A geometric algebra reformulation of geometric optics

A geometric algebra reformulation of geometric optics Quirino M. Sugon Jr. and Daniel J. McNamara Citation: American Journal of Physics 72, 92 (2004); doi: 10.1119/1.1621029 View online: http://dx.doi.org/10.1119/1.1621029 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/72/1?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Using Nonprinciple Rays to Form Images in Geometrical Optics Phys. Teach. 53, 497 (2015); 10.1119/1.4933155 Eyeglasses in the Classroom Phys. Teach. 48, 12 (2010); 10.1119/1.3274350 The rainbow as a student project involving numerical calculations Am. J. Phys. 77, 795 (2009); 10.1119/1.3152991 Novel optical properties of a submerged light bulb Am. J. Phys. 76, 856 (2008); 10.1119/1.2955794 Interferential coupling effect on the whispering-gallery mode lasing in a double-layered microcylinder Appl. Phys. Lett. 80, 3250 (2002); 10.1063/1.1476399

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A geometric algebra reformulation of geometric optics Quirino M. Sugon, Jr.a) and Daniel J. McNamarab) Ateneo de Manila University, Loyola School of Science and Engineering, Physics Department, Loyola Heights, Quezon City, Philippines

共Received 5 February 2003; accepted 27 August 2003兲 We present a tutorial on the Clifford 共geometric兲 algebra Cl3,0 and use it to reformulate the laws of geometric optics. This algebra is essentially a Pauli algebra, with the Pauli sigma matrices interpreted as unit rays or vectors. In this algebra, the exponentials of imaginary vectors act as vector rotation operators. This property lets us rewrite the laws of reflection and refraction of light in geometric optics in exponential form. The reformulated laws allow easy translation of symbols to words and to diagrams. They also are shown to be equivalent to standard vector formulations. These coordinate-free laws can be shown to simplify the analysis of geometric optics problems such as the tracing of meridional and skew rays in lenses and optical fibers. © 2004 American Association of Physics Teachers.

关DOI: 10.1119/1.1621029兴

I. INTRODUCTION The reflected and refracted rays in geometric optics may be viewed as rotations of the incident ray in the plane defined by the incident ray and the normal vector to the interface. This description lets us think of the ray-interface interaction as a scattering problem and ascribe to the interaction diagram a set of Feynman-like rules:1 replace the rays by unit vectors and the vertex 共ray-interface intersection兲 by a rotation operator, which either reflects or refracts the incident ray. There are many ways to express rotation operators, but the most concise forms employ the exponential function. This conciseness is evident in Euler’s theorem in complex ˆ matrices in Cayley–Klein parameters3 for analysis,2 the Q finite rotations,4 and the vector exponentials in Hamilton’s quaternions.5,6 The first is contained in the third because complex numbers form a subalgebra of quaternions. The second is also equivalent to the third, because the Pauli ˆ matrix are intimately related to matrices7 in the rotation Q quaternions. This relation is provided by a new language of mathematical physics: Clifford 共geometric兲 algebra.8 –10 In geometric algebra, the Pauli matrices ␴ˆ 1 , ␴ˆ 2 , and ␴ˆ 3 are interpreted as the orthonormal basis of a threedimensional Euclidean space. This interpretation makes the dyadic11 multiplication of the Pauli sigma matrix ␴ˆ by vectors superfluous, so that the Pauli identity in quantum mechanics becomes12 共 ␴ˆ •a兲共 ␴ˆ "b兲 ⬅ab⫽a"b⫹ î 共 aÃb兲 .

共1兲

ˆ matrix as This identity lets us rewrite the Q ˆ ⫽e î ␴ˆ "␪/2⫽e î ␪/2. Q

共2兲

ˆ On the other hand, the connection of the rotation operator Q in Eq. 共2兲 to quaternions is easily shown. If we define the imaginary number î as î ⫽ ␴1 ␴2 ␴3 ,

共3兲

we can show that the imaginary basis vectors î ␴1 , î ␴2 , and ˆ in Eq. î ␴3 form a quaternion basis.13 Thus, the quantity Q 共2兲 is a quaternion.

At present, geometric algebra has only been applied to reflection and rotation; there is yet no known formulation for refraction. If we use the notation and terminology that we shall adopt, the reflected vector is related to the incident vector ␴ and the unit normal vector ␩ of a convex interface by14,15

␴⫽⫺ ␩␴␩.

