Precise recursive formula for calculating spot size in optical waveguides and accurate evaluation of splice loss

Yasuo Kokubun; Shuichi Tamura

doi:10.1364/AO.34.006862

Applied Optics
Vol. 34,
Issue 30,
pp. 6862-6873
(1995)
•https://doi.org/10.1364/AO.34.006862

Precise recursive formula for calculating spot size in optical waveguides and accurate evaluation of splice loss

Yasuo Kokubun and Shuichi Tamura

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Yasuo Kokubun and Shuichi Tamura, "Precise recursive formula for calculating spot size in optical waveguides and accurate evaluation of splice loss," Appl. Opt. 34, 6862-6873 (1995)

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Abstract

The spot size of a single-mode waveguide was defined from the viewpoint of a least-squares fit of the field profile to a Gaussian profile. The field profile was expanded in terms of Hermite–Gaussian functions, and a new precise recursive formula for calculating the spot size was derived. It was shown that our formula is equivalent to the best fitting of the offset coupling loss to that of a Gaussian profile and keeps its form against the Fourier transform that corresponds to the diffraction in the same manner as the Gaussian profile. The accuracy of conventional formulas and our new spot-size formula was compared with the exact value defined from the viewpoint of a least-squares fit to a Gaussian profile, and it was shown that our recursive formula is the most accurate of the approximate formulas. Next we proposed a new formula for calculating the splice loss between two waveguides and showed that our formula is more accurate than the conventional one, which involves only the spot size.

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Formula	NFP	FFP
Villuendas et al.8	$w_{V} = {\frac{\int_{- \infty}^{\infty} {[f (x)]}^{2} d x}{\int_{- \infty}^{\infty} {[\frac{d f (x)}{d x}]}^{2} d x}}^{1 / 2}$	$W_{V} = {[\frac{4 \int_{- \infty}^{\infty} Ω^{2} ∣ F (Ω) ∣^{2} d Ω}{\int_{- \infty}^{\infty} ∣ F (Ω) ∣^{2} d Ω}]}^{1 / 2}$
Second moment	$w_{q} = {\frac{4 \int_{- \infty}^{\infty} x^{2} {[f (x)]}^{2} d x}{\int_{- \infty}^{\infty} {[f (x)]}^{2} d x}}^{1 / 2}$	$W_{q} = {[\frac{\int_{- \infty}^{\infty} ∣ F (Ω) ∣^{2} d Ω}{\int_{- \infty}^{\infty} {∣ \frac{d F (Ω)}{d Ω} ∣}^{2} d Ω}]}^{1 / 2}$
This paper	$w_{K} = {\frac{4 \int_{- \infty}^{\infty} x^{2} exp [- {(\frac{x}{w})}^{2}] f (x) d x}{\int_{- \infty}^{\infty} exp [- {(\frac{x}{w})}^{2}] f (x) d x}}^{1 / 2}$	$W_{K} = {\frac{4 \int_{- \infty}^{\infty} Ω^{2} exp [- {(\frac{Ω}{W})}^{2}] F (Ω) d Ω}{\int_{- \infty}^{\infty} exp [- {(\frac{Ω}{W})}^{2}] F (Ω) d Ω}}^{1 / 2}$

V	C ₀	C ₂	C ₄
0.7	0.9907	−0.6490 × 10⁻⁵	0.1260
1.2	0.9965	−0.6625 × 10⁻⁶	0.08133

Formula	NFP	FFP
Villuendas et al.8	$w_{V} = {\frac{\int_{- \infty}^{\infty} {[f (x)]}^{2} d x}{\int_{- \infty}^{\infty} {[\frac{d f (x)}{d x}]}^{2} d x}}^{1 / 2}$	$W_{V} = {[\frac{4 \int_{- \infty}^{\infty} Ω^{2} ∣ F (Ω) ∣^{2} d Ω}{\int_{- \infty}^{\infty} ∣ F (Ω) ∣^{2} d Ω}]}^{1 / 2}$
Second moment	$w_{q} = {\frac{4 \int_{- \infty}^{\infty} x^{2} {[f (x)]}^{2} d x}{\int_{- \infty}^{\infty} {[f (x)]}^{2} d x}}^{1 / 2}$	$W_{q} = {[\frac{\int_{- \infty}^{\infty} ∣ F (Ω) ∣^{2} d Ω}{\int_{- \infty}^{\infty} {∣ \frac{d F (Ω)}{d Ω} ∣}^{2} d Ω}]}^{1 / 2}$
This paper	$w_{K} = {\frac{4 \int_{- \infty}^{\infty} x^{2} exp [- {(\frac{x}{w})}^{2}] f (x) d x}{\int_{- \infty}^{\infty} exp [- {(\frac{x}{w})}^{2}] f (x) d x}}^{1 / 2}$	$W_{K} = {\frac{4 \int_{- \infty}^{\infty} Ω^{2} exp [- {(\frac{Ω}{W})}^{2}] F (Ω) d Ω}{\int_{- \infty}^{\infty} exp [- {(\frac{Ω}{W})}^{2}] F (Ω) d Ω}}^{1 / 2}$

V	C ₀	C ₂	C ₄
0.7	0.9907	−0.6490 × 10⁻⁵	0.1260
1.2	0.9965	−0.6625 × 10⁻⁶	0.08133

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Equations (57)

Applied Optics