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In this paper, the artificial generation of elementary catastrophe optics having odd codimensions K = 1, 3 and 5 such as the Fold, the Swallowtail and the Wigwam diffraction caustics is investigated theoretically. It is shown that the integral catastrophes with odd polynomials phase functions can be reduced to the well-known Airy-Hardy cosine integrals. In this connection, the caustic functions of the Fold, Swallowtail and Wigwam caustic beams are expressed in closed-form in terms of Airy-Hardy cosine functions. An optical method based on the Fourier transform similar to that described by Lohmann et al. [Optics Comm. 109 (1994) 361-367] is proposed for the generation of the Fold, Swallowtail and Wigwam caustic beams. The displaying of the catastrophe patterns with K = 1, 3 and 5 is optically implemente d in the Fourier transform device by using simple binary screens with tailored polynomials transmission.

In recent years, a lot of researches have been devoted to the nonconventional applications of optical computing [

ψ n ( C ) = 1 2 π ∫ − ∞ ∞ e i ϕ n ( s ; C ) d s , (1a)

where

ϕ n ( s ; C ) = s n + ∑ j = 1 n − 2 C j s j , (1b)

where s represents the state variable, C_{j} are the n-2 control parameters and ϕ n is the catastrophe phase polynomial of n-th order that depends on the state variable s and the control parameters C_{j} with the codimension K = n-2.

The three members of the hierarchy catastrophe structures that we will examine in this paper are given in

By using the change of the variable x = s n 1 / n , one obtains the relationship

between the control parameters C and Δ in the Refs. [

C = a 3 1 / 3 . (2)

and the structure function ψ 1 ( C ) is proportional to the well-known Airy function

Diffraction catastrophe | Codimension K | Berry’s unfolding ϕ ( s ; C ) [ | Stewart’s unfolding ϕ ( x ; Δ ) [ |
---|---|---|---|

Fold | 1 | s 3 3 + C s | x 3 + a x |

Swallowtail | 3 | s 5 5 + C 3 s 3 3 + C 2 s 2 2 + C 1 s | x 5 + a x 3 + b x 2 + c x |

Wigwam | 5 | s 7 7 + C 5 s 5 5 + C 4 s 4 4 + C 3 s 3 3 + C 2 s 2 + C 1 s | x 7 + a x 5 + b x 4 + c x 3 + d x 2 + e x |

ψ 1 ( C ) = 1 2 π ∫ − ∞ + ∞ d s ⋅ e i ( s 3 3 + C s ) = 2 π A i ( C ) , (3a)

where the Airy function is defined as [

A i ( x ) = 1 π ∫ 0 + ∞ d t ⋅ cos ( t 3 3 + x t ) . (3b)

For K = 3, the phenomenon is called Swallowtail diffraction catastrophe [

C 1 = 1 5 1 / 5 c , C 2 = 2 5 2 / 5 b , C 3 = 3 5 3 / 5 a , (4)

and the caustic function reads

ψ 3 ( C 1 , C 2 , C 3 ) = 1 2 π ∫ − ∞ + ∞ d s ⋅ e i ( s 5 5 + C 3 s 3 3 + C 2 s 2 2 + C 1 s ) . (5)

The case K = 5 corresponds to the so-called Wigwam diffraction catastrophe [

C 1 = 1 7 1 / 7 e , C 2 = 1 7 2 / 7 d , C 3 = 3 7 3 / 7 c , C 4 = 4 7 4 / 7 b and C 5 = 5 7 5 / 7 a . (6)

and

ψ 5 ( C 1 , C 2 , C 3 , C 4 , C 5 ) = 1 2 π ∫ − ∞ + ∞ d s ⋅ e i ( s 7 7 + C 5 s 5 5 + C 4 s 4 4 + C 3 s 3 3 + C 2 s 2 2 + C 1 s ) . (7)

It is worth noting that, apart from the case K = 1, the caustic functions ψ n above haven’t been expressed, to the best of our knowledge, in terms of well-known mathematical functions. In the remainder of the paper, we will show the relationship between these catastrophe functions and the Airy-Hardy integrals, and propose an optical method by using tailored binary screens, for the creation of the elementary optical catastrophes of odd codimensions (K = 1, 3 and 5).

One can note that for particular values of the control parameters, the phase functions in Equations (6) and (7) are reduced to odd polynomials, and thus the associated catastrophe functions are proportional to the Airy-Hardy cosine integrals [

ψ 1 ( C ) = 2 π ∫ 0 + ∞ d s ⋅ cos ( s 3 3 + C s ) , (8)

ψ 3 ( C 1 , C 3 ) = 2 2 π ∫ 0 + ∞ d s ⋅ cos ( s 5 5 + C 3 s 3 3 + C 1 s ) , (9)

and

ψ 5 ( C 1 , C 3 , C 5 ) = 2 π ∫ 0 + ∞ d s ⋅ cos ( s 7 7 + C 5 s 5 5 + C 3 s 3 3 + C 1 s ) . (10)

These last integrals belong to the Airy-Hardy integrals class and then they can be expressed in the canonical form [

C h n ( α ) = ∫ 0 + ∞ d t ⋅ cos [ T n ( t , α ) ] , (11a)

where C h n ( α ) denotes the Airy-Hardy function and

T n ( t , α ) = t n F 2 1 ( − 1 2 n , 1 − n 2 ; 1 − n ; − 4 α t 2 ) , for n ≥ 2 (11b)

where F 2 1 ( . ) is the hypergeometric function defined by [

F 2 1 ( a , b ; c ; z ) = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n z n n ! , (12a)

where ( . ) n is the Pochhammer symbol.