共4兲

On the other hand, the new vector r⬘ generated by rotating r counterclockwise about ␪/ 兩 ␪兩 by an angle ␪ is given by14,16 ˆ

ˆ

r⬘ ⫽e ⫺ i ␪/2re i ␪/2.

共5兲

In this paper, we shall reformulate the laws of geometric optics in exponential form using geometric algebra. But we will not use the double exponential form in Eq. 共5兲. Instead, we shall employ a single exponential function that encodes the axis of rotation, the angle of incidence, and the concavity of the interface. The power of this formulation will be shown in future work on tracing of meridional and skew rays in optical fibers and lenses. Except for reflected rays,17 these problems cannot be handled by matrix methods,18 –20 because the angles are not paraxial and the transformations between the coordinates are nonlinear.21,22 The body of the paper is divided into four parts. The first part is a short introduction to the Clifford algebra Cl3,0 . The succeeding parts are on the propagation, reflection, and refraction of light. The treatment of reflection and refraction is similar. First we derive the exponential form of the law that is valid for both concave and convex interfaces. Then we expand the exponential using Euler’s theorem and show that the formulation is equivalent to the standard vector formulations.17,23

II. THE GEOMETRIC ALGEBRA Cl3,0 We shall introduce the geometric algebra Cl3,0 and show that it is the same as the Pauli algebra, with the Pauli sigma matrices interpreted as vectors in three-dimensional Cartesian space. We shall discuss the orthonormality axiom, the geometric product, and the exponential function for planar rotations. Because ␴ will be used later to denote the direc-

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tion of propagation of a light ray, we will adopt the standard notation of e1 , e2 , and e3 for the basis vectors instead of the Pauli matrices. A. The orthonormality axiom The algebra Cl3,0 is a group algebra over the field of real numbers, with three vector generators e1 , e2 , and e3 that satisfy the orthonormality relation e j ek ⫹ek e j ⫽2 ␦ jk ,

共6兲

where ␦ jk is the Kronecker delta function. This relation may be broken into two: e21 ⫽e22 ⫽e23 ⫽1,

共7兲

e1 e2 ⫽⫺e2 e1 ,

共8a兲

e2 e3 ⫽⫺e3 e2 ,

共8b兲

e3 e1 ⫽⫺e1 e3 .

共8c兲

ˆ

Fig. 1. Vector rotation expressed by a⬜⬘ ⫽a⬜ e ⫹ i ␪. In words, a⬜⬘ is (⫽) the vector a⬜ rotated (e) counterclockwise (⫹) by an angle ␪ about ( î ) the vector ␪ pointing out of the paper.

and

Equation 共7兲 is the normalization axiom 共the length of the vectors e1 , e2 , and e3 is normalized to unity兲; while Eq. 共8兲 is the orthogonality axiom 共mutually orthogonal vectors anticommute兲. Notice that Eqs. 共7兲 and 共8兲 define the Pauli algebra if we replace e1 , e2 , and e3 by the Pauli spin matrices ␴1 , ␴2 , and ␴3 , respectively. But regardless of their matrix representations, the vectors e1 , e2 , and e3 retain their geometrical interpretation as vectors or rays 共oriented line segments兲. This fact means that the familiar geometric interpretation of the dot and cross products in Gibbs–Heaviside24 vector algebra still holds. Indeed, geometric algebra may be defined simply as the extension of the familiar vector algebra by defining a new, associative product of vectors 共juxtaposition multiplication兲. We shall see later how this geometric product of two vectors is expressed in terms of their dot and cross products.

Let a and b be two vectors spanned by the Cartesian vectors e1 , e2 , and e3 : a⫽a 1 e1 ⫹a 2 e2 ⫹a 3 e3 ,

共9a兲

b⫽b 1 e1 ⫹b 2 e2 ⫹b 3 e3 ,

共9b兲

where the coefficients a i and b i are real numbers. We can use the distributive property and the orthonormality axiom in Eq. 共6兲 to show that 共10兲

where î ⫽e1 e2 e3

共11兲

is the unit imaginary number that commutes with both vectors and scalars in Cl3,0 . There are two properties of the geometric product in Eq. 共10兲 that will become useful later. First, notice that if a is parallel to b, then aÃb⫽0, so that Eq. 共10兲 becomes ab⫽a"b⫽b"a⫽ba.

ab⫽ î 共 aÃb兲 ⫽⫺ î 共 bÃa兲 ⫽⫺ba.

