Optimum optical phase conjugation parameters for a holographic laser oscillator A. De la Piedad, S. González-Martínez, E. Rosas* Laboratorio de Optoelectrónica, Instituto Tecnológico de Toluca, Av. Tecnológico s/n, Ex-Rancho La Virgen, 52140, Metepec, México. ABSTRACT We present an ABCD transfer matrix spatial mode analysis of a holographic laser oscillator (HLO), carried out in order to determine the parameters for optimum optical phase conjugation of the Gaussian beam spot size. Those parameters allow a large mode size and efficient energy extraction operation of the studied HLO configurations. The HLO is formed by a nonlinear medium in a self-intersecting loop geometry incorporating a mobile intracavity lens and a feedback mirror acting as output coupler. Keywords: Optical Phase Conjugation, Gaussian Beams, Holographic Laser Oscillators.
1. INTRODUCTION The techniques of Optical Phase Conjugation1 (OPC) have been incorporated in some resonators in order to correct for aberrations in gain media which cause degradation of spatial beam quality in solid-state lasers2. With this aim, many studies of modes in stable/unstable conventional3,4 and in OPC resonators5 have been done and the self-consistency of fundamental modes has also been analyzed in holographic laser oscillators6-8, (HLO) 9. In this work we present a transfer matrix study of the transient evolution of the fundamental mode and determine the parameters for optimum OPC of the spot size for an HLO formed by a nonlinear Four Wave Mixing medium (FWM) with interacting beams E1-E4, a lens of focal length f, an amplifier and a plane output coupler (OC), Fig. 1.a. Plane Output Coupler (OC)
Plane Output Coupler (OC)
Amplifier
E2
E1
q~0
E3
E4
Non-linear FWM Medium
L=Loop length
x q~1
q~2
q~3 d
(a)
Non-linear FWM Medium
q~4
(b)
Fig. 1. a) Schematic diagram and, b) FWM interaction for the studied holographic laser oscillator.
*
Present address: División de Óptica y Radiometría, Centro Nacional de Metrología, km 4,5 Carretera a Los Cués, C. P. 76241, El Marqués, Querétaro, México, E-mail:
[email protected] Fifth Symposium Optics in Industry, edited by E. Rosas, R. Cardoso, J. C. Bermudez, O. Barbosa-García, Proceedings of SPIE Vol. 6046, 60462B, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.674626
Proc. of SPIE Vol. 6046 60462B-1
2. FWM INTERACTION AND TRANSFER MATRIX METHOD FOR GAUSSIAN BEAMS We begin by considering the beams involved in the FWM interaction shown in Fig. 1.b, as:
Ej =
1 2
~ A j exp[ i (ωt − kz ) ] + c.c. ,
(1a)
⎡ ikr 2 ⎤ ~ A j (r , z ) ∝ exp ⎢− ⎥, ⎢⎣ 2q~ j ( z ) ⎥⎦
1 1 λ = −i , ~ q j ( z) R j ( z) πw j 2 ( z )
(1b) j=1..4,
(1c)
~ is the complex where ω is the angular frequency, k = 2π λ is the wave vector magnitude with wavelength λ , and A j
amplitude with symmetric Gaussian variation with respect to the radial coordinate r, q~ j is the complex beam parameter for a beam with radius of curvature Rj (ROC) and spot size wj; and z is the distance along the propagation direction. Thus the beam generated in the FWM interaction, E4, is given by10:
~ ~ ~ ~ A4 ∝ A1 A2 A3∗ .
(2)
By substituting equation (1b) in relation (2) we find that the complex beam parameters of the interacting beams, as well as their corresponding ROC and spot sizes, are related by:
1 1 1 1 = ~ +~ − ∗ , ~ ~ q 4 q1 q 2 q 3
1 1 1 1 = + − , R4 R1 R2 R3 1 1 1 1 = 2 + 2 + 2 . 2 w4 w1 w2 w3
(3a) (3b) (3c)
It is known that in order to obtain a perfect OPC it is required that A~4 ∝ A~3∗ 10, ( q~4 = −q~3∗ ), which can be achieved when A1 and A2 are great aperture waves with conjugated wavefronts, or when the beam A3 has a small ROC compared to those of A1 and A2; although in experiments there is a spot size reduction, (w4
