Classical optics representation of the quantum mechanical translation operator via ABCD matrices
aa r X i v : . [ phy s i c s . op ti c s ] M a y Classical optics representation of the quantum mechanicaltranslation operator via ABCD matrices
Marco Ornigotti and Andrea Aiello , Max Planck Institute for the Science of Light,G ¨ u nther-Scharowsky-Strasse 1/Bau24, 91058 Erlangen, Germany and Institute for Optics, Information and Photonics,University of Erlangen-Nuernberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany ∗ (Dated: July 28, 2018)The ABCD matrix formalism describing paraxial propagation of optical beamsacross linear systems is generalized to arbitrary beam trajectories. As a by-productof this study, a one-to-one correspondence between the extended ABCD matrix for-malism presented here and the quantum mechanical translation operator is estab-lished. I. INTRODUCTION
Rays of light propagate along rectilinear trajectories in air. Therefore, at the genericposition x a ray, which may be represented by the linear function f ( x ) = a + bx , iscompletely determined by a pair of numbers solely: f ( x ) and f ′ ( x ). Such a pair may berepresented in a vector-like form as follows: f ( x ) = f ( x ) f ′ ( x ) . (1)The simple linear relation existing between f ( x ) and f ( x ) at two arbitrarily chosenpositions x and x = x + L , with L >
0, is usually written in optics textbooks [1–4] in thefollowing matrix form: ∗ Electronic address: [email protected] f ( x ) f ′ ( x ) = L f ( x ) f ′ ( x ) . (2)The 2 × x − x ≡ δx ≪
1. In this case, a simple approach like the one given by the ABCD matricesfails to be efficient and one has to embrace a more complicated method capable of dealingwith the medium inhomogeneities [7]. However, formally, the ABCD matrix approach canstill be used if an appropriate generalization of this method is constructed by observing thatthe ABCD matrix formalism is nothing but the consequence of a linearization of the trajec-tory of a ray of light around the initial point x [8, 9], namely a Taylor expansion truncatedup to and including first order terms. Such a linearization procedure, however, is physicallymeaningful only for x close enough to x : x = x + δx . But what if x is no longer closeto x ? Does the first order Taylor expansion break down? If so, can such expansion besuitably extended? If higher order terms must be retained, what is their physical meaning?To answer these question we intend to proceed in a two-step reasoning. Firstly, in Sect.2, we put on rigorous basis this linearization procedure showing that a 2 × ∞ × ∞ matrix describing the full nonlineardynamics of a curvilinear ray of light. In Sect. 3 we then discuss the physical meaningof the proposed generalization scheme, pointing out that the generalized ABCD matrixis nothing but a physical representation of the well known quantum translation operator e L ( d/dx ) in one dimensional quantum mechanics. II. GENERALIZED ABCD MATRICES FOR NON-RECTILINEAR LIGHTPROPAGATION
To begin with, let us first re-derive Eq.(2) for an arbitrary linear function f ( x ) that nowwe write in the following manner: f ( x ) = a + a x ≡ a i + a i ( x − x i ) , (3)where x i is an arbitrary point belonging to the domain of the function f ( x ). If we choose x i = 0 then we retrieve the previous expression f ( x ) = a + bx with a = a and a = b .However, for x i = 0, the last equality in Eq.(3) gives: a i = a + a x i = f ( x i ) , (4a) a i = a = f ′ ( x i ) , (4b)where, for the sake of simplicity, we have introduced the notation a ik = a k ( x i ). Since thepoint x i is arbitrarly chosen, we can pick out a different point x j = x i + L and write: f ( x ) = a i + a i ( x − x i ) = a j + a j ( x − x j ) . (5)By equating the factors with the same powers of x at the second and third terms in theequation above, we obtain a j = a i + ( x j − x i ) a i and a j = a i . This can relation be rewrittenin the following matrix form: a j a j = x j − x i a i a i . (6)This result is equivalent with the one written in Eq. (2), if we identify a j = f ( x ), a j = f ′ ( x ), a i = f ( x ) and a i = f ′ ( x ). Obviously, the formal derivation of Eq.(2) via the steps(3-6) it is highly redundant for the linear-function case. However, it has the virtue to begeneralizable to the case of non-rectilinear ray propagation.Now, in order to describe a ray that propagates in an arbitrary inhomogeneous mediumwe need a generic smooth non-linear function f ( x ) which can be expanded in a Taylor seriesaround x = 0 as follows: f ( x ) = a + a x + a x + · · · . (7)For any x i ∈ R we can write x = x − x i + x i and insert this relation into Eq.(7) to obtain f ( x ) = a + a ( x − x i + x i ) + a ( x − x i + x i ) + · · · = ∞ X n =0 a in ( x − x i ) n , (8)where the a in coefficient are given by: a i = f ( x i ) , (9a) a in = 1 n ! d n f ( x ) dx n (cid:12)(cid:12)(cid:12) x = x i = ∞ X k = n (cid:18) kn (cid:19) a k x k − ni . (9b)Now we can repeat the same reasoning that lead to Eq.