共13兲

Thus, parallel vectors commute and perpendicular vectors anticommute.

C. Planar rotations The geometric algebra Cl3,0 is especially suited for the description of rotations. In geometric algebra, the exponential of a bivector 共an imaginary vector or an oriented plane兲 is a vector rotation operator. Let ␪ be a vector in Cl3,0 , so that the oriented plane perpendicular to it is î ␪. This plane has a negative square, 共 î ␪兲 2 ⫽ 共 î ␪兲共 î ␪兲 ⫽ î 2 ␪2 ⫽⫺ ␪ 2 .

共14兲

This property lets us employ Euler’s theorem with the unit bivector î ␪/ 兩 ␪ 兩 as the unit imaginary number:

B. The geometric product

ab⫽a"b⫹ î 共 aÃb兲 ,

On the other hand, if a is perpendicular to b, then a"b⫽0, so that

共12兲

ˆ e i ␪⫽cos ␪ ⫹ î

␪ sin ␪ . 兩␪兩

共15兲

Let a⬜ be a vector in Cl3,0 perpendicular to the vector ␪. We can use the expansion of the exponential in Eq. 共15兲 and the anticommutation property of perpendicular vectors in Eq. 共13兲 to show that a⬜ flips the sign of the exponential argument: ˆ

ˆ

a⬜ e i ␪⫽e ⫺ i ␪a⬜ .

共16兲

If we expand the left side of Eq. 共16兲 and apply the identity in Eq. 共10兲, we obtain ˆ

a⬜ e i ␪⫽a⬜ cos ␪ ⫺a⬜ ⫻

␪ sin ␪ , 兩␪兩

共17兲

because a⬜ is perpendicular to ␪, so that a⬜ "␪⫽0. Equation ˆ

共17兲 has a simple geometrical interpretation: e i ␪ rotates a⬜ counterclockwise by an angle ␪ about the vector ␪/ 兩 ␪兩共see Fig. 1兲.

93 Am. J. Phys., Vol. 72, No. 1, January 2004 Q. M. Sugon, Jr. and D. J. McNamara 93 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.125.102.33 On: Tue, 10 Nov 2015 13:26:46

Fig. 2. The law of propagation expressed by rk⫹1 ⫽rk ⫹s k⫹1 ␴k⫹1 . In words, the new position rk⫹1 of a ray is (⫽) its initial position rk plus (⫹) its displacement of length s k⫹1 in the direction of its unit propagation vector ␴k⫹1 .

D. Summary of geometric algebra In summary, all results in Cl3,0 may be derived from the orthonormality axiom in Eq. 共6兲. But if we want to work with vectors without reference to a coordinate system, we must keep five points in mind: 共1兲 The unit trivector î ⫽e1 e2 e3 commutes with all scalars and vectors in Cl3,0 . 共2兲 The geometric product ab is expressible as ab⫽a"b ⫹ î (aÃb). 共3兲 Parallel vectors commute; orthogonal vectors anticommute. ˆ ˆ 共4兲 Euler’s formula for e i ␪ is e i ␪⫽cos ␪⫹iˆ␪/ 兩 ␪兩 sin ␪. ˆ

共5兲 If a⬜ is a vector perpendicular to ␪, then a⬜ e i ␪ is the vector a⬜ rotated in the counterclockwise direction by an angle ␪ about the vector ␪/ 兩 ␪兩 .

Fig. 3. The law of reflection for a convex interface expressed by ␴k⬘ ⫽ ˆ

⫺ ␴k e ⫺2 i ␤ k e␴ ␩ k . In words, the reflected ray ␴k⬘ is (⫽) opposite (⫺) the incident ray ␴k rotated (e) clockwise (⫺) about ( î ) the axis e␴ ␩ k 共out of the paper兲 by twice 共2兲 the incident angle ␤ k .

the photon from its initial position. This interpretation is trivial for propagation. But when we discuss reflection and refraction, the transformation between the incident and reflected or refracted rays will not require vector addition and stretching operations. Instead, it will require pure rotation about an axis. The exponential function will act as the vector rotation operator. IV. REFLECTION