The first odd-order functions T n ( t , α ) are given by

{ T 3 ( t , α ) = t 3 + 3 α t T 5 ( t , α ) = t 5 + 4 α t 3 + 5 α 2 t T 7 ( α ) = t 7 + 7 α t 5 + 14 α 2 t 3 + 7 α 3 t (12b)

It is well-known that the Airy-Hardy functions C h n ( . ) are solutions of the following differential equation [

d 2 C h n ( α ) d α 2 − n 2 α n − 2 C h n ( α ) = 0. (13)

Hobbs et al. [

C h n ( α ) = π | α | 1 / 2 2 n sin ( π 2 n ) × { I − 1 / n ( 2 α n / 2 ) − I 1 / n ( 2 α n / 2 ) for α > 0 J − 1 / n ( 2 | α | n / 2 ) + J 1 / n ( 2 | α | n / 2 ) for α < 0 (14)

where J ν and I ν are the υ-th order Bessel and the modified Bessel functions of the first kind, respectively.

For α = 0 , the approximate value of C h n ( α ) reads

C h n ( 0 ) ≈ 2 ( 2 n + 1 ) 2 n / ( 2 n + 1 ) Γ ( 1 2 n + 1 ) cos [ π 2 ( 2 n + 1 ) ] , for n = 1, 2 and 3 (15)

By putting α = C 3 2 / 3 and using the similarity between the Equations (11b)

and (16a), we obtain

ψ 1 ( α ) = 3 1 / 3 2 π C h 3 ( α ) . (16)

The integral expression of Equation (9) can be rewritten

ψ 3 ( C 1 , C 3 ) = 2 ⋅ 5 1 / 5 2 π ∫ 0 + ∞ d t ⋅ cos ( t 5 + C 3 3 ⋅ 5 3 / 5 t 3 + C 1 ⋅ 5 1 / 5 t ) . (17a)

If we take C 3 3 5 3 5 = 4 α and C 5 1 5 = 5 α 2 , the last diffraction integral reduces

to

ψ 3 ( α ) = 2 ⋅ 5 1 / 5 2 π ∫ 0 + ∞ d t ⋅ cos [ T 5 ( t , α ) ] , (17b)

or equivalently

ψ 3 ( α ) = 5 1 / 5 2 π C h 5 ( α ) . (17c)

The Wigwam catastrophe integral can also be expressed as

ψ 5 ( α ) = 2 ⋅ 7 1 / 7 2 π ∫ 0 + ∞ d t ⋅ cos ( t 7 + C 5 5 7 5 / 7 t 5 + C 3 3 7 3 / 7 t 3 + C 1 7 1 / 7 α 3 t ) . (18a)

By putting 7 α = 7 5 / 7 5 C 5 , 14 α 2 = 7 3 / 7 3 C 3 , 7 α 3 = 7 1 / 7 C 1 , this leads to

ψ 5 ( α ) = 7 1 7 2 π C h 7 ( α ) . (18b)

We inferred from the above procedure that the caustics of odd codimensions K = 1, 3 and 5 are connected to the Hardy-Airy functions of orders n = 3, 5 and 7, respectively. In the forthcoming section, we will be interested in displaying in optical way these non-conventional functions.

Based on the fact that the complex caustic structures are expressed in terms of polynomial phase functions, we adopt an optical technic similar to that employed by Lohmann et al. [

T ( x , y ) = δ ( y − F ( x ) ) , (19)

where F ( x ) is the curve profile and δ ( . ) is the Dirac function.

The amplitude distribution of the Fraunhofer diffraction pattern in the back focal plane of the lens (L) is obtained by taking the Fourier transform of Equation (19). This leads to

T ^ ( ν , μ ) = ∬ T ( x , y ) e − 2 i π ( ν x + μ y ) d x d y . (20a)

Substituting from Equation (19) into Equation (20a) and using the integral property of the Dirac function yields

T ^ ( ν , μ ) = ∫ − ∞ + ∞ d x e − 2 i π ( ν x + μ F ( x ) ) . (20b)

It is worth noting that Equation (20b) is a typical integral representation for many special functions employed is lasers physics, e.g., the Airy, Bessel, Laguerre functions [

T ^ ( ν , μ ) = 2 ∫ 0 + ∞ d x ⋅ cos 2 π ( ν x + μ F ( x ) ) . (21)

In the following, we propose the transmission profiles that may create oscillating integral patterns encountered in many fields, e.g, in physics, chemistry and biology.

