Invited Paper

Optical Connections by GRIN lenses C.Gómez-Reino, M.V.Pérez, C.Bao and M.T.Flores-Arias GRIN Optics Group, Applied Physics Department, Faculty of Physics and Optics and Optometry School, Campus Sur, University of Santiago de Compostela, E-15782 Santiago de Compostela, Spain.

ABSTRACT Devices for optical connections by GRIN lenses have been reported in literature. In particular, two optical configurations designed for beam size control and light deflection by tapered GRIN lenses and for correction of astigmatic gaussian beams from laser diodes by anamorphic selfoc lenses are presented. Keywords: Optical connections, GRIN optics, index profiles, photonic components.

1. INTRODUCTION A GRIN lens consists of a cylinder of inhomogeneous dielectric material with a refractive index distribution that has a maximun at the cylinder axis and decreases continuously from the axis to the periphery along the transverse direction. The focusing and transforming capabilities of a GRIN lens come from a quadratic variation in refractive index with radial distance from axis. In a GRIN lens, rays follow sinusoidal trajectories as if they were bent by a force toward the higher refractive index. [1-5]. Taking advantages of the typical functions such as on-axis and off-axis imaging and Fourier transforming of the GRIN lenses, a wide variety of devices have been designed and fabricated for applications in science and technology [6] . In particular, many optical configurations for connecting fibers, sources or detectors by GRIN lenses have been reported for coupling purposes. We will present two configurations designed for beam-size control and light deflection by tapered GRIN lenses and for correction of astigmatic gaussian beams from laser diodes by anamorphic selfoc lenses.

2. BEAM SIZE CONTROL AND LIGHT DEFLECTION BY TAPERED GRIN LENSES Tapered GRIN lenses can be used in applications requiring beam-size control and light deflection for coupling purposes. A simple study of light propagation by geometrical optics in a tapered GRIN lens illuminated by a tilted plane beam is presented. We will consider a GRIN lens of thickness d and semiaperture a whose refractive index profile is given by n 2 [1 − g 2 (z )r 2 ] for 0 ≤ r ≤ a and 0 ≤ z ≤ d n 2 (r, z ) = 0 otherwise 1

(1)

where g(z) is the taper function which describes the evolution of the transverse index distribution along the z axis. When the lens is illuminated by a tilted plane beam not all rays reaching the input face will be confined through it so it is necessary to define an input effective semiaperture for the upper and lower marginal rays [7]. In order to evaluate the input effective semiaperture we can proceed as follows: for a GRIN lens, the position and slope of a ray at any z > 0 (fig. 1), in matrix formalism, are given by r (z ) H f (z ) H a (z ) r0 = r(z ) H f (z ) H a (z ) r0

104

(2)

Photonic Devices and Algorithms for Computing IV, Khan M. Iftekharuddin, Abdul Ahad S. Awwal, Editors, Proceedings of SPIE Vol. 4788 (2002) © 2002 SPIE · 0277-786X/02/$15.00

where dot denotes derivative with respect to z, r0 and r0 are the position and the slope, respectively, of a ray at the input of the GRIN lens, and H a ,f and H a ,f are the position and the slope of the axial and field rays, respectively. Eq.(2) indicates that the trajectory of any ray through the GRIN lens can be expressed as a linear combination of the axial and field rays. These rays are two linearly independent particular solutions of the paraxial ray equation [1,5,8] (z )+ g 2 (z )H a (z ) = 0 H a

f

(3)

f

with initial conditions (0 ) 0 H a (0 ) = H = f

;

(0 ) 1 H f (0 ) = H = a

(4)

Axial and field rays are related by Lagrange’s invariant and at any z it is given by ( ) H a z H f (z ) − H a (z )H f (z ) = 1

(5)

r0 r0 r(z)

r(z)

G R IN lens z= d

z= 0

Figure. 1. Arbitrary ray trajectory through a GRIN lens.

Both rays, for refractive index that changes very slightly over distances comparable with the wavelength(lens with weak inhomogeneity), are expressed as z (z ) H a (z ) = [g 0 g(z )] -1/2 sin ∫ g (z′) dz′ = −[g 0 g (z )] -1 H f 0 g H f (z ) = 0 g(z )

1/2

z g (z ) cos ∫ g (z′) dz′ = 0 H a ( ) g z 0

(6a)

(6b)

where g0 is the value of g(z) at z = 0. On the other hand, at a distance du from the input face of the lens where the upper marginal ray achieves the maximum deviation from axis (fig.2), the position and the slope of this ray can be written as. r (d u ) = a eu F(d u ) = a

(7a)

r(d u ) = a eu F (d u ) = 0 ⇒ F (d u ) = 0

(7b)

where eq.(2) has been used and

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(⋅)

(⋅)

F (d u ) =

H

f

(d u ) +

H a (d u )

(8)

n 0 d 0u

d 0u being the distance from the input face of the lens to the cut off point of the upper marginal ray with the axis and a eu being the upper effective semiaperture.

r0 a

u θ0 a e

d u0

u

z

a el n (r,z)

l

z= d u

z= 0

z= d l

Figure. 2. Effective semiaperture for the upper and the lower marginal rays in a tapered GRIN lens illuminated by a tilted plane beam.

