Supergaussian beams of continuous order as GRIN modes

Optics Communications 102 ( 1993 ) 21-24 North-Holland

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Supergaussian beams of continuous order as GRIN modes J. O j e d a - C a s t a f i e d a 1, G . S a a v e d r a Departamento de Optica, Universitat de Valkncia, 46100 Burjassot, Spain

and E. L 6 p e z - O l a z a g a s t i Facultad de Ciencias, Universidad Aut6noma de Puebla, Apdo. Postal 1651, Puebla 72000, Pue., Mexico

Received 2 February 1993; revised manuscript received 1 April 1993

It is recognized that for certain planar waveguides, and for certain cylindrical GRIN fibers, some diffraction modes are supergaussian beams of continuous order.

I. Introduction The impulse response o f optical systems can be i m p r o v e d by using t a p e r e d a m p l i t u d e transmittances, or reflectances, at the exit pupil window. This optical procedure finds useful results in imaging devices working either under coherent illum i n a t i o n [ 1 ] or u n d e r incoherent i l l u m i n a t i o n [ 2 ], as well as in unstable optical resonators [ 3 ]. The use o f laser windows tapered with supergaussian profiles, p ( r ) = A e x p [ - ( r / R ) 2~ ] ,

(la)

has recently received attention by several authors [ 3 6]. In a different context a novel analytical formulation has been r e p o r t e d for describing the wavefield generated by a window, whose a m p l i t u d e transmittance has a supergaussian profile [7]. The 1D version o f the above supergaussian functions can be expressed in m a t h e m a t i c a l terms as p ( x ) = A exp[ - ( x / X ) z'~ ] .

(lb)

In fig. 1 we show a 3D plot o f any o f the two famt Permanent address: INAOE, Apdo. Postal 216, Puebla 72000, Pue., M4xico.

Fig. 1. Supergaussian profile of continuous order against the normalized coordinate. In this graph, a varies from 1 to 50. ilies o f profiles in eq. ( 1 ), for several values o f the order c~, against the n o r m a l i z e d coordinate r / R , or equivalently x / X . O u r aim here is to show that a supergaussian b e a m is a diffraction m o d e o f certain optical fibers. This result is shown for both p l a n a r waveguides, a n d cylindrical G R I N fibers. In other words, we answer the

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question: given a beam of cross sectional shape as in eq. ( 1 ), is there a refractive index profile that would support such a light wave. The answer is yes and the refractive index profile for the slab as well as the round fiber case are given in this paper. For the sake of completeness of our formulation, we discuss in sect. 2 the eigensolutions of the paraxial version of the Helmholtz equation. Then, in sect. 3, we consider the diffraction modes of a 1D waveguide, with refractive index variation to the a power, where a is a real number. And in sect. 4, we discuss the 2D case with cylindrical symmetry.

N(x) =No+ [a(2aX { 1 - [2 a / ( 2 a -

Under the paraxial approximation, the reduced form of the Helmholtz equation is

[02 +02 +i2kNoOz + 2kZNoN(x, y) ] u(x, y, z) = 0 .

(2) In eq. (2) we denote by k the wave number, 2n/2, and N(x, y) describes the variation of the refractive index around the constant value No. Note that eq. (2) governs the wave propagation along the z-axis. See for example refs. [ 8 ] and [9 ]. Now, the diffraction modes are defined as

(6)

-

(x/X)

(7)

2or ] ,

[02 + 2k2NoN(x) ] v(x) = 2 k 2 N 2 v(x) .

(8)

Hence, by comparison between eqs. (5) and (8), we have that the eigenvalue fl is equal to 2kZN2; and consequently, from eqs. (3) and (8) we obtain that the wavefield is

u(x, z) =exp( ikNoz) exp[ - ( x / X) 2~'] ,

(9)

which represents a diffraction mode, or a nondif/A

(3)

where v(x, y) is the boundary condition at z = 0 . And consequently, for the diffraction modes the lateral irradiance remains invariant at any distance z. That is

lu(x,y, z)12= Iv(x,y)12

1 ) ] ( x / X ) 2'~} .

then it is straightforward to show that

]

u(x, y, z) = exp(iflz/2kNo) v(x, y) ,

1 ) /k2No x2 ] ( x / X ) 2a-2

Note that in eq. (6) a is a real number equal to, or greater than, unity. In fig. 2 we show a 3D graph that displays the variation of N ( x ) against the dimensionless parameters x / X a n d a. For plotting fig. 2 we set, in eq. (6), No= 1.50; x=52. We will show next that the 1D version of the supergaussian function in eq. ( l b ) is an eigensolution of eq. (5), for the refractive index profile in eq. (6). If we assume that the initial wavefield, at z = 0, is the complex amplitude

v(x) = e x p [

2. Eigensolutions to the Helmholtz equation

15 September 1993

"'~i

i~i~

(4)

Therefore, for a given profile of refractive index variation, N(x, y), the diffraction modes are characterized by satisfying the differential equation

[02+02+2k2NoN(x,y)] v(x,y)=flv(x,y).

