is also practical to reduce the spherical aberration by setting a sufficiently large value for the apical radius R. 4. Applications of the Fermat’s principle on the moving curved mirrors Now let us study the behaviour of a mirror system where the object, the image and the curved mirror are all moving parallel to the principal axis at a constant speed v to the right relative to a rest observer, as shown in Figure 5. The light ray travels from point A, is reflected at point C on the surface of the mirror, and finally reaches point B. During the movement of the mirror system, it is important to keep in mind the two important matters : • the speed of light is remains the same as observed by either the rest or moving observers, as stated by the second postulate of the special theory of relativity [7]. • the length that is parallel to the direction of the movement appears shorter with respect to the rest observer due to the Lorentz contraction. Based on the two matters above, we construct the OPL of the light from A to C and from C to B as v #2 #2 " u" u h − h0 vta − ¯l(h) t ¯ + , (19) la (h) = S0 1 + S¯0 S¯0

and

v u" u ¯t

lb (h) = Si

vta + ¯l(h) 1− S¯i

#2

"

#2

hi − h + , S¯i

(20)

The Gaussian formula and spherical aberrations from Fermat’s principle

A

8

v

ho

t=0 Principal axis

S0

(a)

A

A’’ v.ta

l  h

la

C

ho

h t = ta Principal axis

S0

v.ta (b)

A’’’

A’’ v.tb

So C

B

lb

l  h

h

hi

t = ta + tb Principal axis

Si

v.tb (c)

Figure 5. The schematic diagram of the moving curved mirror system. The figures (a), (b) and (c) are arranged in chronological order. The OPL from A to B is shown by the dashed-double dotted line. During the movement of the system, the initial point is moving from A to A”’, so the distance between the initial point and the mirror is held constant at all time.

respectively. The S¯0 , S¯i and ¯l(h) are the Lorentz contracted version of S0 , Si and l(h), respectively, i.e. S0 S¯0 = , γ Si S¯i = , γ ¯l(h) = l(h) , (21) γ 

where γ ≡ 1 −

v2 c2

− 1 2

.

The Gaussian formula and spherical aberrations from Fermat’s principle

9

The traveling time from A to C can be obtain from Eq. (19) by dividing it with the speed of light c and solving the resulting equation for ta . The result is v

¯l(h) S¯0

u S¯0 2 u ta (h) = γ t 1 − c

!2

1 γ2

+

h − h0 S¯0

!2



¯l(h)  + β 1 − ¯ , S0 !

(22)

where β ≡ vc . Similarly, the traveling time from C to B can be obtain from Eq. (20), leading to v

¯l(h) S¯i

u S¯i 2 u tb (h) = γ t 1 − c

!2

+

1 γ2

hi − h S¯i

!2



¯l(h) −β 1− ¯  . Si !

(23)

The total traveling time is then t(h) = ta (h) + tb (h). Applying the Fermat’s principle by vanishing the first derivative of the total traveling time t(h), we obtain     ¯ ¯ l(h) ¯ l(h) ¯ hi −h ˙ ˙ h−h0 − 1 − + 1 − l(h) l(h) 2 2 γ S¯ γ S¯i S¯0 S¯i r 0 r (24)   2 −    = 0. ¯ 2 ¯ l(h) 2 hi −h 2 h−h0 1 1 1 − l(h) 1 − + + γ2 γ2 S¯0 S¯0 S¯i S¯i

For v ≪ c, Eq. (24) leads to Eq. (6). As in the static case, Eq. (24) is very difficult to solve for h, and we shall use the Taylor expansion (7) and the relation (8) to analyze it properly. 4.1. The Gaussian formula of the moving mirrors The first two terms of the relation (8) obtained from Eq. (24) give     ¯ ¯ l(0) ¯ l(0) ¯ hi ˙ ˙ h0 − 1 − + 1 − l(0) l(0) 2 2 ¯ ¯ ¯ ¯ γ S γ Si S0 Si r 0 r   2 +   2 = 0, ¯ 2 ¯ l(0) 2 hi 1 1 h0 + + 1 − l(0) 1 − γ 2 S¯0 γ 2 S¯i S¯0 S¯i 1 S¯0