III. PROPAGATION We are now equipped to rewrite the laws of propagation and refraction of light in vector form using Cl3,0 . Except for a slight change in notation, we employ the same standard vector equation given in Born and Wolf25 for propagation. But for reflection and refraction, the geometric algebra reformulation in terms of the exponential function becomes more advantageous. In this formulation, the reflected or refracted vectors are related to the incident propagation vector by a rotation about an axis perpendicular to both the incident ray and the vector normal to the interface. The exponential function acts as the vector rotation operator. Its vector argument encodes the axis, angle, and sense of rotation as functions of the concavity of the surface and the angle of incidence. The angle of incidence is expressed in terms of the cross product between the incident and normal vectors. Let us begin with the propagation of light. The position vector rk⫹1 of a ray of light with respect to a chosen origin is completely specified by its initial position rk , its unit propagation vector ␴k⫹1 , and the distance s k⫹1 traveled from its initial position 共see Fig. 2兲: rk⫹1 ⫽rk ⫹s k⫹1 ␴k⫹1 .

共18兲

Equation 共18兲 is equivalent to the formulation of Ref. 25. The vector formulation in Eq. 共18兲 lets us read the law of propagation as if it were written in words: the final position rk⫹1 of a photon is (⫽) its initial position rk connected head-to-tail (⫹) the unit propagation vector ␴k⫹1 stretched by 共juxtaposition multiplication兲 a distance s k⫹1 traveled by

We now reformulate the law of reflection using geometric algebra. We shall divide this section into four parts. First, we derive the law of reflection for a convex interface and define the rotation axis e␴ ␩ k . Next, we derive this axis for a concave interface. Then we shall combine these two laws into one using the concavity function c ␴ ␩ k , which has the values ⫾1. Finally, we shall verify that our formulation reduces to the standard vector formulation for reflection.17,23 A. Convex interface Suppose that a light ray is moving rectilinearly in the direction described by the unit propagation vector ␴k . If the interface k is convex with respect to the incident light, then the outward normal vector ␩k at the point of contact makes an interior angle ␣ k with ␴k . From Fig. 3, the angle of incidence ␤ k is

␤ k⫽ ␲ ⫺ ␣ k .

共19兲

If ␤ ⬘k is the angle of reflection, then the law of reflection states that

␤ k ⫽ ␤ k⬘ .

共20兲

Our aim is to rewrite Eq. 共20兲 in exponential vector form in terms of the unit vectors ␴k and ␩k . To rewrite the law of reflection in Eq. 共20兲, we must relate the unit incident vector ␴k and the unit reflected vector ␴⬘k . From Fig. 3, we see that we can arrive at ␴k⬘ by rotating ␴k clockwise by an angle 2 ␤ k about the unit vector e␴ ␩ k ,


e␴ ␩ k ⫽

␴k Ã␩k , 兩 ␴k Ã␩k 兩

共21兲

and then multiply this result by ⫺1 to reverse the direction of the resulting vector. If we use the exponential form in Eq. 共17兲, we may express these geometrical operations as ˆ

␴k⬘ ⫽⫺ ␴k e ⫺2 i ␤ k e␴ ␩ k .

共22兲

In standard vector analysis, we know that if ␣ k is the interior angle between the unit vectors ␴k and ␩k , then 兩 ␴k Ã␩k 兩 ⫽sin ␣ k .

共23兲

We solve for ␣ k in Eq. 共19兲 and substitute the result in Eq. 共23兲 to obtain 兩 ␴k Ã␩k 兩 ⫽sin共 ␲ ⫺ ␤ k 兲 ⫽sin ␤ k .

共24兲

Hence,

␤ k ⫽sin⫺1 兩 ␴k Ã␩k 兩 .

共25兲

Equation 共29兲 is the law of reflection for both concave and convex interfaces. It says that the reflected ray ␴⬘k is (⫽) and opposite (⫺) to the incident ray ␴k rotated (e) counterclockwise or clockwise 共depending on the concavity c ␴ ␩ k of the interface兲 by twice 共2兲 the incident angle ( ␤ k ) about ( î ) the axis e␴ ␩ k . If the interface is concave, c ␴ ␩ k ⫽⫹1, and the rotation is counterclockwise. If convex, c ␴ ␩ k ⫽⫺1 and the rotation is clockwise.

