V-parameter

Lateral-mode control of vertical-cavity surface-emitting lasers (VCSELs) is one of the key issues in realizing high performance optical communication systems, in which single-mode operation is necessary for long and short wavelength regions. High-power single-mode operation is also required for free-space data communication applications.

Recently, a two-dimensional photonic crystal (2D PC) structure formed on a VCSEL surface has been investigated as a lateral-mode control method. The most attractive feature of this structure is the enlargement of the emission area, thereby permitting higher power output. The large area can be realized because of strong wavelength dependence of the refractive index in the 2D photonic crystal structure, analogous to the situation in a

photonic crystal fiber. Although good single mode operation has been reported for a specific structure, the optimized design of 2-D photonic crystal structure was not clear, especially when considering a finite etching depth. Since the mode control mechanism utilized in this technology is the effective index control achieved by forming a 2D photonic crystal structure, a parameter representing this control must have a strong dependence, both theoretically and experimentally, of the effective index change in a VCSEL structure.

A two-dimensional (2D) photonic crystal structure formed on a VCSEL surface has been investigated as a control method of lateral mode. D. S. Song et al. first applied 2-D photonic crystal on oxide-confined VCSEL, and the schematic structure was shown in Figure 2-4 [3]. This concept was from the endless single-mode from photonic crystal fibers [4]. By introducing the single defect photonic crystal to the VCSEL, a waveguide is expected to be formed around the central defect region where the effective index is larger than the surrounding region. The effective index model [5, 6] is used to understand the VCSEL with lateral structural variation. According to the effective index model, in ref. [3], the number of guided mode of an oxide-confined VCSEL is determined by V parameter, which has the form [6]

2 where kcore,z , kclad,z , are longitudinal resonance wave vectors of core and cladding region, respectively, and hcore , hclad are transverse wave vector in the medium. d is a diameter of the core. The〈ε represents the dielectric constant weighted by the longitudinal standing 〉 wave. For the photonic crystal VCSEL shown in Figure 2-3, the resonance wavelengths would be different for central core region and rthe surrounding region. This situation is approximated as a simple step-index waveguide

Figure 2-4 Schematic of the 850 nm PC-VCSEL. Note that the first generation PC-VCSEL structure has no oxide current aperture. The oxide aperture is added to the second generation devices for current confinement.

surrounding region. This situation is approximated as a simple step-index waveguide as shown in Figure 2-4. With the aid of this model, the difference of effective indices in the two regions can be estimated by measuring the resonant wavelengths in the core and surrounding regions. The overall effects of the etch diameter, and pitch of the holes show up experimentally as the shift of the resonant wavelength. This may be written compactly as

core clad

core core

eff clad core

eff

n

n

/

≈ Δ λ

/ λ

Δ

PC-VCSEL were Fabricated on selective oxide VCSEL, single-mode output with higher output power were realized from larger aperture [1]. However, those photonic crystal PC-VCSELs exhibit relatively high threshold current (Ith) due to large oxide confined apertures.

The 2D photonic crystal structure with finite etching depth incorporating a single

point or a seven-point defect is formed in the DBR. It is know that the normalized frequency or V-parameter is useful in evaluating the number of guided modes in cylindrical wave guides, an important example being step index optical fibers. The cutoff condition of the first higher mode leads to V

n

n

n

n

Figure 2-5 The Schematic of lateral effective index variation provided by the photonic crystal.

eff = 2.405, and thus a wave guide with Veff < 2.405 is considered to be single mode. In a photonic crystal VCSEL, the effective V-parameter can be expressed by

(2-3)

where λ is an operating wavelength, γ is an equivalent defect radius, neff is the effective index of the VCSEL cavity [9] without a photonic crystal structure present, Dn

is the refractive index reduction introduced by the photonic crystal structure, and r is

the hole depth dependence factor that accounts for finite etching depth of the photonic crystal holes in actual photonic crystal VCSELs. The r factor can be understood qualitatively as proportional to the spatial overlap between the photonic crystal structure. Thus, γ=0 for vanishing etching depth, γ=1 for infinite etching depth, and γ=0.5 for holes reaching the middle of the cavity. In the following discussion, the equivalent defect radii of a single point defect and seven-point defect structures are assumed to be Λ-d/2 and √3Λ-d/2, respectively, where L is lattice constant and d is the hole diameter of a circular hole of the 2-D photonic crystal structure. Since we need to investigate larger d/L ratios than those of PCFs [10], Veff is slightly modified from its appearance in Ref. 10, with the introduction of the Veff-parameter and the modified r in Eq. (2-3). The refractive index variation Dn can be obtained from the photonic band diagram of an out-of-plane propagation mode[11,12], it calculated by assuming that the photonic crystal structure is infinite both in lateral and vertical directions.

Reference

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Koyama, and K. lga, “Design and fabrication processes consideration of GaN-based surface emitting lasers“, Trans. IEICE, J81-C-II, pp.97-104,1998.

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[10] R.Brand, “10 gigabit Ethernet interconnection with wide area networks,” 10GEA, March 2003.

[11] A.A. Maradudin and A.R. McGurn, J. Mod. Opt. 41, 275~1994.

[12] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystal-Molding the Flow of Light~Princeton University Press, Princeton, NJ, 1995.