GNOSEOLOGY OF FRACTALITY -

FRACTAL OPTICS.

G.S. Melnikov

State Enterprise

Federal Research Center "Vavilov State Optical Institute"

199034 Birzhevaya liniya, 12, St. Petersburg Tel.:(812) 3112201, Fax: (812) 2188065

St. Petersburg

1997.

GNOSEOLOGY OF FRACTALITY -

FRACTAL OPTICS.

G.S. Melnikov

State Enterprise

Federal Research Center "Vavilov State Optical Institute"

199034 Birzhevaya liniya, 12, St. Petersburg Tel.:(812) 3112201, Fax: (812) 2188065

E-mail: liader@soi.spb/su

ABSTRACT

The paper deals with the theory and experiment of fractal optics. Described are results of experimental tests of fractal properties of photon paths propagating in light conducting cavities of cylindrical type along modal paths (geometric longitudinal, transversal, longitudinal-transversal and longitudinal-radial modes). Basing upon analysis of these properties and attracting results of development of the theory of numbers (integers, rational, irrational) the author suggests his interpretation of gnoseology of fractality. Discussed are equations of movement of photons in light conductors. Properties of new optical elements, multimode multilevel Fabry-Perot interferometers, are considered.

Key words: topological concept of fractality, geometric and functional spaces, optics of multiple reflections, "macrowave" equations, optical analog of VLSI

INTRODUCTION

Discrete digital methods of optical transformations opened up by the potentials of fractal optics trace back to the first descriptions of digital phenomena of extraparaxial optics in elements of cylindrical group discovered in the Vavilov State Optical Institute in 1986 [1,2 ]. Statement of the first applied problems that can be solved by the methods of fractal optics is described in the review paper [3]. Some applied problems are solved and solutions are described in applications for RF patents [4, 5].

DEFINITIONS The mathematical term "fractality" and principles of its interpretation were introduced by Benoit Mandelbrot in 1975 [6,7].

Fractality covers a vast class of natural and artificial space and topological forms of fractional dimensionality, that up to now have not had a reliable body of mathematics. Idea of fractality appeared to explain such mathematical and physical pathologies (monsters) as, for example: - Koch "island" (1904), - Richardson paradox (measurement of coastline length, when it is mapped at different scales); - functions without derivatives and geometric objects without tangents. For mathematical description of the phenomenon of self-similarity found in fractal structures Mandelbrot following Richardson suggested the general formula determining properties of self-similar fractal structures

N 1R D= (1)

where N is the number of similar parts in scale R; D is the corresponding dimensionality (amount of coordinates). It was found that in general case dimensionality index D can be of not integer value. The most exact definition of fractality has been given by the authors of work [8]. Objects of both usual geometric and functional spaces can have fractal topology. Self-similar hierarchic structure is the basic feature of fractals. The given definitions and results of great volume of theoretical and experimental work carried out by the author make it possible to introduce a new class of optical elements and light conducting cavities under the name "fractal optics". Fractal optics is optics using conventional optical elements operating in extraparaxial region, where light rays undergo partial or total multiple reflection during their propagation in light conducting cavities restricted by reflecting surfaces. Or even shorter: - fractal optics is optics of multiple reflections. ( Figs. 1a and b show examples of such multiple reflections) Actually, as shown below, it is impossible to describe the processes of multiple reflection photon propagation in the system of three space coordinates without additional parameters. These additional parameters are connected with interfaces and characterize special features of space, where there takes place multiple reflection propagation of partial light rays grouped before in an initial beam. For simplicity of the following description assume that the initial light beam is obtained from a coherent radiation source. Elements of fractal optics similar to elements of wave and geometric optics can be divided into two large classes: class of centered elements and class of decentered elements. Elements having a form of solid or hollow sphere as well as toric elements fit into the class of centered fractal elements. Cylindrical elements and a number of elements with reflection surfaces similar to surfaces of degenerated cylinder (planes) fit into the class of decentered elements. Unique features of fractal optics of cylindrical group are most studied and described by the author. Why are the above optical elements named fractal? The point is that from results of theoretical and experimental studies [1 ... 5, 9 ... 12 ] it is found that the space of parameters describing geometrical paths of partial rays (photon trains) in their movement within the range of multiple reflections from cylindrical surfaces always has larger than 3 + 1 dimensions in independent space-time coordinates. In other words for their theoretical description in addition to three geometrical coordinates, "length", "height" and "width", and one time coordinate it is necessary to add a number of parameters connected with orthogonal to them interfaces of light conducting elements. By virtue of the fact that for different optical elements not all interfaces and not to all space

coordinates are orthogonal there take place differences in both theoretical and experimental descriptions of different elements included in fractal classes. In this case parametric dimensionality index D for fractal elements of centered optics is an integer and for decentered optics it can be fractional. Proceeding from necessity to observe orthogonality of interfaces to coordinate axes and, on this basis, finding general parametric dimensionality, one can use for calculation of the dimensionality index the following formula:

D=( D(q)+D(p))+D(t); (2)

where D(q) is the number of independent space coordinates involved in description of movement of light particles. D(t) is the number of independent time coordinates required for description of movement (at this stage it is assumed that there is only one time coordinate). D(p) is the number of added parametric coordinates required for movement description. In its turn D(p) is defined as D(p) = p/q where p is the number of orthogonal intersections of lines (surfaces) with which the added parameters are connected by space coordinates q required for description of movement considered. As a result for certain calculations we have

D = (q+p/q)+1 (3)

Or in general form

D=(q2+ττ⋅q+p)/q (3*′) where ττ is the quantitative characteristic of time coordinates. As a result we have the following: The most perfect fractal optical elements are spherical and toric single-fractal elements and bispherical and bitoric bifractal elements. Really, there is only one interface in sphere orthogonal to all three space coordinates, i.e. for sphere the parametric dimensionality index in fractal region equals 5

D sp = (3+1)+1=5 (4) In this case for torus one also can consider that its parametric dimensionality index D t = 5. However, it is worth noting that an interface in torus is locally-orthogonal to coordinate axes. For bifractal sphere and torus, i.e. for figures of rotation with two interfaces (external and

internal ones) we obtain the parametrical dimensionality index equal to 6

Dbf = (3 + 2) + 1 = 6 (5) As shown below for segmented optical elements in case, if cutting planes of cylinder segments are perpendicular to one or several space coordinates and these planes passes through the centre of rotation of the considered figure, i.e. if they contain Z axis, the space dimensionality index has the same numerical value, because planes will finish the mirror reflection of total figure of rotation for all rays involved in multiple reflection.

Indices of space dimensionality for cylindrical elements according to the accepted classification are fractional. In cylindrical elements where Z coordinate is not perpendicular to a curvilinear intereface, but parallel to it, the parametric dimensionality index is most likely equal to 4 2/3

Dc = 3(+2/3) +1 = 4 2/3 (6) For bifractal cylindrical elements (tubular cylinders) we have: Dbc = (3+2/3+2/3)+1=5 1/3 (7) All said above about segmented elements is valid in this case as well. Studies revealed also that fractality coefficients, i.e. coefficients of divisibility of restricting curvilinear surfaces by points of multiple reflections as integer, fractionary-integer, irrational or transcedent ratios, can be successfully used as values characterizing the added parameters. In the general case such points are apexes of fractal polygons mapping geometric modal photon paths propagating inside the light conducting elements in the mode of multiple reflections according to the laws of geometric optics. The polygons that have n-apexes (reflection points during m-"revolutions" around reflecting surface curvature centre, or a curve in two-dimensional consideration), i.e. polygons having fractality coefficient

k = n/m (8) can be named fractal polygons. This definition eliminates a "mistery" of the Koch island.* *The "Koch island" is a mathematical construction comprising geometric figures with finite surface area and a length of broken line restricting it tending to infinity. If fractality coefficient is presented in the general case as k=n/m, we obtain a polygon closed at n-th reflection, when k is fractionary-rational. In this case a polygon side ( in a light conducting element it is a free path from one reflection to another) "runs" a finite area during m revolutions. If k is transcedent, movement of apexes of such a polygon and "running" of the finite area will take place, when n and m tends to infinity and the current apexes never fall strike the first reflection point. In other words the transcedent fractal polygon is an unclosed figure. Illustration of hierarchic structure of self-similar polygonal figures for various single and bifractal rational constructions are given in works [3,9]. Figs. 1c and d show examples of such polygons.