D. Verification of reflection law One way to verify the exponential reflection law in Eq. 共29兲 is to show that it is equivalent to a standard vector formulation. If we use Euler’s theorem in Eq. 共15兲, the exponential reflection law in Eq. 共29兲 becomes

␴k⬘ ⫽⫺ ␴k cos共 2c ␴ ␩ k ␤ k 兲 ⫺ î ␴k e␴ ␩ k sin共 2c ␴ ␩ k ␤ k 兲 .

共30兲

Thus, Eq. 共22兲 is the exponential vector form of the law of reflection for a convex interface, with its rotation axis given in Eq. 共21兲 and its incident angle ␤ k defined in Eq. 共25兲. The exponential form is advantageous because its symbols may be substituted directly by words: for a convex interface, the reflected ray ␴⬘k is (⫽) opposite (⫺) of the incident ray ␴k rotated (e) clockwise (⫺) by twice (2) the incident angle ( ␤ k ) about ( î ) the axis e␴ ␩ k .

Because the incident vector ␴k is perpendicular to the rotation axis e␴ ␩ k defined in Eq. 共21兲, then ␴k "e␴ ␩ k ⫽0, so that

B. Concave interface

where we used Eq. 共10兲 and a familiar vector cross product identity. If we substitute Eq. 共31兲 into Eq. 共30兲, we obtain

On the other hand, if the surface is concave, instead of Eq. 共19兲, we have

␤ k⫽ ␣ k .

共26兲

Also, the unit axis vector in Eq. 共21兲 reverses direction, so that the reflected vector ␴⬘k is now obtained by a counterclockwise rotation of the incident vector ␴k followed by a direction reversal because of the (⫺1) factor. With these changes, the reflection law in Eq. 共22兲 becomes ˆ

␴k⬘ ⫽⫺ ␴k e 2 i ␤ k e␴ ␩ k .

共27兲

Equation 共27兲 is the exponential vector form of the law of reflection for a concave interface. The interpretation of Eq. 共27兲 is similar to that for the convex interface in Eq. 共22兲, except that the rotation is now counterclockwise (⫹) about the axis e␴ ␩ k . C. The law of reflection To combine Eqs. 共22兲 and 共27兲 into a single equation, we need to define the concavity function c ␴ ␩ k : c ␴␩k⫽

␴k "␩k . 兩 ␴k "␩k 兩

共28兲

If c ␴ ␩ k ⫽1, we say that the interface is concave; if c ␴ ␩ k ⫽⫺1, we say it is convex. By using the definition of the concavity function in Eq. 共28兲, we may combine the laws of reflection in Eqs. 共22兲 and 共27兲 into one: ˆ

␴k⬘ ⫽⫺ ␴k e 2 i c ␴ ␩ k ␤ k e␴ ␩ k .

共29兲

î ␴k e␴ ␩ k ⫽⫺ ␴k Ãe␴ ␩ k

␴k Ã␩k ⫽⫺ ␴k Ã 兩 ␴k Ã␩k 兩 ⫽

1 共 ␩ ⫺ 共 ␴k "␩k 兲 ␴k 兲 , 兩 ␴k Ã␩k 兩 k

共31兲

␴k⬘ ⫽⫺ ␴k cos共 2c ␴ ␩ k ␤ k 兲 ⫹

共 ␴k "␩k 兲 ␴k ⫺ ␩k sin共 2c ␴ ␩ k ␤ k 兲 . 兩 ␴k Ã␩k 兩

共32兲

Because c ␴ ␩ k ⫽⫾1, we have cos共 2c ␴ ␩ k ␤ k 兲 ⫽cos共 2 ␤ k 兲 ,

共33a兲

sin共 2c ␴ ␩ k ␤ k 兲 ⫽c ␴ ␩ k sin共 2 ␤ k 兲 .

共33b兲

Equation 共33兲 lets us rewrite Eq. 共32兲 as

␴k⬘ ⫽⫺ ␴k cos共 2 ␤ k 兲 ⫹c ␴ ␩ k

共 ␴k "␩k 兲 ␴k ⫺ ␩k sin共 2 ␤ k 兲 . 兩 ␴k Ã␩k 兩共34兲

Now, because ␣ k is the interior angle between the incident angle ␴k and the normal vector ␩k , we have

␴k "␩k ⫽cos ␣ k .