Then, the upper effective semiaperture is given by (d ) a eu = aH a u

(9)

where eqs.(7) and (5) have been used. Likewise, the input slope of the marginal ray can be expressed as r0 =

au θ0 ( = e u = −aH f du ) n 0 n 0d 0

(10)

where eqs.(7b-9) have been considered. From eqs.(9-10) it follows that the effective semiaperture and the input slope of the upper marginal ray are proportional to the slope of the axial and field rays at du, respectively. Eq.(9) can be written, in terms of the taper function and the input slope, as a eu

g (d ) r u = a − 0 g0 ag 0

2

1/ 2

(11)

where trigonometric relationship and eqs.(6) and (10) have been used. In the same way, we can evaluate the input effective semiaperture for the lower marginal ray. In this case, we have

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a le

=−

aH

g (d ) r l − 0 ( ) d a = − a l ag g0 0

2

1/ 2

(12)

where (d ) r0 = −aH f l

(13)

(d ) H H f u = f (d l )

(14)

verifying

and dl being the GRIN lens length for which the slope of the lower marginal ray is parallel to the z axis (fig 2). On the other hand, from eqs.(2) and (6) it follows that any ray leaving the input of a GRIN lens describes a sinusoidal path through it. This result shows that a GRIN lens carries out optical operations such as imaging and transforming [9]. Taking advantages of such operations we can design a device in order to deflect and contract or expand a tilted beam by a GRIN lens. We consider now lens lengths z = zm, where m is an integer, such that (z ) 0 H a (z m ) = H f m =

(15)

Then, ray equations at these lengths for the upper and lower marginal rays become u l r

(z m ) =

u

l r

u l a e H f

(z m )

(16a)

(z m ) = r0 H a (z m )

(16b)

where eq.(2) has been used. Equation (16a) represents the output effective semiaperture for the upper and lower marginal rays and eq.(16b) gives the output beam slope at zm. Likewise, equations (16) indicate that the position and the slope of the marginal rays at zm are proportional, respectively, to the position and the slope of rays at the input. In other words, input rays, all of one direction, produce a parallel beam at zm in some other direction whose scaling factor for position (transverse magnification) and for slope (angular magnification) are given by g M t = H f (z m ) = (− 1)m 0 g (z m )

1/ 2

(z ) ( 1)m g (z m ) Ma = H a m = − g 0

(17a) 1/ 2

(17b)

verifying Ma Mt = 1

(18)

where eqs.(6) have been considered. In short, cutting the tapered GRIN lens at zm lengths we can obtain a device for changing the light propagation direction and for controlling the size of a light beam.

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We will apply the above results to a GRIN lens whose taper function is given by [8,10] g(z ) =

g0 1− z

(19)

2

L2

Figure. 3. Equi-index surface of a parabolic GRIN lens.

The equi-index surfaces are revolution paraboloids around the z-axis with common apices at z = ± L (fig.3). In this kind of lens, axial and field rays can be written as

(L H (z ) =

2

a

− z2 g0L

z2 H f (z ) = 1 − 2 L

)

1/ 2

1/ 2

z sin g 0 L tan −1 L

(20a)

z cosg 0 L tan −1 L

(20b)

and the zm lengths verifying eq.(15) are given by mπ z m = L tanh g0L

(21)

For this profile, the input and output effective semiapertures of the upper and lower marginal rays and the output beam slope can be expressed as u l a e

u l r

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2 2 r L = ±a 2 − 0 L − d 2u ag 0 l u m l a e sech

(z m ) = (− 1)

1/ 2

mπ g0L

(22)

(23)

re (z m ) =

θe mπ = (− 1)m r0 cosh n0 g0L

(24)

where θe is the deflection angle of the output beam and where eqs.(11-12), (16), (17), (19) and (21) have been used.

a)

b) 0.50

θe( o )

20

0.45

10

a ue [m m ]

0.40 0.35

0

1

2

3

4

5

6

m

0.30 -10 0.25 -20

0.20 0

1

2

3

4

5

6

7

8

θ0 ( o )

Figure. 4. Dependence of (a) upper effective semiaperture on the input beam slope and (b) the deflection angle of the output beam on m. Calculations have been made for a = 500µm , g 0 = 0.192mm −1 , n 0 = 1.6 and L = 20mm (a) and L = 50mm (b).

Figure 4a shows the dependence of the effective semiaperture on the input beam slope, θ 0 = n 0 r0 , for this kind of lens. As expected, semiaperture decreases with the input slope. Figure 4b represents the hyperbolic variation of the deflection angle of the output beam with integer m for the first six zeros of Ha.

r0

a ue+ a el

z n (r,z) z= 0

zm =1

Figure. 5. Light deflector and beam size controller obtained cutting the lens at z1. Ray-tracing has been made for a = 500µm , L = 20mm , g 0 = 0.192mm −1 , n 0 = 1.6 , θ0 = 5o and an effective aperture of a eu + a le ≅ 918µm

Figure 5 depicts a device designed by cutting a parabolic tapered GRIN lens at length z1 for which the first zero of Ha is obtained. The deflection angle and the output beam size depend on the integer m and on the lens parameters. Output

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beam size of 678 µm at z1 and of 344.5 µm at z2 with deflection angles of –6.8o and 13.3o, respectively can be obtained when the GRIN lens of parameters a = 500µm , L = 20mm , g 0 = 0.192mm −1 and n 0 = 1.6 is illuminated by a tilted plane beam of θ 0 = 5 o . Note that for the principle of reversibility of rays, this device can also be used as a beam expander and for beam-slope reduction [11]. Finally, when a collimated beam impinges normally on the input face of the tapered GRIN lens, the physical semiaperture becomes the input effective semiaperture. Therefore, the emerging collimated beam propagates along the same direction as the input . The lens works as a beam-size controller to contract a collimated beam. The relationship between sizes of input and output beams is given by eq.(17a), and the lens can be applied for butt-joining coupling between two multimode fibers of different core sizes.