1. HIERARCHIC STRUCTURE OF NUMERICAL CONTINUUM The essense of theoretical prerequisites comes to the fact, that for description of structural order of numerical continuum and derivation of the law of sequence of composite natural and rational numbers, it is necessary to consider finite sums of intervals of dual-infinite recurrent sequences having a form

U-∞,...U-1,U0,U1,...U+∞ (9)

or, in brief, {U±p} and these requirements are met in the most by expansion of analytical functions into the Laurent series [12]

(10) ( ) ( ) ( )f z akk 0z a k bkk 1

z a k=

=

∞∑ − +

=

∞∑ −

−

where the first series is an ordinary power series (in the general case it is an alternating series). The second series is named the main part of the Laurent series. In general case formula (10) can be presented as follows

(f z c=( ) )kkz a k

=−∞

+∞−∑ , (11)

where ( )( )

c ifa

dck k 1=

−∫ +

12π

ζ

ζζ

As is seen expression (11) describes the dual-infinite sequences having form (9). Function f(z) analytical within a certain ring between two concentric circles with centre "a" can, by definition, be presented by the Laurent series in a single way: 0 < r ≤ z - a ≤ R, a ≠ ∞ (12)≤ Search for the physical model subject to condition (12) led to necessity of detail consideration of properties of billiard ray paths in a circle. Really, a light ray entered in a hollow reflecting cylinder or solid optically transparent cylinder in plane Z = 0 will undergo multiple reflections within a ring zone from r to R, where r = R cos ( π π/k) is a coordinate of intersection of the ray normally to axis X(z, y=0) and R is a reflecting curvilinear surface radius. All these definitions led to consideration of the laws of ray propagation in elements of fractal optics introduced before. Hierarchic structuring resulting from theoretical description of the laws of distribution of composite numbers (integers, rational numbers, surds) in numerical continuum served as an additional help in derivation of laws of new discrete optics.

1.1. STUDY OF HIERARCHIC STRUCTURE IN FRACTAL OPTICS.

Work [2] has been devoted to consideration of spectral expansion of numerical continuum specifying space distribution in numerical axis of unit radius R and in unit circle 2 π R = 1 of points of entrance and reflection of light beams, respectively, whose coordinates depend on fractality coefficients. These cyclic functions are connected by expressions (13, 14) with circular trigonometric functions

( )( )

( )sink, tcos

ksin t

cos t2 p 1k 0

ϕω

πω

ωπ

ϕ

=

⋅

−+

−

(13)

( )( )

( )cosk, tcos

kcos t

cos t2 p 1k 0

ϕω

πω

ωπ

ϕ

=

⋅

−+

−

(14)

with the given initial and boundary conditions: 0≤ϕ0≤π/k, p = 0,1,2,...;

ωt ∈ [2πp/k + ϕ0; 2π(p+1)/k + ϕ0]; k ≥ 2.

Examples of introduced non-circular trigonometric functions are given in Fig. 1a,b,c.. For description of longitudinally-transversal modes one should add parameter l connected with inclination β in plane XY. In this case Z can be determined by the formula Zk,l = 2τRcos(α)tg(β)=2τlrcos(α)sin(β)=2τlRsin(π/k)sin(β) (14^)

where l = 1/cos(β), k = n/m, τ∈0,1,2.3...

Procedure of propagation of light rays in bifractal light conductors (tubular cylinders) gets complicated due to the fact that extraparaxial light rays undergo multiple reflections not only from the external curvilinear surface of radius R restricting the element, but they can be reflected also from the internal surface of radius r. In case when there is no reflection of rays from the internal surface, or rays are reflected tangentially to the interface of radius r, the light rays are described by conventional expressions given above for single-fractal light conducting systems characterizing transversal and longitudinally-transversal modal paths. When rays are reflected in turn from surfaces of radius R and r, we have the general case of description of movement along radially-transversal modal paths. For derivation of general equations of ray propagation we consider example of rational division of circles of radius R and r by a finite amount of points of light ray reflection. In other words for derivation of general expressions specifying position of transversal and radially-transversal modal paths we proceed from consideration of simplified case, i.e. from the case when the R-r relationship and fractality coefficients k = n/m and p = η/µ determine divisibilities of these circles as integer and fractionary- rational ratios. The given case is shown in Fig. 1d. In this case we may introduce an angular parametric function G(m,n,µ,η)=cos(π/k)⋅cos(π/p)/(cos(π/k)-sin(π/p)⋅-sin(π⋅(k-p)/k⋅p)) (15)

connected with fractality coefficients. For the circle of radius R:

κ(m,n)=n/m (16)

For the circle of radius r:

ρ(µ,η)=η/µ (17)

Commutation function at reflecting surfaces (function of current time is expressed in current change of reflection points)