共35兲

If we use Eqs. 共19兲 and 共26兲, we can show that

␣ k ⫽c ␴ ␩ k ␤ k ⫹ 共 1⫺c ␴ ␩ k 兲

␲ . 2

共36兲

We substitute Eq. 共36兲 into Eq. 共35兲 and arrive at

␴k "␩k ⫽c ␴ ␩ k cos ␤ k ,

共37兲

by virtue of the trigonometric identity cos(␲⫺␤k) ⫽⫺cos ␤k . We employ the results in Eqs. 共25兲 and 共37兲, so that the expanded reflection law in Eq. 共34兲 simplifies to


the incident propagation vector ␴k by a counterclockwise rotation by an angle ␤ k ⫺ ␤ k⫹1 about the axis e␴ ␩ k defined in Eq. 共21兲: ˆ

␴k⫹1 ⫽ ␴k e ⫹ i e␴ ␩ k ( ␤ k ⫺ ␤ k⫹1 ) .

Fig. 4. The law of refraction for a concave interface expressed by ␴k⫹1 ˆ

⫽ ␴k e i ( ␤ k ⫺ ␤ k⫹1 )e␴ ␩ k . In words, the refracted ray ␴k⫹1 is 共⫽兲 the incident ray ␴k rotated (e) counterclockwise (⫹) about ( î ) the axis e␴ ␩ k 共out of the paper兲 by the difference (⫺) between the incident angle ␤ k and the refracted angle ␤ k⫹1 . Note that because ␤ k ⬍ ␤ k⫹1 , the difference ␤ k ⫺ ␤ k⫹1 is negative, so that the rotation is not counterclockwise but clockwise.

␴k⬘ ⫽⫺ ␴k cos共 2 ␤ k 兲 ⫹c ␴ ␩ k 共 c ␴ ␩ k ␴k cos ␤ k ⫺ ␩k 兲

sin共 2 ␤ k 兲 . sin ␤ k

共38兲

If we use the trigonometric double angle identities together with the Pythagorean theorem, Eq. 共37兲 reduces to

␴k⬘ ⫽ ␴k ⫺2c ␴ ␩ k ␩k cos ␤ k .

共39兲

Equation 共39兲 is equivalent to that given in Klein and Furtak,23 except that the authors’ definition of the normal vector ␩k to the interface is inward instead of outward. V. REFRACTION We now reformulate the law of refraction using geometric algebra. Similar to the previous discussion on reflection, we shall divide our discussion into four main parts, though in slightly different order for pedagogical reasons. We begin with the derivation of the law of refraction for a concave interface and followed by that for the convex interface. Then we shall combine the two formulations into one using the concavity function c ␴ ␩ k defined in Sec. IV. Finally we verify our result by converting it to the standard vector formulation for refraction.17,23 A. Concave interface Suppose that an interface k separates medium k and medium k⫹1, characterized by the indices of refraction n k and n k ⫹1, respectively. Suppose also that the incident light is in medium k and the refracted light is in medium k⫹1. If ␤ k and ␤ k⫹1 are the incident and refracted angles, respectively, then we know that

␤ k⫹1 ⫽sin⫺1

冉

nk n k⫹1

冊

sin ␤ k .

共40兲

Our aim is to rewrite Eq. 共40兲 in terms of the unit incident vector ␴k and the unit normal vector ␩k of the interface k at the point where the light hits the interface. If the interface is concave with respect to the incident 共see Fig. 4兲, the refracted propagation vector ␴k⫹1 is related to

共41兲

Because the interface is concave, the incident angle ␤ k is equal to the interior angle ␣ k between ␴k and ␩k , as given in Eq. 共25兲. Thus, Eq. 共41兲 is the law of refraction for a concave interface, with the rotation axis e␴ ␩ k given in Eq. 共21兲, the incident angle ␤ k given in Eq. 共25兲, and the refracted angle ␤ k⫹1 given in Eq. 共40兲. The law of refraction in Eq. 共41兲 for a concave interface may be read as if it were written in words: the refracted ray ␴k⫹1 is (⫽) the incident ray ␴k rotated (e) counterclockwise (⫹) about ( î ) the axis e␴ ␩ k by the incident angle ␤ k minus (⫺) the refracted angle ␤ k⫹1 .

