Phase-matched electric-field-induced second-harmonic generation is demonstrated in single-mode germania-doped silica fibers. A periodic second-order nonlinearity is induced by a simple interdigitated electrode structure, which can be rotated to permit phase matching between all propagating modes. The most efficient mode interaction between HE_{11}^{ω} and HE_{11}^{2}^{ω} is achieved at 1.064 μm by using a Q-switched Nd^{+3}:YAG laser. In principle, phase matching at any propagating wavelength is possible. This technique could be applied to planar as well as cylindrical waveguides and can be used with many non-χ^{(2)} materials. The asymmetry in the applied electric field enhances the optical-field overlaps between modes of dissimilar orders, and this is also demonstrated. A conversion efficiency of 4.0 × 10^{−4}% has been obtained in unoptimized devices. Device optimization is also discussed.

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Sensitivity of the Phase-Matching Pitch Λ on the Core Radius a, Core–Cladding Index Difference Δn, and Wavelength λ, for a = 4.0 μm, Δn = 4.5 × 10^{−3}, and λ = 1.064 μm

Mode Interaction

Parameter

LP_{01}^{ω} → LP_{01}^{2}^{ω}

LP_{01}^{ω} → LP_{11}^{2}^{ω}

∂Δ/∂a

2

5 × 10^{−3}

∂Λ/∂Δn (m)

−10^{−3}

14.55 × 10^{−6}

∂Λ/∂λ (μm/nm)

55 × 10^{−3}

79.4 × 10^{−3}

Phase-match length limited by above parameters (mm) [see Eq. (8)]^{a}

1237

446

Length limited by pitch tolerance alone (mm) (see text)^{b}

90

54.95

δΔn/Δn of 10^{−3}% and δa/a of 10^{−2}% have been used to calculate the data.
A tolerance of ±2.5 × 10^{−2}% assumed.

Table 2

Magnitudes and Phases of the Active χ_{ijkl}^{(3)}E_{l}^{0} Tensor Elements as a Result of the Components of the Applied dc Electric Field Relative to χ_{xxxx}^{(3)}E_{x}^{0} and Grouped as in Eqs. (21a) and (21b)^{a}

The subscripts of χ^{(3)} are grouped from terms such as those in Eqs. (18a)–(19b).
Dependent on electrode mark/space ratio, b/c. Values shown assume that the static field components are equal to E_{x}^{0}.

Table 3

Dimensions of Periodic Electrode Structures Used in Experiments

Mask

Electrode Width, b (μm)

Electrode Space, c (μm)

Pitch, Λ Λ (μm)

A

15.6

4.4

40

B

4.4

15.6

40

C

4.4

11.6

32

Table 4

Measured and Computed Coherence Lengths and Overlap Integrals Ratios for Seven Mode Interactions Are Compared Using Data from Device D1^{a}

LP_{11}^{ω} → LP_{21}^{2}^{ω} and LP_{01}^{ω} → LP_{01}^{2}^{ω} are coincident in the phase-matched data shown in Fig. 12. The former, weaker interaction was therefore not resolved. LP_{11}^{ω} → LP_{01}^{2}^{ω} could not be phase matched because the grating pitch used in the experiment was too large. Interaction LP_{01}^{ω} → LP_{31}^{2}^{ω}, could not be measured. a = 4 μm, Δn = 0.0045.

Table 5

Measurement Data on Devices Used in the Experiments

For the computation, values of R have been estimated from measurements of attenuation using oil overlays (see Ref. 32).
Using relation (26). The computed efficiency assumes that all the fundamental wavelength power was in the LP_{01}^{ω} mode, which was not the case.

Tables (5)

Table 1

Sensitivity of the Phase-Matching Pitch Λ on the Core Radius a, Core–Cladding Index Difference Δn, and Wavelength λ, for a = 4.0 μm, Δn = 4.5 × 10^{−3}, and λ = 1.064 μm

Mode Interaction

Parameter

LP_{01}^{ω} → LP_{01}^{2}^{ω}

LP_{01}^{ω} → LP_{11}^{2}^{ω}

∂Δ/∂a

2

5 × 10^{−3}

∂Λ/∂Δn (m)

−10^{−3}

14.55 × 10^{−6}

∂Λ/∂λ (μm/nm)

55 × 10^{−3}

79.4 × 10^{−3}

Phase-match length limited by above parameters (mm) [see Eq. (8)]^{a}

1237

446

Length limited by pitch tolerance alone (mm) (see text)^{b}

90

54.95

δΔn/Δn of 10^{−3}% and δa/a of 10^{−2}% have been used to calculate the data.
A tolerance of ±2.5 × 10^{−2}% assumed.

Table 2

Magnitudes and Phases of the Active χ_{ijkl}^{(3)}E_{l}^{0} Tensor Elements as a Result of the Components of the Applied dc Electric Field Relative to χ_{xxxx}^{(3)}E_{x}^{0} and Grouped as in Eqs. (21a) and (21b)^{a}

The subscripts of χ^{(3)} are grouped from terms such as those in Eqs. (18a)–(19b).
Dependent on electrode mark/space ratio, b/c. Values shown assume that the static field components are equal to E_{x}^{0}.

Table 3

Dimensions of Periodic Electrode Structures Used in Experiments

Mask

Electrode Width, b (μm)

Electrode Space, c (μm)

Pitch, Λ Λ (μm)

A

15.6

4.4

40

B

4.4

15.6

40

C

4.4

11.6

32

Table 4

Measured and Computed Coherence Lengths and Overlap Integrals Ratios for Seven Mode Interactions Are Compared Using Data from Device D1^{a}

LP_{11}^{ω} → LP_{21}^{2}^{ω} and LP_{01}^{ω} → LP_{01}^{2}^{ω} are coincident in the phase-matched data shown in Fig. 12. The former, weaker interaction was therefore not resolved. LP_{11}^{ω} → LP_{01}^{2}^{ω} could not be phase matched because the grating pitch used in the experiment was too large. Interaction LP_{01}^{ω} → LP_{31}^{2}^{ω}, could not be measured. a = 4 μm, Δn = 0.0045.

Table 5

Measurement Data on Devices Used in the Experiments

For the computation, values of R have been estimated from measurements of attenuation using oil overlays (see Ref. 32).
Using relation (26). The computed efficiency assumes that all the fundamental wavelength power was in the LP_{01}^{ω} mode, which was not the case.