3. ASTIGMATIC-CORRECTING SELFOC LENS FOR LASER DIODES Laser diodes offer significant advantages over other laser sources in efficiency, size and cost, but suffer from inferior optical characteristics. Their beam diverge, have asymmetric cross section and are highly astigmatic. These deficiencies must be corrected to comply with many of the current applications of laser diodes in communications, data storage and imaging. Refractive-optics elements such as cylindrical lenses [12-14] and anamorphic prisms [15] are often used in correcting the output of the laser diodes. Because these elements are bulky and expensive, other approaches were proposed: integrated optics lenses or gratings [16] and holographic elements [17-18]. We present a design for correcting astigmatic gaussian beams from laser diodes by an anamorphic GRIN lens, to obtain at its output face a rotationally symmetric Gaussian beam, both in phase and amplitude.

Figure. 6. Refractive index profile in the anamorphic GRIN lens. Calculations have been made for a=1mm,. n0=1.5, gx=0.12mm-1, gy=0.08mm-1

We consider a GRIN lens whose refractive index profile is given by (fig.6)

[

n 2 (x , y ) = n 02 1 − g 2x x 2 − g 2y y 2

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]

(25)

where n0 is the index along the z optical axis, and gx and gy are the gradient parameters describing the evolution of the transverse parabolic index distribution along x and y axes, respectively [19]. The equi-index surfaces are cylinders of elliptical basis around z axis. At a distance d1 from the output face of the laser diode, the emited beam can be closely described by a spherocylindrical Gaussian beam

[

]

π exp{iϕ(d1 )}expi U x (d1 )x 02 + U y (d1 )y 02 w x (d1 )w y (d1 ) λ w 0x w 0y

ψ(x 0 , y 0 ) =

(26)

where ϕ(d1 ) is the on-axis phase at d1 and the beam parameters are given by the complex wavefront curvatures U x (d1 ) = y

1

R x (d1 ) y

+i

λ πw 2x y

(27)

(d1 )

where R x and w x are the principal radii of curvature and the beam half-widths at d1, respectively. The emitted beam y

y

has its waist w0x and w0y, both perpendicular and parallel to the junction at the output face of the laser diode as shown in fig.7. The relationship between the waists and the half-widths at d1 and the radii of curvature are given by

a)

2 (d + D )λ dλ w 2y (d1 ) = w 02 y 1 + 1 2 ; w 2x (d1 ) = w 02 x 1 + 1 2 πw πw 0y 0x

2 πw 02 y πw 02 x R y (d1 ) = (d1 + D ) 1 + ; R x (d1 ) = d1 1 + (d1 + D )λ d1λ

x y

w 0x

(28a)

2

(28b)

z

y0

wx

x

b)

x0

2

D

wy

y

w 0y z

Figure. 7. Astigmatic gaussian beam produced by a laser diode (a) D is the axial astigmatism and (b) the difference between the waist w0x and w0y will be referred as transverse astigmatism.

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At the output face of the GRIN lens of thickness d, the mode field can be easily evaluated by a integral transform where the kernel of the linear transformation is separable in both transverse axes and is written as [5,8] K (x 0 , y 0 , x ′, y ′; d ) =

kn 0 exp{ikn 0 d}

[

]

i 2π H ax (d )H ay (d )

1/ 2

[

]

kn 0 2 x′2H exp i ax (d ) + x 0 H fx (d ) − 2 x ′x 0 2H ax (d )

[

kn 0 2 y′ 2 H exp i ay (d ) + y 0 H fy (d ) − 2 y ′y 0 2H ay (d ) (⋅)

(29)

]

(⋅)

Where H a x and H f x are the position and the slope of the axial and field rays at d in both transverse directions, given y

y

by sin g x d H f x (d ) y y = − 2 H a x (d ) = g x g x y

y

(30a)

y

H f x (d ) = cos g x d = H a x (d ) y y y

(30b)

Performing the integral transformation, the field at the output face of the GRIN lens can be expressed as ψ(x ′, y′; d ) =

[

]

π exp{iϕ(d )}expi U x (d )x ′ 2 + U y (d )y′ 2 Ω x (d )Ω y (d ) λ w 0x w 0y

(31)

where ϕ(d ) is the on-axis phase at d and the complex curvatures are given by U x (d ) = y

λ 1 +i ρ x (d ) πΩ 2x (d ) y

(32)

y

where the radii of curvature and the beam half-widths are written as 2 2 λR x (d1 )H a x (z ) 1 d y y ρ x (d ) = ln H a x (z ) + n 0 H f x (z )R x (d1 ) + 2 2 dz ( ) π w d y y y 1 x y y z =d

2 2 H a x (d ) λH a x (d ) 2 2 y y + Ω x (d ) = w x (d1 ) H f x (d ) + n 0 R x (d1 ) πn 0 w 2x (d1 ) y y y y y