B. Convex interface If the interface is convex, the rotation axis e␴ ␩ k defined in Eq. 共21兲 would reverse direction, so that e␴ ␩ k is replaced by ⫺e␴ ␩ k in Eq. 共43兲. Thus, we have ˆ

␴k⫹1 ⫽ ␴k e ⫺ i e␴ ␩ k ( ␤ k ⫺ ␤ k⫹1 ) .

共42兲

Equation 共42兲 is the law of refraction for a convex interface. The interpretation of Eq. 共42兲 is similar to that for the convex interface, except that the rotation is now clockwise about the axis e␴ ␩ k .

C. The law of refraction To combine the laws of refraction in Eqs. 共41兲 and 共42兲 into one, we employ the concavity function c ␴ ␩ k defined in Eq. 共28兲: ˆ

␴k⫹1 ⫽ ␴k e i c ␴ ␩ k e␴ ␩ k ( ␤ k ⫺ ␤ k⫹1 ) .

共43兲

Equation 共43兲 is the exponential form of the law of refraction for both convex and concave interfaces. It implies that the refracted ray ␴k⫹1 is (⫽) the incident ray ␴ k rotated (e) counterclockwise or clockwise 共depending on the concavity c ␴ ␩ k of the interface兲 about ( î ) the axis e␴ ␩ k by the incident angle ␤ k minus (⫺) the refracted angle ␤ k⫹1 . If the interface is concave, c ␴ ␩ k ⫽⫹1, and the rotation is counterclockwise. If convex, c ␴ ␩ k ⫽⫺1, and the rotation is clockwise.

D. Verification of refraction law Just like in Sec. IV D, we may verify the exponential refraction law in Eq. 共43兲 by showing its equivalence to the standard vector formulation. We use Euler’s theorem in Eq. 共15兲 and expand the exponential refraction law in Eq. 共43兲 to

␴k⫹1 ⫽ ␴k cos共 c ␴ ␩ k 共 ␤ k ⫺ ␤ k⫹1 兲兲 ⫹ î ␴k e␴ ␩ k sin共 c ␴ ␩ k 共 ␤ k ⫺ ␤ k⫹1 兲兲 .

共44兲

If we employ the identities in Eqs. 共33兲 and 共31兲 together with the definitions of the incident angle ␤ k in Eqs. 共25兲 and 共37兲, Eq. 共44兲 may be expressed as


冉

␴k⫹1 ⫽ cos共 ␤ k ⫺ ␤ k⫹1 兲 ⫺

冊

cos ␤ k sin共 ␤ k ⫺ ␤ k⫹1 兲 ␴k sin ␤ k

sin共 ␤ k ⫺ ␤ k⫹1 兲 ⫹c ␴ ␩ k ␩k , sin ␤ k

共45兲

after rearranging terms. Equation 共45兲 may be further simplified to

␴k⫹1 ⫽

sin ␤ k⫹1 ␴ sin ␤ k k

冉

⫹c ␴ ␩ k cos ␤ k⫹1 ⫺cos ␤ k

冊

sin ␤ k⫹1 ␩k , sin ␤ k

ACKNOWLEDGMENTS This research was supported by the Physics Department of the Loyola School of Science and Engineering of the Ateneo de Manila University and by the Manila Observatory. The authors would like to thank Felix Muga II for the REVTEX 4 class file and other packages, Jerrold Garcia for his Linux tutorial, and Reginald Marcelo for debugging the graphics importation. a兲