(8), but with a different expansionpoint x j = x i + L , and write the following equality: ∞ X n =0 a in ( x − x i ) n = ∞ X n =0 a jn ( x − x j ) n , (10)which simply states the independence of f ( x ) from the expansion points x i and x j . Byexpanding both sides of this equation with the help of the Newton’s binomial formula oneobtains: ∞ X n =0 a in n X k =0 (cid:18) nk (cid:19) x k ( − x i ) n − k = ∞ X n =0 a jn n X k =0 (cid:18) nk (cid:19) x k ( − x j ) n − k . (11)This expression can be turned into a recursive relation by equating terms with the samepower of x . Then, for k = 0 we have: ∞ X n =0 ( − n a in x ni = ∞ X n =0 ( − n a jn x nj , (12)which can be rewritten, after isolating the n = 0 term, as: a j = a i + ∞ X n =1 ( − n ( a in x ni − a jn x nj ) . (13)For k = 1 the same operation yields: a j = a i + ∞ X n =2 ( − n − ( a in x n − i − a jn x n − j ) . (14)This procedure can be iterated for arbitrary values of k thus generating the following recur-sive relation: a jk = a ik + ∞ X n = k +1 (cid:18) nk (cid:19) ( − n − k ( a in x n − ki − a jn x n − kj ) , (15)with k = 0 , , · · · , n . The equation above can be seen as a linear algebraic system relatingthe variables a in to the quantities a jn . This result can be then written in matrix form asfollows: b j (0) b j (0) b j (0) · · · b j (1) b j (1) · · · b j (2) · · · ... ... ... . . . a j a j a j ... = b i (0) b i (0) b i (0) · · · b i (1) b i (1) · · · b i (2) · · · ... ... ... . . . a i a i a i ... , (16)where a j = ( a j , a j , · · · ), a i = ( a i , a i , · · · ) and b jn ( k ) = (cid:0) nk (cid:1) ( − n − k x n − kj . If we call B ( j ) thematrix on the left-side of the previous equation and B ( i ) the one on the right-side, Eq. (16)can be written in the compact form: B ( j ) a j = B ( i ) a i , (17)where again the shorthand notations B ( i ) = B ( x i ) and B ( j ) = B ( x j ) are used for the sakeof clarity. Solving for a j by multiplying on the left both sides of the previous equation by B ( j ) − and defining A = B − ( j ) B ( i ), we can write the relation between the vectors a j and a i as a j = Aa i . (18)The matrix A is our sought generalized ABCD matrix, whose expression is the following: A = B − ( j ) B ( i ) = L L L · · · L L · · · L · · · · · · ... ... ... ... . . . ≡ A ( L ) , (19)where L = x j − x i . Note that this matrix contains the usual (i.e. linear) ABCD matrixdefined in Eq.(2) as the first 2 × L L = L + L , (20)as can be checked by a straightforward calculation. Simliarly, for the 3 × L L L L L L = L + L ( L + L ) L + L )0 0 1 . (21) III. CONNECTION WITH THE TRANSLATION OPERATOR
The composition properties of the various sub-matrices, as given for the linear andquadratic order by Eq. (20) and Eq. (21) respectively, have a straightforward physicalmeaning: they illustrate the fact that propagation across two consecutive distances L and L can be described as a single propagation along the distance L + L . From amathematical point of view, this is a signature of the semigroup property of our generalizedABCD matrices [4]. With the help of a suitable mathematical software for algebraicmanipulation, it is not difficult to verify via explicit N × N matrix multiplications, thatEq.(21) is valid for arbitrary N . Thus, by iteration, one can easily convince oneself thatthe matrix A satisfies the following relation [2]: N Y n =1 A ( L n ) = A (cid:16) N X n =1 L n (cid:17) . (22)The physical implications of this relation are immediately understood: the propagation ofthe function through the total distance L + L + · · · + L N can be achieved by consecutivepropagation across the distances L , L , · · · , L N .This analogy is not accidental. A closer inspection to Eq.(18) reveals in fact that thisequation tells us how the value of the function f ( x ) in a point x j can be calculated knowingthe value of the same function in a point x i < x j . With this in mind, we can calculate thederivative of f ( x ) as follows: df ( x ) dx = lim ∆ x → f ( x + ∆ x ) − f ( x )∆ x = lim x j → x i (cid:16) a j − a i x j − x i (cid:17) = lim L → (cid:16) A − I L (cid:17) a i ≡ Da i , (23)where we have chosen ∆ x = x j − x i ≡ L in order to represent f ( x + ∆ x ) as a j and f ( x ) as a i . Note that this does not cause any loss of generality, since the definitionof derivative involves only the concept of neighboring points and, as discussed previ-ously, the quantities a i and a j represent the value of the function f ( x ) in two arbitraryneighboring points. Note, moreover, that in the last equality we used Eq.(18) to write a j as a function of a i . Here, D is the matrix representation of the differential operator d/dx [11] D = lim L → (cid:16) A − I L (cid:17) = · · · · · · · · · · · · ... ... ... ... . . . . (24)At this point, it is not difficult to see, via an explicit calculation, that the generalizedABCD matrix is related to the differential operator by the following formula: A ( L ) = ∞ X k =0 ( L D ) k k ! = e L D . (25)Equation (25) gives therefore an actual physical representation of the well-known translationoperator e L ( d/dx ) [12], such that: e L ddx f ( x ) | x =0 = ∞ X k =0 f k (0) k ! L k = f ( L ) . (26)In our case, in fact, the action of the A matrix completely defines the vector F ( x ) ≡ a j atthe point x knowing the expression of the F ( x ) ≡ a i at the point x , i.e. F ( x ) = e L D F ( x ) = A ( L ) F ( x ) . (27) IV. CONCLUSIONS
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