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−1

(33a)

(33b)

So, in order to correct astigmatism in a laser diode by this kind of lens, the index along the axis n0, gradient parameters gx and gy, and thickness d of the lens must be chosen in such a way that ρx = ρy

(34a)

Ωx = Ωy

(34b)

to correct axial astigmatism, and

to correct transverse astigmatism at any plane behind the GRIN lens. By solving the above set of equations, a rotationally symmetric Gaussian beam of waist w0 at distance d1′ from the output face of the GRIN lens is obtained as shown in fig.8.

x0

a)

w0

w 0x

b)

d' 1

d D

y0

w 0y

w0

d Figure. 8. Anamorphic selfoc lens to correct the astigmatism in a diode laser; (a) direction perpendicular to the junction, (b) direction parallel to the junction.

Some applications require only circular symmetry of the beam, without regard to position and size of the output beam waist [20]. For this case, it is interesting to study a particular design of the GRIN lens in which the set of transcendental equations (34) presents an analytical solution that takes into account the following conditions: (i) the lens must be attached to the output face of the laser diode so that one of its transverse directions fits with the direction perpendicular to the junction where the waist w0x of the emitted beam is located, and (ii) the index along the axis and the gradient parameters in the x direction must verify λ w 0x = πn 0 g x

1/ 2

(35)

The waist of the beam in this direction matches the fundamental mode of the lens and propagates without distorsion for x direction in such a way that the beam propagates inside the GRIN lens as adiffractional beam, that is ρ x → ∞ ; Ω x = w 0x

(36)

for any value of the lens thickness.

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Since a spherical Gaussian beam of waist w0x must be obtained at the output face of the GRIN lens, the thickness d and the gradient parameter gy can be evaluated by the following conditions ρ y → ∞ or

dΩ 2y dz

=0

(37a)

z =d

Ω y = w 0x

(37b)

2g y 1 −1 d= tan + mπ 2 2g y 2 λ 1 n 0 R y (D )g y − 2 2 ( ) − 2 2 4 ( ) π n R D n w D 0 y 0 y

(38a)

From eq.(37a) it follows that

Inserting above equation into eq.(37b), taking into account eqs.(30) we obtain g 2y =

λ2 R 2y (D ) w 2y (D ) 1 + n 02 R 2y (D ) π 2 w 02 x w 2y (D ) w 02 x − w 2y (D )

(38b)

where R(D) and wy(D) are the radius of curvature and beam half-width for d1 = 0 , that is, at the input face of the lens. Eqs.(38) give the values of d and gy for which a spherical Gaussian beam of waist of w0x will be obtained at the output face of the tapered GRIN lens (fig.9).

a) x0 w 0x

w 0x

y0

b)

w 0x

w 0y

d Figure 9. Anamorphic selfoc lens to correct the astigmatism in a diode laser by matching to the fundamental mode; (a) direction perpendicular to the junction, (b) direction parallel to the junction.

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CONCLUSIONS A design was presented to correct an astigmatic gaussian beam from laser diodes by an anmorphic selfoc lens in order to obtain rotationally symmetric gaussian beam, both in amplitude and phase, at the output face of the lens. A device to control the beam size and to deflect light by a tapered GRIN lens was designed for coupling purposes.

ACKNOWLEDGMENTS This work was supported by the Ministerio de Educación y Cultura, Spain, under contract TIC99-0489.

REFERENCES 1. R.K.Luneburg, Mathematical Theory of Optics, University of California Press, 1964. 2. E.W.Marchand, Gradient Index Optics, Academic Press, New York, 1978. 3. K.Iga, Y.Kokibun and M.Oikawa, Fundamentals of Microoptics, Academic Press, New York, 1984. 4. Y.A.Kravtsov and Y.I.Orlov, Geometrical Optics of Inhomogeneous Media,, Springer-Verlag, Berlin,1990. 5. C.Gomez-Reino, M.V.Perez and C.Bao, Gradient-Index Optics: Fundamentals and Applications, Springer-Verlag, Berlin, 2002. 6. W.J.Tomlinson, “Application of GRIN-rod lenses in optical fiber communication systems”, Appl.Opt.19, pp.1127-1138, 1980. 7. J.M.Rivas-Moscoso, C.Gomez-Reino, M.V.Perez and C.Bao, “Marginal rays in tapered gradient-index lenses”, Opt. Eng. 41, 303-313, 2002. 8. C.Gomez-Reino, “GRIN optics and its application in optical connections”, Int.J.Optoelectron 7, pp. 607-680, 1992. 9. C.Gomez-Reino and E.Larrea, “Paraxial imaging and transforming in a medium with gradient-index: transmittance function”, Appl.Opt. 24, pp.4375-4378, 1985. 10. C.Gomez-Reino and J.Sochacki, “Imaging and transforming capabilies of GRIN rods with non cylindrical surfaces of constant index: a family of exact solutions”, Appl.Opt.24, pp.4271-4275, 1985. 11. C.Bao, C.Gomez-Reino and M.V.Perez, “Light deflector and beam size controller design for tilted plane illumination by a GRIN planar taper medium” Opt.Comm. 172, pp.1-4, 1999. 12. I.Gorg, P.V.Goedertier, J.D.Knox, I.Ladany, J.P.Wittke and A.H.Firester, “Information scanning technology: application of CW AlGaAs injection lasers” Appl.Opt. 15, pp.1425-1430, 1976. o

