Laser Diode to Single-Mode Circular Core Parabolic Index Fiber Coupling via Upside-Down Tapered Hyperbolic Microlens on the Tip of the Fiber: Prediction of Coupling Optics by ABCD Matrix Formalism

Angshuman Majumdar; Chintan Kumar Mandal; Sankar Gangopadhyay

doi:10.1515/joc-2017-0040

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Laser Diode to Single-Mode Circular Core Parabolic Index Fiber Coupling via Upside-Down Tapered Hyperbolic Microlens on the Tip of the Fiber: Prediction of Coupling Optics by ABCD Matrix Formalism

Angshuman Majumdar , Chintan Kumar Mandal und Sankar Gangopadhyay

Veröffentlicht/Copyright: 12. August 2017

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Abstract

We employ ABCD matrix formalism in order to investigate the coupling optics involving laser diode to single-mode circular core parabolic index fiber excitation via upside-down tapered hyperbolic microlens on the fiber tip. Analytic expressions for coupling efficiencies both in absence and in presence of transverse and angular mismatches are formulated. The concerned investigations are made for two practical wavelengths namely 1.3 µm and 1.5 µm. The execution of the prescribed formulations involves little computation. It has been found that the wavelength 1.5 µm is more efficient in respect of coupling. It is also seen that the present coupling device at both the wavelengths shows more tolerance with respect to angular mismatch. As regards tolerance with respect to transverse mismatch, the result is poor at both the wavelengths used. Consequently, it is desirable that designers should not to exceed transverse mismatch beyond 1 μm.

Keywords: laser diode; upside-down tapered hyperbolic microlens; single-mode circular core parabolic index fiber; optical coupling; coupling losses

Acknowledgments

The authors are grateful to the anonymous reviewer for the constructive suggestion.

Appendix

The laser beam input and output parameters(q1,q2) are connected by the relation given below

(19)q2=Aq1+BCq1+D

where,

(20)1q1,2=1R1,2−jλ0πw1,22n1,2

Here, R, n, w and λ0represent the radius of curvature of wavefront, refractive index, spot size and the wavelength in free space respectively.

The transformation matrix T₁ of the graded-index taper lens with parallel ends is given by [8, 9]

(21)T1=R2zR1z/nconcodR2zdzdR1zdz

Further, the transformation matrix T₂ of the hyperbolic end face has been found as [16, 17]

(22)T2= (101−nconco(b2/a)1nco)

Accordingly, the transformation matrix T=(T₁T₂) of the system can be expressed as [8, 9]

(23)T=[R2(z)R1(z)/nconcodR2(z)dzdR1(z)dz] (101−nconco(b2/a)1nco)

=A1B1C1D1

The ray matrix M for the upside down tapered hyperbolic microlens on the fiber tip is given by [9, 16, 17]

(24)M=A1B1C1D11u01=ABCD

Here,

(25)A1=R2(z)−(nco−1)nco(b2/a)R1(z),B1=1ncoR1(z),C1=ncodR2(z)dz−(nco−1)(b2/a)dR1(z)dz,D1=dR1(z)dz,

Here, b and a denote lengths of semi-axes of the hyperbolic interface in the plane of the paper

Further,

(26)R1z=−LηGz12SinηlnGzdR1zdz=1Gz12CosηlnGz+12ηSinηlnGzR2z=Gz12CosηlnGz−12ηSinηlnGzdR2zdz=A02a02Ld2Gz12SinηlnGz

where z is the axial length along the tapered region, L is axial length from the end face of the fiber to the geometrical vertex of the tapered profile, d is the radius of the aperture.

Further, the used parameters G (z), η, A₀, z are given by

(27)G(z) = 1 − zLη = A02 a02 L2d2 − 14A0 =d (1 − zL) (1 − n2cln2co)12a02z = L (d − a0)d}

Further, from eq. (21)

A=A1;B=A1u+B1;C=C1;D=C1u+D1

The lens transformed spot sizes w_2x,2y and radii of curvature R_2x,2y are obtained by using eqs (19), (20) and the ABCD matrix given by (21) and those are given below

(28)w2x,2y2=A22w1x,1y2+(λ12B2)/(π2w1x,1y2)nA2D−BC2

(29)1R2x,2y=A2C2w1x,1y2+λ12BD/(π2w1x,1y2)A22w1x,1y2+λ12B2/(π2w1x,1y2)

where λ1=λ0/n1,A2=A+B/R1;C2=C+D/R1andn=n2n1

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Received: 2017-03-11

Accepted: 2017-07-20

Published Online: 2017-08-12

Published in Print: 2019-07-26

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Schlagwörter für diesen Artikel

laser diode; upside-down tapered hyperbolic microlens; single-mode circular core parabolic index fiber; optical coupling; coupling losses