+

1 S¯i





1 γ2

2 + ¯l˙ (0) − 1 −

r 1 γ2



1− 2



 ¯ l(0) 2 S¯0 

+

r

1−

 ¯ l(0) 2 ¯ Si

+

−

1 S¯0

h20 γ 2 S¯2

 ¯ l(0) ¨ ¯l(0) ¯ Si

h2i 2 γ S¯i2



1−



1−

0

+ ¯l˙ (0) − 1 − 

 ¯ l(0) ¨ ¯l(0) ¯ S0

−

1 S¯i





1− 1−

(25)

 ¯ l(0) ¯ ˙ l(0) ¯ S0  ¯ l(0) 2 ¯ S0

h20 γ 2 S¯02

+

 ¯ l(0) ¯ ˙ l(0) S¯i  ¯ l(0) 2 S¯i

+

h0 γ 2 S¯0

+

+

3 2

hi γ 2 S¯i

h2i γ 2 S¯2 i

3 2

2

2

= 0. (26)

Since ¯l(h), like l(h), is an even function of h, and using the paraxial ray approximation (11), Eqs (25) and (26) lead to ! hi Si S¯i 1 h0 hi + = 0 ⇐⇒ = − = − , (27) 2 γ S¯0 S¯i h0 S¯0 S0 1 1 + ¯ = 2γ 2¨¯l(0). ¯ S0 Si

(28)

The Gaussian formula and spherical aberrations from Fermat’s principle

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Comparing Eqs. (12) and (27), it is clear that the lateral magnification formula is preserved, i.e. the magnification rule appears the same in either static or relativistic cases. The Gaussian formula, however, suffers a modification on the focal length of the mirror, i.e. f 1 1 = , =⇒ f¯(v) ≡ (29) f≡ ¨ ¨ γ 2l(0) 2γ 2 ¯l(0) based on comparison between Eqs. (13) and (28). Hence, the focal length of the relativistic mirror is Lorentz contracted according to the rest observer. Inserting Eq. (15) into Eq. (29), we obtain the focal length of the relativistic curved mirror with the shape of a conic section as ¯ = R. (30) f(v) 2γ Due to the quadratic term of the speed v in γ, the Gaussian formula (28) and the focal length (29) are independent of the direction of movement. 4.2. Spherical aberrations in the moving mirrors Using the same arguments as used in the discussion of spherical aberration in Section 3.2, we shall start from Eq. (24), take S¯0 → ∞ and hi = 0, and solve the resulting equation for S¯i . The result is   2 ˙ 1 ¯ h γ 2 − l (h) S¯i (h) = f¯(h, v) = ¯l(h) + ˙ 2¯l(h)    h 1 − l˙2 (h) f (h) 1 = l(h) + . (31) = ˙ γ γ 2l(h)

Eq. (31) indicates that the spherical aberration relation (17) is Lorentz contracted, which is consistent to Eq. (29). Figure 6 shows the spherical aberration of the relativistic spherical concave mirror for the purpose of illustration of Eq. (31). Again, Eq. (31) is independent of the direction of movement due to the quadratic term of the speed v in γ. In order to obtain a clearer analysis on Eq. (31), let us take the derivative of f (h, v) with respect to h, i.e. ˙¯ v) = 1 f˙(h). f(h, (32) γ where f˙ ≡ df . Eq. (32) indicates that the change of f¯(h, v) with respect to the change of h dh

decreases as the moving speed of the mirror system increases, due to the multiplication of a ˙ It means that when the mirror system is moving at a considerably small value of γ1 on f(h). high speed, the shift of the intersection point between the reflected ray and the principal axis is relatively small so that the position of the intersection point may be considered as a fixed point. Hence the spherical aberration of the relativistic mirror system may be reduced by increasing its moving speed. However, since the spherical aberration relation suffers the Lorentz contraction, the focal point of the mirror system moving with high speed will also be reduced according to Eq. (30). As an illustration, Figure 6 for the relativistic spherical concave mirror clearly indicates this effect. For the relativistic parabolic mirror, no spherical

