Chapter 4 Airy Functions - SPIE

1 Chapter 4 airy IntroductionAiry Functions are named after the English astronomer George Biddell airy (1801 1892). airy s first mathematical work was on the diffraction phenom-enon, namely, theAiry disk the image of a point object by a telescope which is familiar to all of us in optics. The name airy is connected with manyphysical phenomena and includes, besides the airy disk, the airy spiral, anoptical phenomenon visible on quartz crystals, and the airy stress function was very interested in optics and in fact studied the formation ofrainbows. A good qualitative summary of the rainbow is given by paper Adam shows how the optical rainbow can be studied at many levels:(i) geometrical optics (rays), (ii) the airy approximation, (iii) Mie scattering,(iv) complex angular momentum, and (v) catastrophe theory.

2 airy s analysis isapproximate but applies well to large raindrops that make up the commonrainbow (for small drops, catastrophe theory has been used). Details of airy sanalysis can be found in the Chapter on the optics of raindrops in the bookby van de Hulst41(see also Berry42). airy also analyzed the intensity of lightnear a caustic wavefront. During his investigation utilizing the scalardiffraction integral, he introduced a functionW(m) defined by the integralW m Z`0cos p2 v3 mv dv( )as a solution of the differential equationd2 Wdv2 p212mW 0:( )Jeffreys43introduced the modern notation currently used:Ai x 1pZ`0cos t33 xt dt:( )71 Equation ( ) is referred to as the airy integral and can be shown to be thesolution to a homogeneous differential equation of the typed2ydx2 xy:( )This equation is generally known as the airy equation or the airy differentialequation.

3 However, caution must be exercised in differentiating Eq. ( )under the integral, since the integral would become indeterminate ast!`.For a rigorous proof of this solution, we need to use complex variabletechniques. Here, we show a simple intuitive technique. Let s define a functionFs(x) as follows:Fs x Zs0cos tx 13t3 dt,( )wheresis a large but finite number. Substituting thisFsforyin Eq. ( ), weobtaind2 Fsdx2 xFs x 1pZs0 t2 x cos tx 13t3 dt 1psin sx 13s3 :( )Ass!`, the function oscillates rapidly between 1pand 1p. Therefore, wecan set the mean value of the function equal to zero and show thatFsis asolution of Eq. ( ) in the limits!`. In this limitFsbecomes the Airyintegral [Eq. ( )]. (x) andBi(x) FunctionsAi(x) can be given by a power series expansionAi x 132 3G 23 1 13!

4 X3 1 4 6x6 1 4 7 9!x9 131 3G 13 x 24!x4 2 5 7!x7 2 5 8 10!x10 :( )The airy differential equation [Eq. ( )] is a second-order differentialequation; it must, therefore, have a second independent solution. This isdenoted asBi(x) and is given byBi x 1pZ`0 exp 13t3 xt sin 13t3 xt dt:( )72 Chapter 4 The power series expansion ofBi(x) is given asBi x ffiffiffi3p32 3G 23 1 13!x3 1 4 6!x6 ffiffiffi3p31 3G 13 x 24!x4 2 5 7!x7 :( )FunctionsAi(x) andBi(x) are the airy Functions are available as airy in in Python. Thisfunction returns four arrays,Ai,Ai0,Bi, andBi0in that order. Figure the plots of airy is usual, let us write a power series solution of the formy x a0 a1x a2x2 ( )to solve the airy equation.

5 Substituting Eq. ( ) in Eq. ( ) and simplifying,we obtain2a2 3 2 a3x 4 3 a4x2 n n 1 anxn 2 a0x a1x2 a2x3 an 3xn 2 :( )Figure airy functionsAi(x) (solid line) andBi(x) (dotted line).73 airy FunctionsEquating coefficients, we find thata2 0,a3 a0 2 3 ,a4 a1 3 4 ..an an 3n n 1 ,( )and consequently, we obtain for the solution,y x a0 1 x2 2 3 x6 2 3 5 6 a1 x x4 3 4 x7 3 4 6 7 ,( )wherea0anda1are the two arbitrary constants that need to be fixed applyingthe appropriate boundary conditions. If we were to writef x 1 13!x3 1 4 6!x6 1 4 7 9!x9 ( )g x x 24!x4 2 5 7!x7 2 5 8 10!x10 ,( )we notice that Eq. ( ) could be written asy a0 1 13!x3 1 4 6!x6 1 4 7 9!x9 a1 x 24!

