Abstract

A simple noniterative electric field retrieval method using a sinusoidally driven optical phase modulator is demonstrated. We derive an analytic relation between the two-frequency correlation function of the electric field of the optical signal under test and the derivatives of the optical spectrum of the phase-modulated signal with respect to the amplitude of the phase modulation. This relation is used to algebraically reconstruct the electric field of the signal under test. We validate the technique by demonstrating accurate and sensitive characterization of picosecond pulses used in telecommunication.

© 2007 Optical Society of America

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References

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2005 (2)

2003 (1)

2002 (4)

1998 (1)

IEEE Photon. Technol. Lett. (1)

L. P. Barry, S. Del Burgo, B. Thomsen, R. T. Watts, D. A. Reid, and J. Harvey, IEEE Photon. Technol. Lett. 14, 971 (2002).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Lett. (5)

Other (3)

R. P. Feynmann and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

A. Yariv, Optical Electronics (Holt, Reinhart and Winston, 1985).

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Figures (3)

Fig. 1
Fig. 1

Left, schematic of the measurement setup. Right, timing between the pulses and cosine modulations (top) and sine modulations (bottom).

Fig. 2
Fig. 2

Top, pulses from the lithium niobate modulator: the solid (dashed) curve is the calculated power (phase), and the solid (hollow) squares are measured power (phase). Bottom, pulses from the semiconductor mode-locked laser: the solid (dashed) curve is the measured power (phase) using the spectrogram method, and the solid (hollow) squares are the measured power (phase) using the current technique.

Fig. 3
Fig. 3

Top, pulses from the lithium niobate modulator measured at 50 μ W (curves), 1 μ W (solid symbols), and 100 nW (hollow symbols). Bottom, MLL laser pulses after a 25 ps delay-line interferometer measured at 250 μ W (curves), 10 μ W (solid symbols), and 2.5 μ W (hollow symbols).

Equations (8)

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Δ ( ω ) d I ( c ) ( ψ , ω ) d ψ ψ = 0 + i d I ( s ) ( ψ , ω ) d ψ ψ = 0 = i E ̃ ( ω + Ω ) E ̃ * ( ω ) sinc ( β ( 2 ) L [ ( ω + Ω ) 2 ω 2 ] 4 ) i E ̃ ( ω ) E ̃ * ( ω Ω ) sinc ( β ( 2 ) L [ ω 2 ( ω Ω ) 2 ] 4 ) ,
E ̃ ( ω ) = E ( ω ) exp [ i β ( 2 ) ω 2 L 4 ] ,
Δ ( ω ) = i [ E ( ω + Ω ) E * ( ω ) E ( ω ) E * ( ω Ω ) ] .
E ( t , z ) z i β ( 2 ) 2 2 E ( t , z ) t 2 = i ψ L cos ( Ω t ) E ( t , z ) .
E ( c ) ( ψ , ω ) = exp [ i ψ cos ( Ω t ) ] E ( t ) e i ω t d t .
E ( c ) ( t , z ) = G ( t , z ; t 0 , z 0 ) E ( t 0 , z 0 ) d t 0 + i ψ L G ( t , z ; t 1 , z 1 ) cos ( Ω t 1 ) G ( t 1 , z 1 ; t 0 , z 0 ) E ( t 0 , z 0 ) d t 1 d z 1 d t 0 .
G ( t , z ; t 0 , z 0 ) = 1 i β ( 2 ) ( z z 0 ) exp [ i ( t t 0 ) 2 2 β ( 2 ) ( z z 0 ) ] θ ( t t 0 ) ,
d I ( c , s ) ( ψ , ω ) d ψ ψ = 0 I ( c , s ) ( ψ , ω ) I ( c , s ) ( ψ , ω ) 2 ψ .

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