First, let us consider a mask transmission profile of the form T ( x , y ) = δ ( y − σ x 3 ) , where σ is an arbitrary constant (

The Fourier transform of this input amplitude can be written as

T ^ f ( ν , μ ) = 2 ( σ μ ) − 1 / 3 C h 3 ( α 3 2 / 3 ) , (22)

with α = ν 3 ( σ μ ) − 1 / 3 . The irradiance distribution of the Fraunhofer pattern in the

output plane is depicted in

By placing a narrow slit in the output plane along the υ-axis, at μ = η just behind the slit, the output diffracted amplitude reads

T ^ f ( ν , η ) = T ^ f ( ν , μ ) δ ( μ − η ) = 2 ( σ η ) − 1 / 3 C h 3 ( ( σ η ) − 1 / 3 3 5 / 3 ν ) . (23)

It follows from this last equation that by displacing the slit to arbitrary values

of η , one can produce the Hardy-Airy function C h 3 ( . ) along the υ-axis.

Now, by considering the curve function F s ( x ) of the form (

F s ( x ) = p x 3 + q x 5 , (24a)

where p and q are arbitrary constants, the Fourier transform of the associated input amplitude reads

T ^ S ( ν , μ ) = 2 ∫ 0 + ∞ d x cos ( 2 π ν x + 2 π μ p x 3 + 2 π μ q x 5 ) . (24b)

Making the change of the variable x = a − 1 / 5 t , with a = 2 π μ q , this leads to

T ^ S ( ν , μ ) = 2 a − 1 / 5 ∫ 0 + ∞ d t cos ( t 5 + 2 π μ p a − 3 / 5 t 3 + 2 π μ q a − 1 / 5 t ) . (25)

Putting 4 α = 2 π μ p ⋅ a − 3 / 5 and 5 α 2 = 2 π ν ⋅ a − 1 / 5 , and after some algebraic operations, Equation (25) is reduced to

T ^ s ( ν , μ ) = T ^ ( α ) = 2 a − 1 / 5 C h 5 ( α ) . (26)

Reciprocally, one can writes

C h 5 ( α ) = 1 2 ( 2 π μ q ) 1 / 5 T ^ s ( ν , μ ) . (27)

The irradiance distribution of the Fraunhofer pattern of the input amplitude is shown in

T ^ s ( ν , ξ ) = T ^ s ( ν , ξ ) δ ( μ − ξ ) = 2 ( 2 π ξ q ) − 1 / 5 C h 5 ( 5 4 ν ξ p ( 2 π ξ q ) 2 / 5 ) . (28)

Hence, one can visualize the profile of the Hary-Airy function C h 5 ( α ) along the υ-axis.

By considering the mask transmission function F_{w}

F W ( x ) = p x 7 + q x 5 + r x 3 , (29)

where p, q and r are arbitrary constants, its Fourier transform is given by

T ^ w ( ν , μ ) = 2 ∫ 0 + ∞ d x cos ( 2 π μ r x 7 + 2 π μ q x 5 + 2 π μ p x 3 + 2 π ν x ) . (30)

Now, making the change of variable x = b − 1 / 7 t , with b = 2 π μ r , and by taking

7 α = C 5 5 7 3 / 7 = 2 π μ q b − 5 / 7 , 14 α 2 = C 5 3 7 3 / 7 = 2 π μ p b − 3 / 7 ,

7 α 3 = C 1 7 3 / 7 = 2 π ν b − 1 / 7 , Equation (30) can be expressed as

T ^ w ( ν , μ ) = 2 ⋅ b − 1 / 7 C h 7 ( α ) , (31)

or equivalently

C h 7 ( α ) = 1 2 ( 2 π μ q ) 1 / 7 T ^ s ( ν , μ ) . (32)

Following the same procedure as described in Sections (3.1) and (3.2) and using the binary mask of Equation (29), we have displayed the irradiance distribution of the Fraunhofer pattern associated with C h 7 ( . ) in

worth noting that the patterns of

The parameter σ can be regarded as a scaling length for controlling the Fraunhofer pattern structure of

In summary, based on the fact that the integral diffraction catastrophes can be reduced in the case of odd polynomials phase functions into Airy-Hardy cosine integrals, we obtained the closed-form expressions for the related caustic functions. We showed that by the use of an optical Fourier Transformer device with tailored binary screens one can obtain the displaying of caustic beams such as the Fold, the Swallowtail and the Wigwam catastrophes. Furthermore, it is shown that the mapping caustics for Swallowtail and Wigwam caustics can be controlled by the scaling parameters p, q and r of the binary mask transmittance. The result of this work can be useful in shaping lasers with catastrophe amplitude function and may have application opportunities in micromachining and light guiding paths. This study can be extended to higher oscillatory integral functions.

The authors declare no conflicts of interest regarding the publication of this paper.

Belafhal, A., Hricha, Z. and El Halba, E.M. (2019) Generation of Some Catastrophes in Optics by Binary Screens. Open Access Library Journal, 6: e5958. https://doi.org/10.4236/oalib.1105958