6 X4 2 5 7!x7 2 5 8 10!x10 :( )If we seta0 132 3G 23 ( )a1 131 3G 13 ,( )74 Chapter 4we obtain the series expansionAi(x) [Eq. ( )] andBi(x) [Eq. ( )] asAi x a0f x a1g x ( )Bi x ffiffiffi3p a0f x a1g x :( )Equations ( ) and ( ) are the ways these two functionsAi(x) andBi(x)are traditionally written. It is easy to show, from Eqs. ( ) and ( ), thatAi 0 a0,Ai0 0 a1,Bi 0 ffiffiffi3pa0,Bi0 0 ffiffiffi3pa1,( )where the prime (0) is used to denote the derivative, andf0(0) is a short formfordfdxjx 0. For higher derivatives we can show thatAi n 0 1 ncnsin p n 1 3 ( )Bi n 0 cn 1 sin p 4n 1 6 ( )withCn 1p3 n 2 3G n 13 :( )Here, the superscriptndenotes then-th derivative. We can get the ascendingseries of the derivatives by differentiatingf(x) andg(x) in Eq.

7 ( ) term byterm, as shown below:Ai0 x a0f0 x a1g0 x Bi0 x ffiffiffi3p a0f0 x a1g0 x f0 x x22 1 2 3 x55 1 2 3 5 6 x88 g0 x 1 1 1 3 x33 1 1 3 4 6 x66 1 1 3 4 6 7 9 x99 :( ) Relationship with Bessel FunctionsIn Sec. , we mentioned that the beta function could be written in terms ofthe Bessel function. Similarly, through Eqs. ( ), we related the Fresnelintegral to the Bessel Functions . In a similar way, we deal with the Besselfunction before it makes its appearance in this book (see Ch. 5). This isbecauseAi(x) andBi(x) can be expressed in terms of the Bessel function, and75 airy Functionsusing the asymptotic forms of the Bessel function, it is possible to get theasymptotic form of the airy function.

8 Readers may choose to skip this sectionand come back to it after working through Ch. Eq. ( ),d2y x dx2 xy x 0( )in the regionx,0. Let us definez x. Therefore, , we obtaind2y z dz2 zy z 0:( )Lety(z) z1/2f(z). The above equation, therefore, becomesz1 2d2fdz2 z 1 2dfdz 14z 3 2 z3 2 f z 0:( )Now, we make another transformation of the variablez 23z3 2. This resultsin the following transformations of the derivatives:z 23z3 2( )dfdz dfdzdzdz z1 2dfdz( )d2fdz2 z1 2ddz dfdz dzdz 12z 12dfdz zd2fdz2 12z 1 2dfdz:( )Substituting these expressions for the derivatives into Eq. ( ) andsimplifying, we end up withz3 2d2fdz2 32dfdz 14z 3 2 z3 2 f 0:( )Using the transformationz 23z3 2, we can simplify the above equation toobtainz2d2fdz zdfdz z2 19 f 0:( )As will be seen in Ch.

9 5 [Eq. ( )], the solution of the differential equationx2d2ydx2 xdydx x2 n2 y 0( )76 Chapter 4is the Bessel function of the ordern. Therefore, Eq. ( ) represents a Besseldifferential equation of the order13. Since it is a second-order differentialequation, it has two solutions, namely Bessel Functions of the order 13. Thetwo independent solutions of the equation arey jxj1 2J1 3 z ,andy jxj1 2J 1 3 z ,wherez 23z3 2 23jxj3 2. The appropriate linear combinations of the Airyfunctions forx,0 are, therefore,A1 x 13ffiffiffixp J 1 3 z J 1 3 z ,( )andB1 x ffiffiffix3r J 1 3 z J 1 3 z ,( )with the understanding thatz 23jxj3 can do a similar analysis for the and obtain as twoindependent solutions,y x1 2I1 3z,andy x1 2I 1 3 z ,whereInrepresents the modified Bessel function (see Sec.)

10 As above, theAiry Functions then becomeAi x ffiffiffixp3 I 1 3 z I 1 3 z ( )Bi x ffiffiffix3r I 1 3 z I 1 3 z :( ) airy Functions are thus Bessel Functions or linear combinations of thesefunctions of the order13. Jeffreys44makes an interesting observation about thisrelationship between the Bessel Functions and the airy Functions : Bessel Functions of order13seem to have no application except toprovide an inconvenient way of expressing this [ airy ] function. 77 airy Functions