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Active Fourier-transform spectroscopy combining the direct RF beating of two fiber-based mode-locked lasers with a novel referencing method

Philippe Giaccari, Jean-Daniel Deschênes, Philippe Saucier, Jérôme Genest, and Pierre Tremblay

Open Access Open Access

Abstract

A new approach is described to compensate the variations induced by laser frequency instabilities in the recently demonstrated Fourier transform spectroscopy that is based on the RF beating spectra of two frequency combs generated by mode-locked lasers. The proposed method extracts the mutual fluctuations of the lasers by monitoring the beating signal for two known optical frequencies. From this information, a phase correction and a new time grid are determined that allow the full correction of the measured interferograms. A complete mathematical description of the new active spectroscopy method is provided. An implementation with fiber-based mode-locked lasers is also demonstrated and combined with the correction method a resolution of 0.067 cm-1 (2 GHz) is reported. The ability to use slightly varying and inexpensive frequency comb sources is a significant improvement compared to previous systems that were limited to controlled environment and showed reduced spectral resolution. The fast measurement rate inherent to the RF beating principle and the ease of use brought by the correction method opens the venue to many applications.

©2008 Optical Society of America

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Figures (11)
Fig. 1.
Fig. 1. Electrical field for a dispersion-free and stationary frequency comb in the frequency domain (a and b) and in the time domain (c and d); A(ν) is the envelope distribution, νIII(ν) is a shifted Dirac comb, fr is the repetition rate and f0 is the CEO frequency; a0(τ) is the pulse envelope, a(τ) is the fundamental pulse for p=0, τIII(τ) is the iFT of νIII(ν) (only the amplitude is shown here) and ν0 is the mean frequency of the light source.

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Fig. 2.
Fig. 2. Principle of cFTS; (a) Simplified cFTS example: in the optical band with both the electrical fields of the two frequency comb light sources and in the RF band with the beating components from the interference of neighboring lines from both combs; (b) the RF band is occupied by several replicas of the envelope product S and the internal beatings located exactly at multiple of fr; (c) an adequate change of f0 eliminates the overlap between two replicas if their widths are not larger than fr/2; (d) time domain representation with the relative sliding of the second source pulses (dotted) compared to the center of the pulses of the first source (solid line) that can be viewed as triggers.

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Fig. 3.
Fig. 3. Time replicas q and q+1 from the fundamental RF replica g(τ).

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Fig. 4.
Fig. 4. Experimental setup: solid lines represent optical fiber paths and dashed lines electrical wires; FC1,2 are the two frequency comb light sources, A’s are erbium amplifiers, PS is a polarization controller, FBG1,2 are the two fiber Bragg gratings, X’s are optical couplers, D’s are detectors, C’s are circulators, F’s are electrical low-pass filters (2nd order at 10 MHz) and P’s are band-pass filters around fr.

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Fig. 5.
Fig. 5. (a) Reflection intensity (solid lines) of the FBGs used as metrology for the cFTS system; the dashed lines represent the theoretical spectrum for 5 cm-long uniform gratings; (b) autocorrelation function for both FBGs (solid lines) in amplitude and the phase deviation compared to the phase of the shift term exp(i2πfτ) where f is the Bragg frequency of the grating; the dashed lines represent the theoretical autocorrelation curves.

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Fig. 6.
Fig. 6. (a) Black lines: acquired interferograms for the two reference FBGs; grey: amplitude of the frequency filtered interferograms and white: amplitude of the interferograms after the second frequency filtering; (b) frequency response of FBG1,2 and of the HCN cell in transmission corresponding to the Fourier transform of the acquired interferograms.

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Fig. 7.
Fig. 7. (a) Frequency domain response of FBG1,2 and of the HCN cell in transmission after the phase correction; (b) RF frequency for FBG1, RF frequency difference between the two gratings and linear phase deviation from φ2-φ1.

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Fig. 8.
Fig. 8. (a) Amplitude in logarithmic scale for the corrected interferograms of the four channels of the cFTS method and presented on a distance scale equivalent the optical path length difference (OPD) used in standard FTS; (b) amplitude response of all four channels after the whole correction process of the cFTS measurements.

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Fig. 9.
Fig. 9. (a) Amplitude of the corrected interferograms (average on 10 measurements) for the case with nearly no dispersion (top) and with a moderate amount of dispersion (bottom); (b) direct (top) and normalized (bottom) spectra of the HCN sample in transmission in the case of low dispersion (light grey) and in the case of moderate dispersion (black).

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Fig. 10.
Fig. 10. (a) Normalized spectra for 10 cFTS measurements with a moderate amount of dispersion (thin lines) and average curve (thick line); (b) zoom on the same frequency range around an absorption peak for different window width (and thus resolution) and comparison between the result for the 10 cm width window (solid line) and an external measurement for the same spectral resolution (dashed line).

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Fig. 11.
Fig. 11. Top: transmission spectra measured with the cFTS system and with another calibrated instrument (in mirring 200%-transmission); center: slowly varying error from the normalization process; bottom: difference between the cFTS and the external measurements.