13. D.R.Scifres, R.A.Sprague, W.Streifer and R.D.Burnham, “Focusing of a 7700 A high power phased array semiconductor lens” Appl.Phys.Lett. 41, pp.1121-1123, 1982. 14. P.M.Asbeck, D.A.Ca,,acl. J.J.Daniele, D.Lou, J.P.J.Heemskerk, W.J.Klenters and W.H.Ophey, “High-density optical recording with (Ga,Al)AsDH lasers” Appl.Phys.Lett. 34, pp.838-837, 1979. 15. K.Tatsuno, R.Drenten, C.Van der Poel, J.Opschoor and G.Acket, “Diffraction-limited circular single spot from phased array lasers” Appl.Opt. 28, pp.4560-4564, 1989. 16. J.R.Leger, G.J.Swanson and W.B. Weldkamp, “Coherent laser addition using binary phase gratings” Appl.Opt. 26, pp.4391-4394, 1987. 17. A.Aharoni, J.W.Goodman and Y.Amitai, “Efficient beam-correcting holographic collimator for laser diodes” Opt.Lett. 17, pp.1310-1312, 1992. 18. A.Aharoni, J.W.Goodmand and Y.Amitai, “Beam-correcting holographic doublet for focusing multimode laser diodes” Opt.Lett. 18, pp.179-181, 1993. 19. J.M.Stagaman and D.T.Moore, “Laser diode to fiber coupling using anamorphic gradient-index lenses”, Appl.Opt. 23, 1730-1733, 1984. 20. E.Acosta, R.M.Gonzalez and C.Gomez-Reino, “Design of an anamorphic gradient-index lens to correct astigmatic of Gaussian laser beams” Opt.Lett. 16, pp.627-629, 1991.

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Optical Connections by GRIN lenses C.Gómez-Reino, M.V.Pérez, C.Bao and M.T.Flores-Arias GRIN Optics Group, Applied Physics Department, Faculty of Physics and Optics and Optometry School, Campus Sur, University of Santiago de Compostela, E-15782 Santiago de Compostela, Spain.

ABSTRACT Devices for optical connections by GRIN lenses have been reported in literature. In particular, two optical configurations designed for beam size control and light deflection by tapered GRIN lenses and for correction of astigmatic gaussian beams from laser diodes by anamorphic selfoc lenses are presented. Keywords: Optical connections, GRIN optics, index profiles, photonic components.

1. INTRODUCTION A GRIN lens consists of a cylinder of inhomogeneous dielectric material with a refractive index distribution that has a maximun at the cylinder axis and decreases continuously from the axis to the periphery along the transverse direction. The focusing and transforming capabilities of a GRIN lens come from a quadratic variation in refractive index with radial distance from axis. In a GRIN lens, rays follow sinusoidal trajectories as if they were bent by a force toward the higher refractive index. [1-5]. Taking advantages of the typical functions such as on-axis and off-axis imaging and Fourier transforming of the GRIN lenses, a wide variety of devices have been designed and fabricated for applications in science and technology [6] . In particular, many optical configurations for connecting fibers, sources or detectors by GRIN lenses have been reported for coupling purposes. We will present two configurations designed for beam-size control and light deflection by tapered GRIN lenses and for correction of astigmatic gaussian beams from laser diodes by anamorphic selfoc lenses.

2. BEAM SIZE CONTROL AND LIGHT DEFLECTION BY TAPERED GRIN LENSES Tapered GRIN lenses can be used in applications requiring beam-size control and light deflection for coupling purposes. A simple study of light propagation by geometrical optics in a tapered GRIN lens illuminated by a tilted plane beam is presented. We will consider a GRIN lens of thickness d and semiaperture a whose refractive index profile is given by n 2 [1 − g 2 (z )r 2 ] for 0 ≤ r ≤ a and 0 ≤ z ≤ d n 2 (r, z ) = 0 otherwise 1

(1)

where g(z) is the taper function which describes the evolution of the transverse index distribution along the z axis. When the lens is illuminated by a tilted plane beam not all rays reaching the input face will be confined through it so it is necessary to define an input effective semiaperture for the upper and lower marginal rays [7]. In order to evaluate the input effective semiaperture we can proceed as follows: for a GRIN lens, the position and slope of a ray at any z > 0 (fig. 1), in matrix formalism, are given by r (z ) H f (z ) H a (z ) r0 = r(z ) H f (z ) H a (z ) r0

104

(2)

Photonic Devices and Algorithms for Computing IV, Khan M. Iftekharuddin, Abdul Ahad S. Awwal, Editors, Proceedings of SPIE Vol. 4788 (2002) © 2002 SPIE · 0277-786X/02/$15.00

where dot denotes derivative with respect to z, r0 and r0 are the position and the slope, respectively, of a ray at the input of the GRIN lens, and H a ,f and H a ,f are the position and the slope of the axial and field rays, respectively. Eq.(2) indicates that the trajectory of any ray through the GRIN lens can be expressed as a linear combination of the axial and field rays. These rays are two linearly independent particular solutions of the paraxial ray equation [1,5,8] (z )+ g 2 (z )H a (z ) = 0 H a

f

(3)

f

with initial conditions (0 ) 0 H a (0 ) = H = f

;

(0 ) 1 H f (0 ) = H = a

(4)

Axial and field rays are related by Lagrange’s invariant and at any z it is given by ( ) H a z H f (z ) − H a (z )H f (z ) = 1

(5)

r0 r0 r(z)

r(z)

G R IN lens z= d

z= 0

Figure. 1. Arbitrary ray trajectory through a GRIN lens.