S(τ)=(1+(-1)τ)/2 (18)

and as a result function of current movement

F(m,n,µ,η,τ)=G(m,n,µ,η)+S(τ)⋅G(m,n,µ,η) (19)

whence for current coordinates X and Y we obtain expressions

Y(m,n,µ,η,τ)=R⋅F(m,n,µ,η,τ)⋅sin(τ⋅π/p) (20)

X(m,n,µ,η,τ)=R⋅F(m,n,µ,η,τ)⋅cos(τ⋅π/p) (21)

It is typical that angular positions of partial rays filling the raster are in full conformity with expressions (12), (13), (14) for single-fractal optical elements and expressions (20), (21), (14^) for bifractal elements. From analysis of expressions (13 ... 21) one can see that in bifractal systems fractality coefficient κ(m,n)=n/m determines divisibility of restricting surfaces of radii R and r expressed as integer, rational, irrational and transcendent ratios. Fractality coefficient ρ(µ,η)=η/µconnected with sectoral-angular division of light conducting ring of bifractal cylindrical optical systems and is specified by an angle of entrance in a light conducting cavity of rays in point 0. (Fig. 1, and expression γ = 2⋅π/ρ(µ,η) ;α = π/2 - π/κ(m,n)). For derivation of general equations of ray propagation we consider example of rational division of circles of radius R and r by a finite amount of points of light ray reflection. In other words for derivation of general expressions specifying position of transversal and radially-transversal modal paths we proceed from consideration of simplified case, i.e. from the case when the R-r relationship and fractality coefficients k = n/m and p = η/µ determine divisibilities of these circles as integer and fractionary- rational ratios. The given case is shown in Fig. 1с.

1.2. STUDY OF HIERARCHIC STRUCTURE IN NUMERICAL SEQUENCES

The law of sequence of composite numbers is determined on the basis of the "Eratosthenes sieve" rule [13]. However, in theory of numbers and the most of all in consideration of this part of the theory the negative branch of numerical axis has been dropped and the law of sequence of rational, irrational and transcedent numbers in numerical axis has not been considered at all from the point of view of the Eratosthenes rule. As in the former case it is necessary to use dual-infinite sequences of form (9). Thus for integers: U{i}=i where i∈[-∞,0,+∞] (22) considering three adjacent numbers of numerical axis

( i-1 ) , ( i ) , ( i+1 ) (23)

one can see that paired composite numbers are written as: a. Composite number of the left and the middle members of dual-infinite arithmetic progression with difference d = 1: Ui,-1 =Ui⋅U(i-1)= (i) × ( i-1 )= i2 - i (24)

b. Composite number of the right and the middle members: Ui,1 =Ui⋅U(i+1)=i⋅(i+1) = i2 + i (25)

c. Composite number of the left member distant from the middle member by n positions and the middle member: Ui,-n =Ui⋅U(i-n)= Ui2-⋅n⋅Ui=(i)⋅(i-n)= i2 - n⋅i (26) d. Composite number of the right member distant from the middle member by n positions and the middle member: Ui,n=Ui⋅U(i+n)=Ui2+n⋅Ui=(i)⋅(i + n) = i2 + i⋅n (27)

Analysis of expressions (22 ... 27) makes it possible to conclude the following:

1. Assuming n∈[-∞, 0, ∞∞] we can write a rule of creation of paired composite numbers in the sequence of integers in form of expression (28):

Ui,n =Ui2+n⋅Ui= i2 + n⋅i

when n = =Ui,n/Ui - Ui (28)

2. In hierarchic structural expansion of dual infinite numerical sequences into paired composite numbers with preselected number Ui they in these sequences in conformity with the Eratosthenes rule are located equidistant to the left and to the right from number Ui. Distance between adjacent composite numbers equals i, Ui is the sequence of integers in numerical continuum. 3. Number Ui,0=(i⋅)2 is the zero composite member in i-ary expansion. 4. Number Ui itself is the composite number with Ui= 1 and proves to be a harmonic in this expansion with the number equal to n = - (i - 1) = -i + 1 (29)

Taking into account that the positive zero composite number can be presented as Ui,o and U- i,o i.e.:

Ui,0 =Ui⋅Ui= i⋅i and U-i,0=U-i⋅U-i= (-i)⋅(-i) (30) whence U-i,0=Ui,0 (31) duality of composite numbers follows from this directly. Note: Selecting negative number U-i as an initial number in presentation of composite numbers one should take into account in conformity with (29) that sign n for such numbers is inverted in numerical axis. Due to this number U -i,-n = -i in harmonic expansion into composite numbers has harmonic number equal to -n - n = -(( i - 1) -2) = -i -1 (when U i is the initial number) and n = (-i) + 1 = i + 1 (when U- i is the initial number) (32) 6. When n = i we obtain doubled zero composite number Ui,i = 2i2 Topological properties of numerical continuum are considered below. On gaining the first impression about distribution of paired composite numbers in numerical continuum it is not difficult to understand, that in traditional presentation only prime numbers having as a factor one and prime number ε are non-composite numbers in numerical sequence. This prime number determines also an interval (period) of sequence of paired composite numbers Uε,n along numerical axis. All the above considerations concerning the law of sequence of integer composite paired numbers in numerical axis can be easy applied to classes of rational numbers and surds. Really, to obtain a picture of distribution of composite rational numbers it is necessary to element-by-element divide sequence (23) by scale R. As a result we go from integer sequences to sequences of R-ary rational numbers whence expression (28) takes form (33) U(i/R),(n/R) =(1/R)2⋅(Ui2+n⋅Ui)= (1/R)2⋅(i2 + i⋅n) (33) It is evident that paired composite numbers from R-ary rational sequences are in sequences of smaller digit i/R

2

than the initial rational sequence. All other conclusions made in analysis of sequences of composite integers are valid also for rational numbers with multiplication of calculation results by scale factor 1/R

2.

Deriving the law of sequence in numerical continuum of composite numbers from surds one should understand that when addressing to surds we look at numerical continuum as if through a "non-linear" lens. For example for the sequence of quadratic surds three adjacent surds in the numerical axis are presented as: √(j-1), √(j),√(j+1)...,ãäå j∈[-∞,...-1,0,1,...∞] (34)

They are obtained by means of element-by-element square rooting of integer sequence elements. By this operation we transform the set of integers into the set of quadratic surds with increase of power of the set. In other words the set which is a dual-infinite sequence of quadratic surds comprises the set of integers. Or in more illustrative form: between a lattice of integers there takes place progressive formation of layers of surds. The number of intermediate layers of quadratic surds is determined by expression ∆Nj = i2-(i - 1)2-1 = 2(i - 1) (35) in this case i is the maximum integer within the selected part of sequence of quadratic surds Uj. The abovementioned laws are shown in Table 1. Table 1.

ТJ 0 1

22 3

44 5 6 7 8 9 10 ... 15 16 17 ... ...

224 25

ii 0 1

22

3

44

5

uUi 0 1

22

3

44

5

UUj 0 √√1

2√√2 √√3

4√√4 √√5 √√6 √√7 √√8 √√9 √√

10 ... √√

15 √√16

√√17

... ... √√24

√√25

Т∆Ni

Т0 1

22

2 44

6

8

The number of intermediate layers of z-surds is determined as:

∆Nζ = iz - (i -1)z - 1 (36)

However, we consider z-surds below.

Table 1 shows that in the general case the three-dimensional body of quadratic surds [9]

Uη,µ,j=η +µ⋅√j (37)

for positive values j is "inside" the rational body with intermediate layers of planes of surds. For negative values j inside the three-dimensional body of real numbers there are layers of the six-dimensional body of complex numbers. When going to sequences of dual-infinite quadratic surds and the problem of search for structural hierarchy of paired composite numbers in these sequences, we come to conclusion that these numbers are distributed in sequence Uj=√j

Uj,n= (√j)(√(j+n))=√(j2+nj)=(j2+n⋅j)1/2 (38)

As it is evident from (38) integers i in numerical continuum are zero paired composite numbers of quadratic surds, i.e. any number, including prime number Ui = ε can be presented in form of one of pairs of composite surds Ui =Uj,0=U-j,0=(√√j)⋅(√√j)=√√(-j)√⋅√(-j)=((√√j)⋅i))⋅((√√j)⋅i) (39)

where i=√√(-1), and i,j∈[-∞∞,...-1,0,1..∞.∞] Expression (39) shows that all integers, including prime integers, are paired composite numbers of two real or imaginary quadratic surds. Using the same mathematical procedure as for derivation of the law of sequence of paired composite numbers in numerical axis one can show, that for triple, quartery, N-dimensional composite numbers in the sequence of integers it is also possible to derive the laws of their sequence:

- for triple composite numbers:

Ui,n,m=i3 + (n + m)⋅i2 + (n⋅m)⋅i (40)

- for quartery composite numbers:

Ui,n,m,q=i4+(n+m+q)⋅i3+(n⋅m+n⋅q+m⋅q)⋅i2+(n⋅m⋅q)⋅i (41)

- for D-dimensional composite numbers:

Ui,n,m,...,w=iD+(n+m+....+w)⋅i(D-1)+(n⋅m+ n⋅p+...+nq+ +v⋅w)⋅i(D-2)+

(n⋅m⋅p+n⋅m⋅r+...+u⋅v⋅w)⋅i(D-3)+...+(n⋅m⋅p...u⋅v⋅w)⋅i (42)

where D =∑i≤s≤w (1) is the number of used parameters. As it is evident from (34...42) integers Ui,n,m,...,w is

i n m w N

D p

p

D p

N

D p C

U FF NC C

where sequences of combination

when ND p N D pN

D p

, , ...

( ) ...... .. ..

!( )! ( )!

= ⋅

=

=− ⋅ − +

−

=

−

=

∑∑

= −

1

0

ιξξ

(43)

General recurrent expressions for D dimensional composite fractionary-rational numbers and D-dimensional composite surds can be obtained using similar methods:

for D-dimensional fractionary-rational numbers:

i R n R m R w R D N

D p

p

D p

N

D p C

U R F

F NC C

where sequences of combination

when ND p N D pN

D p

/ , / , / ... /

( ) ...... .. ..

!( )! ( )!

= ⋅

=

=− ⋅ − +

−

=

−

=

∑

∑= −

11

0

ι

ξξ (44)

for three-dimensional quadratic surds:

Uj(2),n,m=Uj⋅U(j+n)⋅U(j+m)=√j×√(j+n)×√(j+m)=

=(j3+(n+m)⋅j2++n⋅m⋅j)1/2 (45)

where Uj=2√√j

for three-dimensional cubic surds:

Uj(3),n,m=Uj⋅U(j+n)⋅U(j+m)=3√√j⋅×3√√(j+n)⋅×3√√(j+m)=

=(j3+(n+m)⋅j2+n⋅m⋅j)1/3 (46)

U j=3√√j ; i,j∈[-∞...-1,0,1,...∞]

And in general form for D-dimensional V-ary composite surds we have the following recurrent expression:

ς ξξς/ , / , / ... /( ) ... ...... .. ..

!( )! ( )!

( )R n R m R w R D N

D p

p

D pV

N

D p C

U R F F N

C C

where sequences of combination

when ND p N D pN

D p

= ⋅ =

=− ⋅ − +

−

=

−

=∑ ∑

= −

11

1

0 (47)

Analysis of derived recurrent equations (22 ... 47) describing principles of sequence of composite numbers in dual-infinite numerical axis for the systems of integers, rational numbers and surds permits to make the following conclusions:

DERIVES

1.The rules of combinatory presentation of numerological sequences of composite numbers whose distribution in numerical axis follows principles of the "Eratosthenes sieve" are at the heart of these periodical changes. 2.The "Eratosthenes sieve" principle is extended not only to integers but it is derived also for composite fractionary-rational numbers and surds. 3.Obtained in the work are recurrent equations describing sequences of composite "polygonal" numbers for the following three classes: - integers; - rational numbers; - surds, 43...47 respectively. 4. Analysis of these equations in its turn permits to state that composite numbers are fractals of numerical continuum. Parametric dimensionality for them is always higher than coordinate dimensionality. Parametric dimensionality for composite integers and rational numbers is always an integer. Parametric dimensionality for surds is always fractional. Strict hierarchy of sequences of composite numbers is the second indication of their fractality. In addition it is shown that prime numbers also have their composite parts. They are combined of various groups of "prime-like" surds. Principle of self-similarity necessary to refer the considered structures to fractals is in the general principle for separation of all numerical sequences of composite numbers (in the "Eratosthenes sieve" rule). 6. Thus one can state that gnoseologic roots of fractality are in hierarchic structure of continuum composite numbers.

CONCLUSIONS

1. Fractal properties of a new class of extraparaxial multiple reflection optical elements are shown. 2. Mathematical simulation confirming multiparametrical fractality of movement of light rays in decentred light conductors is carried out. 3. Developed are elements of mathematical model of propagation and transformation of light rays and wavefronts for centred elements of fractal optics.