(5)

We discuss next two particular profiles for N(x, y).

,

3. Planar waveguide Let us consider a 1D, planar waveguide specified by the following a-power, refractive index variation 22

Fig. 2. c~-power,refractive index variation of a 1D waveguideas a function of the dimensionless parameter x/X, for ct= l until ~=10.

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fracting beam of the planar waveguide, whose refractive index profile obeys eq. (6). In other words, for each value of ot in eq. (6) we conclude that the supergaussian function of order ot is a diffraction mode of the 1D waveguide. The profiles at z = 0 of the above nondiffracting beams are plotted in fig. 1.

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And consequently, from eqs. (5) and ( 11 ) we obtain that some diffraction modes, or nondiffracting beams, for the refractive index variations in eq. (10), have the form

u( r,z) = e x p ( ikNoz) exp[ - ( r / R ) 2~ ] .

(12)

The modulus of the analytical expression in eq. ( 12 ) is shown in fig. 1. 4. Cylindrical fiber We consider now that the a-power, refractive index variation of certain GRIN fiber is

N ( r ) =No + (2/No) ( o t / k g ) E ( r / R ) za-2 × [ 1 - ( r / R ) 2~ ] .

(10)

The refractive index variations in eq. (10) are shown in fig. 3, as a 3D graph for several values of ot and r; r is measured in units of R, with R = 52; and No= 1.50. Now, if we substitute eq. (10) into eq. (5) we have that the supergaussian function in eq. ( l a ) satisfies the following equation

[0 2 + r - l O ~ + 2 k 2 N o N ( r ) ] v(r) = 2 k 2 N ~ v ( r )

5. Particular cases It is instructive to note that our above formulation contains two particular cases that are related to the well-known quadratic, refractive index variation; see ref. [ 10 ]. In other words, a 1D waveguide with quadratic index variation is described by setting ct = 1 in eq. (6) to obtain

N ( x ) =No + ( 1 / k 2 N o X 2 ) { 2 [ 1 - ( x / X ) 2 ] - 1}. (13) In a similar manner by setting t~ = 1 in eq. (10) we have that

. (11)

N(r)=No+(2/No)(1/kR)2[1-(r/R)

2] ,

(14)

which describes a refractive index variation with quadratic profile. We observe that for a = l, the ID supergaussian beam in eq. (9) reduces to the gaussian beam

u(x, z) = e x p ( i k N o z ) exp [ - ( x / X ) 2 ] ,

( 15 )

which describes the wavefield of a diffraction mode along the ID waveguide with quadratic profile. Furthermore, for this value of ct = l, the wavefield in eq. (12) becomes

u(r, z) = e x p ( i k N o z ) exp[ - ( r / R ) 2 ] ,

°

Fig. 3. a-power, refractive index variation of a G R I N fiber as a function of the dimensionless parameter r/R, for a = l until a=10.

(16)

as is well-known; see for example ref. [ 10 ]. However, to the best of our knowledge it is the first time that the wavefields in eqs. (9) and (12) have been identified as diffraction modes of certain refractive index profiles in eqs. (6) and (10), respectively.

6. Conclusions Summaryzing, we have shown that the amplitude 23

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variations o f some diffraction m o d e s o f certain a power, refractive index profiles are supergaussian functions, similar to those that are used as tapering filters in laser windows. Several 3D graphs display the form o f both the refractive index profiles and the supergaussian beams.

Acknowledgements We are indebted to the reviewer for her (his) useful suggestions. J.O.-C. gratefully acknowledges the financial support o f the Direcci6n General de Investigaci6n Cientifica y T6cnica, Ministerio de Educaci6n y Ciencia, Spain. G. Saavedra was supported during this work by the Conselleria de Cultura, Educaci6 i Ci6ncia de la Generalitat Valenciana, Spain.

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References [ 1] J.P. Mills and B.J. Thompson, J. Opt. Soc. Am. A 3 ( 1986 ) 694. [2 ] J. Ojeda-Castafieda, E. Tepichln and A. Pons, Appl. Optics 27 (1988) 5140. [ 3 ] S. de Silvestri, V. Magni, O. Svelto and G. Valentini, IEEE. J. Quantum Electron. 26 (1990) 1500. [4] A. Cutolo, G. Calafiore and S. Solimeno, Optics Comm. 93 (1992) 163. [5] R. Beach, J. Davin, S. Mitchell, W. Benett, B. Freitas, R. Solarz and P. Avizonis, Optics Lett. 17 (1992) 124. [6] M.S. Bowers, Optics Lett. 17 (1992) 1319. [ 7 ] J. Ojeda-Castafiedaand E. L6pez-Olazagasti, Microwaveand Opt. Technology Lett. 6 (1993) 73. [8] D. Marcuse, Light transmission optics (Van Nostrand Reinhold Co., New York, 1972) p. 103. [9]A.E. Siegman, Lasers (University Science Books, Mill Valley, 1986) p. 628. [10] D. Marcuse, Light transmission optics (Van Nostrand Reinhold Co., New York, 1972) ch. 7.