Both rays, for refractive index that changes very slightly over distances comparable with the wavelength(lens with weak inhomogeneity), are expressed as z (z ) H a (z ) = [g 0 g(z )] -1/2 sin ∫ g (z′) dz′ = −[g 0 g (z )] -1 H f 0 g H f (z ) = 0 g(z )

1/2

z g (z ) cos ∫ g (z′) dz′ = 0 H a ( ) g z 0

(6a)

(6b)

where g0 is the value of g(z) at z = 0. On the other hand, at a distance du from the input face of the lens where the upper marginal ray achieves the maximum deviation from axis (fig.2), the position and the slope of this ray can be written as. r (d u ) = a eu F(d u ) = a

(7a)

r(d u ) = a eu F (d u ) = 0 ⇒ F (d u ) = 0

(7b)

where eq.(2) has been used and

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(⋅)

(⋅)

F (d u ) =

H

f

(d u ) +

H a (d u )

(8)

n 0 d 0u

d 0u being the distance from the input face of the lens to the cut off point of the upper marginal ray with the axis and a eu being the upper effective semiaperture.

r0 a

u θ0 a e

d u0

u

z

a el n (r,z)

l

z= d u

z= 0

z= d l

Figure. 2. Effective semiaperture for the upper and the lower marginal rays in a tapered GRIN lens illuminated by a tilted plane beam.

Then, the upper effective semiaperture is given by (d ) a eu = aH a u

(9)

where eqs.(7) and (5) have been used. Likewise, the input slope of the marginal ray can be expressed as r0 =

au θ0 ( = e u = −aH f du ) n 0 n 0d 0

(10)

where eqs.(7b-9) have been considered. From eqs.(9-10) it follows that the effective semiaperture and the input slope of the upper marginal ray are proportional to the slope of the axial and field rays at du, respectively. Eq.(9) can be written, in terms of the taper function and the input slope, as a eu

g (d ) r u = a − 0 g0 ag 0

2

1/ 2

(11)

where trigonometric relationship and eqs.(6) and (10) have been used. In the same way, we can evaluate the input effective semiaperture for the lower marginal ray. In this case, we have

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a le

=−

aH

g (d ) r l − 0 ( ) d a = − a l ag g0 0

2

1/ 2

(12)

where (d ) r0 = −aH f l

(13)

(d ) H H f u = f (d l )

(14)

verifying

and dl being the GRIN lens length for which the slope of the lower marginal ray is parallel to the z axis (fig 2). On the other hand, from eqs.(2) and (6) it follows that any ray leaving the input of a GRIN lens describes a sinusoidal path through it. This result shows that a GRIN lens carries out optical operations such as imaging and transforming [9]. Taking advantages of such operations we can design a device in order to deflect and contract or expand a tilted beam by a GRIN lens. We consider now lens lengths z = zm, where m is an integer, such that (z ) 0 H a (z m ) = H f m =

(15)

Then, ray equations at these lengths for the upper and lower marginal rays become u l r

(z m ) =

u

l r

u l a e H f

(z m )

(16a)

(z m ) = r0 H a (z m )

(16b)

where eq.(2) has been used. Equation (16a) represents the output effective semiaperture for the upper and lower marginal rays and eq.(16b) gives the output beam slope at zm. Likewise, equations (16) indicate that the position and the slope of the marginal rays at zm are proportional, respectively, to the position and the slope of rays at the input. In other words, input rays, all of one direction, produce a parallel beam at zm in some other direction whose scaling factor for position (transverse magnification) and for slope (angular magnification) are given by g M t = H f (z m ) = (− 1)m 0 g (z m )

1/ 2

(z ) ( 1)m g (z m ) Ma = H a m = − g 0

(17a) 1/ 2

(17b)

verifying Ma Mt = 1

(18)

where eqs.(6) have been considered. In short, cutting the tapered GRIN lens at zm lengths we can obtain a device for changing the light propagation direction and for controlling the size of a light beam.

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We will apply the above results to a GRIN lens whose taper function is given by [8,10] g(z ) =

g0 1− z

(19)

2

L2

Figure. 3. Equi-index surface of a parabolic GRIN lens.