The Gaussian formula and spherical aberrations from Fermat’s principle

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Figure 6. The spherical aberration of the relativistic spherical concave mirror of radius R = 5 unit length. The spherical aberration reduces as the mirror system moves at the higher speed.

aberration takes place regardless of the speed of the mirror system. However, the focal length reduces as the speed increases due to the Lorentz contraction. 5. Conclusion and future work We have discussed the properties of the relativistic curved mirrors and compared them with the properties of the static cases. It is shown that the focal lengths of the moving mirrors are shorten according to the Lorentz contraction. Consequently, all focal lengths will tend to zero as speed of the mirror systems increases up to the speed of light (if it is possible to do so). From the Gaussian formula (28), we may conclude that the light rays are getting harder to reach the mirror system as the speed of the mirror system v is approaching the speed of light, and hence the rays are getting harder to be reflected by the surface of the mirror (S¯i → 0). The spherical aberrations are also affected by the movement of the mirror systems. The spherical aberration of the relativistic mirror system reduces as the speed of the system increases, but at the same time, the intersection position of the reflected ray on the principal axis reduces due to the Lorentz contraction. The parabolic mirror does not suffer any spherical aberration caused by any moving state, even though the associated focal length decreases with the increase of speed. It will be interesting to investigate whether the same techniques used in this work can be applied to the curved lenses. It is well known that every lens system suffers the spherical aberration because of the change of the light speed inside the lens’ media. By a logical

The Gaussian formula and spherical aberrations from Fermat’s principle

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thinking, the Gaussian formula and the spherical aberration relation must change due to the change of OPL caused by the movement of the lens system. Using the same techniques discussed here, we may discover how the speed of a lens system affects the associated focal length and spherical aberration, which can provide some new information about the behavior of the relativistic lenses. Acknowledgement The authors would like to thank Professor B. Suprapto Brotosiswojo and Dr. A. Rusli of the Department of Physics, Parahyangan Catholic University for the valuable comments and discussions during the earlier stage of this work. The work is supported by the Directorate General of Higher Education - Ministry of National Education of The Republic of Indonesia through the Competitive Grant Scheme for International Publication under contract number 481/SP2H/PP/DP2M/VI/2009. References [1] See, for examples : R. P. Feynman, Lectures on Physics : Vol. I, Addison-Wesley, pp. 26-1 - 26-8 (1963); E. Hecht, Optics, 4th ed., Addison-Wesley, pp. 106 - 111 (2002). [2] D. S. Lemons, Gaussian thin lens and mirror formula from Fermat’s principle, Am. J. Phys. 62, pp. 376 (1994). [3] V. Lakshminarayanan, A. K. Ghatak, K. Thyagarajan, Lagrangian Optics, Kluwer Academic Pub., pp. 20 - 26 (2001). [4] R. Erb., Curved Mirrors, Phys. Educ. 30, pp. 287 - 289 (1995). [5] A. Gjurchinovski, Reflection of light from a uniformly moving mirror, Am. J. Phys. 72, pp. 1316 (2004). [6] A. Gjurchinovski, Einstein’s mirror and Fermat’s principle of least time, Am. J. Phys. 72, pp. 1325 (2004). [7] A. Einstein, Zur Elektrodynamik bewegter K¨orper, Ann. Phys. 17, 891 (1905). [8] D. M. Watson, Astronomy 203/403 : Astronomical Instruments and Techniques On-Line Lecture Notes, University of Rochester (1999), http://www.pas.rochester.edu/∼dmw/ast203/Lectures.htm, accessed on October 25, 2009. [9] T. Y. Baker, Raytracing through non-spherical surfaces, Proc. Phys. Soc. 55, pp. 361 - 364 (1943). [10] D. Gantinel, et. al. Determination of corneal asphericity after myopia surgery with excimer laser : a mathematical model, Investigative Ophthalmology & Visual Science 42, 1736 (2001). [11] E. Hecht, Optics, 4th ed., Addison-Wesley, p. 226 (2002).