4. It is shown that it is possible to overcome increase of dimensionality of parametric space of light ray propagation in different classes of light conducting elements from the point of view of their theoretical description. In so doing some special properties of fractal optics has been revealed: - hierarchic structure of propagation of ray caustics in light conductors, - possibility to use principles of self-similarity of ray paths for creation of digital illumination systems, - hierarchic distribution of entrance points and angles of rays in segmented elements of fractal optics providing possibility to create the scanning systems with optical reduction of sectors of caustic and raster rays of formed coherent and nonmonochromatic radiation depending on certain requirements.

ACKNOWLEDGEMENTS The present paper opens a series of papers written to a certain extent due to interest in this problem shown by Defense Group Inc. (USA). The author wishes particularly to acknowledge his thanks to Eleonora I. Melnikova and Galina I. Yurinova for long-sufferance and continuous material and moral support. The author wishes to express deep thanks to colleagues and co-authors of some publications Larionov S.A., Mikheev P.A., Tsvetkov E.A. for conformity of ideas in theoretical and experimental interpretation of this new field of physics.

REFERENCES 1. Melnikov G.S., Kosmachev A.F., Shishkin M.Yu. 'Methods of Ray Description of Raster Phenomena in Cylindrical Lens Operating within the Range of Total Internal Reflection', L., GOI, 1986., Abstract. IV All Union Conference "Theoretical and Applied Optics". 2. Melnikov G.S. 'Digital Processing of Signals by Systolic Processors/'/ Preprint No 9-91, 1991, 61p. Lvov, NTC VPVS "Integral". (Russian). 3. Melnikov G.S., Larionov S.A., Mikheev P.A., Tzvetkov E.A., “Discrete Raster Systems of Digital Optical Processing and Transmission of Images by Systolic Methods”, Transactions of RAS, Physics series, Vol., 59, No 12, 1995, p. 143 ... 150. 4. Melnikov G.S., Larionov S.A., Mikheev P.A., Tzvetkov E.A. , “Method of Creation of Light Flux Time Delays”. Application for RF patent. Priority of RF VNIIGPE No 95114222 of 07.08.95. (Russian). 5. Melnikov G.S., Larionov S.A. "Multimode Multilevel Interferometer Fabry-Perot with Controlled Bi and Polystability". Application for RF Patent. "Optico-Electronic Complexes and Systems No 131/2 of 20.03.1996. 6. Mandelbrot B. Fractals: Forms, Chance and Dimension. - San Francisco: W.H. Freeman and Co. 7. Mandelbrot B. The Fractal Geometry of Nature. - San Francisco: W.H. Freeman and Co.

8. V.F. Uritsky, N.I. Muzalevskaya. "Fractal Structures and Processes in Biology "(Review) p.1 and p.2 in collection ed. by R.I. Polonnikov and K.G. Korotkov Biomedical Informatics and Eniology SPIE RAS "Olga" St. Petersburg, 1995. P. 84 ... 129.

9. Melnikov G.S., Larionov S.A. Mikheev P.A., Tzvetkov E.A. // 5th International Conference on Differential Satellite Navigation Systems , v II, Post Sect. 1, №11, St. Petersburg, RIRT, 1996.

10. Melnikov G.S. Abstract. Regional Seminar "Failure Resistance and Survivability of Digital Data Processing Systems" USSR AS, Minsk, 1990, p.102 ... 109. 11. Melnikov G.S. Abstract. Scientific and Technological Conference of CIS Countries "Recognition Analysis of Images"" (ROAI-93), IAPR, Minsk, 1993. 12. Melnikov G.S. Space-Spectral Expansion of Sequences of Digital Data Operands in Position Notations, Lvov, NTC, VPVS "Integral", 1992, Preprint No 5-92 p. 5-13. 13. Diaphant of Alexandria. Arithmetics and Book of Polygonal Numbers, "Nauka", M., 1974, 327 p. 14. Galperin G.A., Zemlyakov A.N. Mathematical Billiards, M.: - "Nauka", 1990, 287 p. 15. Melnikov G. S., Larionov S.A., Mikheev P.A., Tsvetkov E.A. "Control of Autolocked Modes in Waveguide Cylindrical Cavities", Journal SPIE - The International Society for Optical Engineering, Plan of issue 1995 year.

1c. 1d.

Fig. 1a,b,c,d. Fractal and bifractal paths of a light ray with different pairs of fractality coefficients k and ki