The equi-index surfaces are revolution paraboloids around the z-axis with common apices at z = ± L (fig.3). In this kind of lens, axial and field rays can be written as

(L H (z ) =

2

a

− z2 g0L

z2 H f (z ) = 1 − 2 L

)

1/ 2

1/ 2

z sin g 0 L tan −1 L

(20a)

z cosg 0 L tan −1 L

(20b)

and the zm lengths verifying eq.(15) are given by mπ z m = L tanh g0L

(21)

For this profile, the input and output effective semiapertures of the upper and lower marginal rays and the output beam slope can be expressed as u l a e

u l r

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2 2 r L = ±a 2 − 0 L − d 2u ag 0 l u m l a e sech

(z m ) = (− 1)

1/ 2

mπ g0L

(22)

(23)

re (z m ) =

θe mπ = (− 1)m r0 cosh n0 g0L

(24)

where θe is the deflection angle of the output beam and where eqs.(11-12), (16), (17), (19) and (21) have been used.

a)

b) 0.50

θe( o )

20

0.45

10

a ue [m m ]

0.40 0.35

0

1

2

3

4

5

6

m

0.30 -10 0.25 -20

0.20 0

1

2

3

4

5

6

7

8

θ0 ( o )

Figure. 4. Dependence of (a) upper effective semiaperture on the input beam slope and (b) the deflection angle of the output beam on m. Calculations have been made for a = 500µm , g 0 = 0.192mm −1 , n 0 = 1.6 and L = 20mm (a) and L = 50mm (b).

Figure 4a shows the dependence of the effective semiaperture on the input beam slope, θ 0 = n 0 r0 , for this kind of lens. As expected, semiaperture decreases with the input slope. Figure 4b represents the hyperbolic variation of the deflection angle of the output beam with integer m for the first six zeros of Ha.

r0

a ue+ a el

z n (r,z) z= 0

zm =1

Figure. 5. Light deflector and beam size controller obtained cutting the lens at z1. Ray-tracing has been made for a = 500µm , L = 20mm , g 0 = 0.192mm −1 , n 0 = 1.6 , θ0 = 5o and an effective aperture of a eu + a le ≅ 918µm

Figure 5 depicts a device designed by cutting a parabolic tapered GRIN lens at length z1 for which the first zero of Ha is obtained. The deflection angle and the output beam size depend on the integer m and on the lens parameters. Output

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beam size of 678 µm at z1 and of 344.5 µm at z2 with deflection angles of –6.8o and 13.3o, respectively can be obtained when the GRIN lens of parameters a = 500µm , L = 20mm , g 0 = 0.192mm −1 and n 0 = 1.6 is illuminated by a tilted plane beam of θ 0 = 5 o . Note that for the principle of reversibility of rays, this device can also be used as a beam expander and for beam-slope reduction [11]. Finally, when a collimated beam impinges normally on the input face of the tapered GRIN lens, the physical semiaperture becomes the input effective semiaperture. Therefore, the emerging collimated beam propagates along the same direction as the input . The lens works as a beam-size controller to contract a collimated beam. The relationship between sizes of input and output beams is given by eq.(17a), and the lens can be applied for butt-joining coupling between two multimode fibers of different core sizes.

3. ASTIGMATIC-CORRECTING SELFOC LENS FOR LASER DIODES Laser diodes offer significant advantages over other laser sources in efficiency, size and cost, but suffer from inferior optical characteristics. Their beam diverge, have asymmetric cross section and are highly astigmatic. These deficiencies must be corrected to comply with many of the current applications of laser diodes in communications, data storage and imaging. Refractive-optics elements such as cylindrical lenses [12-14] and anamorphic prisms [15] are often used in correcting the output of the laser diodes. Because these elements are bulky and expensive, other approaches were proposed: integrated optics lenses or gratings [16] and holographic elements [17-18]. We present a design for correcting astigmatic gaussian beams from laser diodes by an anamorphic GRIN lens, to obtain at its output face a rotationally symmetric Gaussian beam, both in phase and amplitude.

Figure. 6. Refractive index profile in the anamorphic GRIN lens. Calculations have been made for a=1mm,. n0=1.5, gx=0.12mm-1, gy=0.08mm-1

We consider a GRIN lens whose refractive index profile is given by (fig.6)

[

n 2 (x , y ) = n 02 1 − g 2x x 2 − g 2y y 2

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]

(25)

where n0 is the index along the z optical axis, and gx and gy are the gradient parameters describing the evolution of the transverse parabolic index distribution along x and y axes, respectively [19]. The equi-index surfaces are cylinders of elliptical basis around z axis. At a distance d1 from the output face of the laser diode, the emited beam can be closely described by a spherocylindrical Gaussian beam

[

]

π exp{iϕ(d1 )}expi U x (d1 )x 02 + U y (d1 )y 02 w x (d1 )w y (d1 ) λ w 0x w 0y

ψ(x 0 , y 0 ) =

(26)

where ϕ(d1 ) is the on-axis phase at d1 and the beam parameters are given by the complex wavefront curvatures U x (d1 ) = y

1

R x (d1 ) y

+i

λ πw 2x y

(27)

(d1 )

where R x and w x are the principal radii of curvature and the beam half-widths at d1, respectively. The emitted beam y

y

has its waist w0x and w0y, both perpendicular and parallel to the junction at the output face of the laser diode as shown in fig.7. The relationship between the waists and the half-widths at d1 and the radii of curvature are given by

a)

2 (d + D )λ dλ w 2y (d1 ) = w 02 y 1 + 1 2 ; w 2x (d1 ) = w 02 x 1 + 1 2 πw πw 0y 0x

2 πw 02 y πw 02 x R y (d1 ) = (d1 + D ) 1 + ; R x (d1 ) = d1 1 + (d1 + D )λ d1λ

x y

w 0x

(28a)

2

(28b)

z

y0

wx

x

b)

x0

2

D

wy

y

w 0y z

Figure. 7. Astigmatic gaussian beam produced by a laser diode (a) D is the axial astigmatism and (b) the difference between the waist w0x and w0y will be referred as transverse astigmatism.