Electronic mail: [email protected] Electronic mail: [email protected] 1 J. J. Sakurai, Advanced Quantum Mechanics 共Addison–Wesley, Reading, MA, 1967兲, pp. 215, 216. 2 K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide 共Cambridge U.P., Cambridge, UK, 2002兲, 2nd ed., pp. 95, 98. 3 H. Goldstein, Classical Mechanics 共Addison–Wesley, Reading, MA, 1980兲, 2nd ed., pp. 148 –158. 4 Ref. 3, pp. 164 –166. 5 K. Gurlebeck and W. Sprossig, Quaternionic and Clifford Calculus for Physicists and Engineers 共Wiley, Chichester, 1997兲. 6 J. B. Kuipers, Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality 共Princeton U.P., Princeton, NJ, 2002兲. 7 W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, ‘‘Light polarization: A geometric algebra approach,’’ Am. J. Phys. 61, 534 –544 共1993兲. 8 D. Hestenes and G. Sobcyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics 共Reidel, Dordrecht, 1984兲. 9 Bernard Jancewicz, Multivectors and Clifford Algebra in Electrodynamics 共World Scientific, Singapore, 1988兲. 10 P. Lounesto, Clifford Algebras and Spinors 共Cambridge U.P., Cambridge, UK, 2001兲. 11 Reference 3, p. 194. 12 D. Hestenes, ‘‘Oersted Medal Lecture 2002: Reforming the mathematical language of physics,’’ Am. J. Phys. 71 共2兲, 104 –121 共2003兲. 13 D. Hestenes, New Foundations for Classical Mechanics 共Reidel Dordrecht, 1990兲, 2nd ed., pp. 54 – 61. 14 Reference 13, pp. 277–292. 15 M. Derome, ‘‘Laws of reflection from two or more plane mirrors in succession,’’ in Applications of Geometric Algebra in Computer Science and Engineering, edited by L. Dorst, C. Doran, and J. Lasenby 共Birkhauser, Boston, 2002兲, pp. 249–259. 16 Terje Vold, ‘‘An introduction to geometric algebra with application in rigid body mechanics,’’ Am. J. Phys. 61, 491–504 共1993兲. 17 D. S. Goodman, Geometric Optics, Handbook of Optics: Fundamentals, Techniques, and Design 共McGraw–Hill, New York, 1995兲, Vol. 1, pp. 26 –29. 18 Eugene Hecht and Alfred Zajac, Optics 共Addison–Wesley, Reading, MA, 1974兲, pp. 171–175. 19 F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics 共Prentice–Hall, London, 1996兲, 2nd ed., pp. 62– 84. 20 D. S. Goodman, Geometric Optics, Handbook of Optics: Fundamentals, Techniques, and Design 共McGraw–Hill, New York, 1995兲, Vol. 1, pp. 70– 80. 21 Reference 17, pp. 60– 68. 22 Max Born and Emil Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light 共Pergamon, Oxford, 1964兲, 2nd ed., pp. 190–196. 23 M. V. Klein and T. E. Furtak, Optics 共Wiley, New York, 1986兲, pp. 129– 141. 24 O. Heaviside, Electromagnetic Theory, Part I, unabridged edition of work originally published in 1893, 1899, and 1912 共Chelsea, New York, 1971兲. 25 Reference 22, pp. 121–127. b兲

共46兲

with the help of angle difference formulas for the sine and cosine functions. If we substitute Eq. 共40兲 in Eq. 共46兲 and multiply the result by n k⫹1 , we arrive at n k⫹1 ␴k⫹1 ⫽n k ␴k ⫹c ␴ ␩ k 共 n k⫹1 cos ␤ k⫹1 ⫺n k cos ␤ k 兲 ␩k . 共47兲 Equation 共47兲 is the vector addition form of the law of refraction given in Ref. 23. VI. SUMMARY AND CONCLUSIONS In this paper, we have used the geometric algebra Cl3,0 , also known as the Pauli algebra, to reformulate the laws of geometric optics. For propagation, we adopted the standard vector addition form. But for reflection and refraction, we expressed the laws in exponential form, with the exponential function acting as the rotation operator of the incident vector. The arguments of the exponential are functions of the incident ray and the normal vector to the interface: the incident angle, the concavity of the interface, and the rotation axis. The reformulated law of reflection states that the reflected ray is opposite to the incident ray rotated by twice the incident angle about the axis perpendicular to both the incident and normal vectors. On the other hand, the law of refraction says that the refracted ray is the incident ray rotated by the difference between the incident and refracted angles, about the axis perpendicular to both the incident and normal vectors. The rotation axis is defined as the cross product between the incident and normal vectors. The rotation sense is specified by the concavity function: the rotation is clockwise about the rotation axis if the interface is convex with respect to the incident ray and counterclockwise if the interface is concave. The exponential form of the laws of reflection and refraction are easily converted into words and pictures and vice versa. Thus, the geometric algebra formalism presented in this paper provides a visual aid and a computational tool for the study and teaching of geometric optics. In future work, we shall show how these new formulations facilitate the efficient derivation of exact ray transfer equations for meridional and skew rays in optical fibers and lenses.
