Theoretical analysis and modeling of a photonic integrated circuit for frequency 8-tupled and 24-tupled millimeter wave signal generation: erratum

Mehedi Hasan; Rabiaa Guemri; Ramón Maldonado-Basilio; Frédéric Lucarz; Jean-Louis de Bougrenet de la Tocnaye; Trevor Hall

doi:10.1364/OL.40.005710

Abstract

A novel photonic circuit design for implementing frequency 8-tupling and 24-tupling was presented [Opt. Lett. 39, 6950 (2014) [CrossRef] ], and although its key message remains unaltered, there were typographical errors in the equations that are corrected in this erratum.

In our published article [1], the following typographical errors were noticed. Hence, this erratum is provided for clarification:

1. Error:

D_{2} = \frac{- i A_{0} ζ}{4} (- \cos φ_{A} + \cos φ_{B} + \cos φ_{C} + \cos φ_{D}) .

Correction:

D_{2} = \frac{- i A_{0} ζ}{4} (- \cos φ_{A} + \cos φ_{B} - \cos φ_{C} + \cos φ_{D}) .

2. Error:

\cos (m \cos (ω_{m} t) + π / 4) = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + π / 4) .

Correction:

\cos {m \cos (ω_{m} t + π / 4)} = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + n π / 2) .

3. Error:

\cos (m \cos (ω_{m} t) + π / 2) = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + π / 2) .

Correction:

\cos {m \cos (ω_{m} t + π / 2)} = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + n π) .

4. Error:

\cos (m \cos (ω_{m} t) + 3 π / 4) = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + 3 π / 4) .

Correction:

\cos {m \cos (ω_{m} t + 3 π / 4)} = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + 3 n π / 2) .

5. Error:

D_{2} = \frac{i ζ A_{0}}{4} [J_{4} (m) \cos (4 ω_{m} t) + J_{4} (m) \cos (- 4 ω_{m} t) + J_{12} (m) \cos (12 ω_{m} t) + J_{12} (m) \cos (- 12 ω_{m} t) + \dots] .

Correction:

D_{2} = i ζ A_{0} [J_{4} (m) \cos (4 ω_{m} t) + J_{4} (m) \cos (- 4 ω_{m} t) + J_{12} (m) \cos (12 ω_{m} t) + J_{12} (m) \cos (- 12 ω_{m} t) + \dots] .

REFERENCE

1. M. Hasan, R. Maldonado-Basilio, F. Lucarz, J.-L. de Bougrenet de la Tocnaye, and T. J. Hall, Opt. Lett. 39, 6950 (2014). [CrossRef]

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D_{2} = \frac{- i A_{0} ζ}{4} (- \cos φ_{A} + \cos φ_{B} + \cos φ_{C} + \cos φ_{D}) .

D_{2} = \frac{- i A_{0} ζ}{4} (- \cos φ_{A} + \cos φ_{B} - \cos φ_{C} + \cos φ_{D}) .

\cos (m \cos (ω_{m} t) + π / 4) = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + π / 4) .

\cos {m \cos (ω_{m} t + π / 4)} = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + n π / 2) .

\cos (m \cos (ω_{m} t) + π / 2) = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + π / 2) .

\cos {m \cos (ω_{m} t + π / 2)} = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + n π) .

\cos (m \cos (ω_{m} t) + 3 π / 4) = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + 3 π / 4) .

\cos {m \cos (ω_{m} t + 3 π / 4)} = J_{0} (m) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (m) \cos (2 n ω_{m} t + 3 n π / 2) .

D_{2} = \frac{i ζ A_{0}}{4} [J_{4} (m) \cos (4 ω_{m} t) + J_{4} (m) \cos (- 4 ω_{m} t) + J_{12} (m) \cos (12 ω_{m} t) + J_{12} (m) \cos (- 12 ω_{m} t) + \dots] .

D_{2} = i ζ A_{0} [J_{4} (m) \cos (4 ω_{m} t) + J_{4} (m) \cos (- 4 ω_{m} t) + J_{12} (m) \cos (12 ω_{m} t) + J_{12} (m) \cos (- 12 ω_{m} t) + \dots] .

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