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At the output face of the GRIN lens of thickness d, the mode field can be easily evaluated by a integral transform where the kernel of the linear transformation is separable in both transverse axes and is written as [5,8] K (x 0 , y 0 , x ′, y ′; d ) =

kn 0 exp{ikn 0 d}

[

]

i 2π H ax (d )H ay (d )

1/ 2

[

]

kn 0 2 x′2H exp i ax (d ) + x 0 H fx (d ) − 2 x ′x 0 2H ax (d )

[

kn 0 2 y′ 2 H exp i ay (d ) + y 0 H fy (d ) − 2 y ′y 0 2H ay (d ) (⋅)

(29)

]

(⋅)

Where H a x and H f x are the position and the slope of the axial and field rays at d in both transverse directions, given y

y

by sin g x d H f x (d ) y y = − 2 H a x (d ) = g x g x y

y

(30a)

y

H f x (d ) = cos g x d = H a x (d ) y y y

(30b)

Performing the integral transformation, the field at the output face of the GRIN lens can be expressed as ψ(x ′, y′; d ) =

[

]

π exp{iϕ(d )}expi U x (d )x ′ 2 + U y (d )y′ 2 Ω x (d )Ω y (d ) λ w 0x w 0y

(31)

where ϕ(d ) is the on-axis phase at d and the complex curvatures are given by U x (d ) = y

λ 1 +i ρ x (d ) πΩ 2x (d ) y

(32)

y

where the radii of curvature and the beam half-widths are written as 2 2 λR x (d1 )H a x (z ) 1 d y y ρ x (d ) = ln H a x (z ) + n 0 H f x (z )R x (d1 ) + 2 2 dz ( ) π w d y y y 1 x y y z =d

2 2 H a x (d ) λH a x (d ) 2 2 y y + Ω x (d ) = w x (d1 ) H f x (d ) + n 0 R x (d1 ) πn 0 w 2x (d1 ) y y y y y

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−1

(33a)

(33b)

So, in order to correct astigmatism in a laser diode by this kind of lens, the index along the axis n0, gradient parameters gx and gy, and thickness d of the lens must be chosen in such a way that ρx = ρy

(34a)

Ωx = Ωy

(34b)

to correct axial astigmatism, and

to correct transverse astigmatism at any plane behind the GRIN lens. By solving the above set of equations, a rotationally symmetric Gaussian beam of waist w0 at distance d1′ from the output face of the GRIN lens is obtained as shown in fig.8.

x0

a)

w0

w 0x

b)

d' 1

d D

y0

w 0y

w0

d Figure. 8. Anamorphic selfoc lens to correct the astigmatism in a diode laser; (a) direction perpendicular to the junction, (b) direction parallel to the junction.

Some applications require only circular symmetry of the beam, without regard to position and size of the output beam waist [20]. For this case, it is interesting to study a particular design of the GRIN lens in which the set of transcendental equations (34) presents an analytical solution that takes into account the following conditions: (i) the lens must be attached to the output face of the laser diode so that one of its transverse directions fits with the direction perpendicular to the junction where the waist w0x of the emitted beam is located, and (ii) the index along the axis and the gradient parameters in the x direction must verify λ w 0x = πn 0 g x

1/ 2

(35)

The waist of the beam in this direction matches the fundamental mode of the lens and propagates without distorsion for x direction in such a way that the beam propagates inside the GRIN lens as adiffractional beam, that is ρ x → ∞ ; Ω x = w 0x

(36)

for any value of the lens thickness.

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Since a spherical Gaussian beam of waist w0x must be obtained at the output face of the GRIN lens, the thickness d and the gradient parameter gy can be evaluated by the following conditions ρ y → ∞ or

dΩ 2y dz

=0

(37a)

z =d

Ω y = w 0x

(37b)

2g y 1 −1 d= tan + mπ 2 2g y 2 λ 1 n 0 R y (D )g y − 2 2 ( ) − 2 2 4 ( ) π n R D n w D 0 y 0 y

(38a)

From eq.(37a) it follows that

Inserting above equation into eq.(37b), taking into account eqs.(30) we obtain g 2y =

λ2 R 2y (D ) w 2y (D ) 1 + n 02 R 2y (D ) π 2 w 02 x w 2y (D ) w 02 x − w 2y (D )

(38b)

where R(D) and wy(D) are the radius of curvature and beam half-width for d1 = 0 , that is, at the input face of the lens. Eqs.(38) give the values of d and gy for which a spherical Gaussian beam of waist of w0x will be obtained at the output face of the tapered GRIN lens (fig.9).

a) x0 w 0x

w 0x

y0

b)

w 0x

w 0y

d Figure 9. Anamorphic selfoc lens to correct the astigmatism in a diode laser by matching to the fundamental mode; (a) direction perpendicular to the junction, (b) direction parallel to the junction.

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CONCLUSIONS A design was presented to correct an astigmatic gaussian beam from laser diodes by an anmorphic selfoc lens in order to obtain rotationally symmetric gaussian beam, both in amplitude and phase, at the output face of the lens. A device to control the beam size and to deflect light by a tapered GRIN lens was designed for coupling purposes.

ACKNOWLEDGMENTS This work was supported by the Ministerio de Educación y Cultura, Spain, under contract TIC99-0489.

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