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Equations (34)

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ν ( III ) ( ν ) = III ( ν − f 0 , f r ) = ∑ m = − ∞ ∞ δ ( ( ν m − f 0 ) − m · f r ) .
E τ ( τ ) = iFT { E ν ( ν ) } ( τ ) = ∫ − ∞ ∞ dτ E ν ( ν ) . e i 2 πντ .
τ III ( τ ) = 1 ∣ f r ∣ · III ( τ , 1 f r ) · e i 2 π f 0 τ = e i 2 πf 0 τ ∣ f r ∣ · ∑ m δ ( τ − m f r ) = ∑ m e i 2 π ( m f r + f 0 ) τ .
E τ ( τ ) = 1 ∣ f r ∣ ∑ m a ( τ − m f r ) · e i 2 π f 0 m f r = ∑ m A ( m · f r + f 0 ) · e i 2 π ( m f r + f 0 ) τ .
I D ( τ ) = ∣ E τ ( τ ) ∣ 2 = E τ ( τ ) · E τ * ( τ ) = ∑ m ∑ n A ( m · f r + f 0 ) A * ( n · f r + f 0 ) · e i 2 π ( m − n ) f r τ .
I D ( τ ) = ∑ p e i 2 π p f r τ · [ ∑ m A ( m · f r + f 0 ) A * ( ( m − p ) · f r + f 0 ) ] = ∑ p e i 2 π p f r τ · α p ,
E τ 1 ( τ ) = ∑ m A 1 ( ν 1 ( m ) ) · e i 2 π ν 1 ( m ) τ and E τ 2 ( τ ) = ∑ n A 2 ( ν 2 ( n ) ) · e i 2 π ν 2 ( n ) τ .
I D ( τ ) = ∣ E τ 1 ( τ ) + E τ 2 ( τ ) ∣ 2 = I τ 1 ( τ ) + I τ 2 ( τ ) + 2 · Re { E τ 1 * ( τ ) · E τ 2 ( τ ) } .
I τ ( τ ) = E τ 1 * ( τ ) · E τ 2 ( τ ) = ∑ p e i 2 π p ( f r + Δ f r ) τ · [ ∑ m A 1 * ( ν 1 ( m ) ) · A 2 ( ν 2 ( m − p ) ) · e − i 2 π ( m Δ f r + Δ f 0 ) τ ] .
A 2 ( ν 2 ( m − p ) ) ≅ A 2 ( ν 1 ( m ) ) .
I τ ( τ ) ≅ ∑ p e i 2 π p ( f r + Δ f r ) τ . [ ∑ m U ( v opt ( m ) ) . e − i 2 π v rf ( m ) τ ] = ∑ p e i 2 π p ( f r + Δ f r ) τ . g ( τ ) .
v opt = ( v rf − Δ f 0 ) · f r Δ f r + f 0 .
g ( τ ) = ∑ m S ( v rf ( m ) ) · e − i 2 π v rf ( m ) τ .
G ( v ) = ∑ m S ( v rf ( m ) ) · δ ( v − v rf ( m ) ) = S ( v ) · III ( v − Δ f 0 , Δ f r ) .
g ( τ ) = s ( τ ) * [ e i 2 π Δ f 0 τ ∣ Δ f r ∣ · III ( τ , 1 Δ f r ) ] = ∑ q [ e i 2 π Δ f 0 q Δ f r ∣ Δ f r ∣ ] · s ( τ − q Δ f r ) = ∑ q g q ( τ ) ,
s ( τ ) = ∣ Δ f r f r ∣ · u ( Δ f r f r τ ) · e i 2 π ( Δ f 0 − Δ f r f r f 0 ) τ .
I τ , crop ( τ ) ≅ e iϕ ∣ Δ f r ∣ · Rect ( τ Δ τ crop ) · ∑ p e i 2 πp ( f r + Δ f r ) τ · s ( τ ) ,
I v , crop ( ν ) = e iϕ ∣ Δ f r ∣ · sinc ⁡ ( ν Δ τ crop ) * { ∑ p s ( ν − p · ( f r + Δ f r ) ) } ,
I ν , cFTS ( ν ) = e iϕ ∣ Δ f r ∣ · sinc ⁡ ( ν Δ τ crop ) * S ( ν − p · ( f r + Δ f r ) ) .
I τ , cFTS ( τ ) = u ( Δ f r f r τ ) · e i 2 π ( Δ f 0 − Δ f r f r f 0 + p ( f r + Δ f r ) ) τ · { e i ϕ ∣ f r ∣ · Rect ( τ Δ τ crop ) } .
v opt = f r Δ f r ( v meas − ( Δ f 0 − Δ f r f r · f 0 + p · ( f r + Δ f r ) ) ) = 1 G opt · ( v meas − O opt ) ,
G rf = 1 G opt = Δ f r f r and O rf = O opt G opt = Δ f 0 − Δ f r f r · f 0 + p · ( f r + Δ f r ) .
U fbg ( v opt ) = [ A 1 * ( v opt ) · r * ( v opt ) ] · [ A 2 ( v opt ) · r ( v opt ) ] = U ( v opt ) · R ( v opt ) ,
I τ , fbg ( τ ) = q ( G rf · τ ) · e i 2 π v 0 G rf τ · e i 2 π O rf τ · { e iϕ ∣ f r ∣ · Re ct ( τ Δτ crop ) } .
φ fbg ( τ ) = 2 π ( G rf · ν 0 + O rf ) · τ + ϕ ·
d φ fbg ( τ ) d τ = 2 π ( G rf · ν 0 + O rf ) .
I τ , meas ( τ meas ) = u ( τ opt ( τ meas ) ) · e iθ ( τ meas ) · { e iϕ ∣ f r ∣ · Rect } .
d τ opt d τ meas = f r Δ f r = G rf ( τ meas ) ,
d θ d τ meas = 2 π · ( Δ f 0 − f 0 · Δ f r f r + p · ( f r + Δ f r ) ) = 2 π · O rf ( τ meas ) .
I τ , meas ( τ meas ) = u ( ∫ 0 τ meas dτ · G rf ( τ ) ) · exp ⁡ ( i 2 π · ∫ 0 τ meas dτ · O rf ( τ ) ) · { e iϕ ∣ f r ∣ · Rect } .
dφ 1,2 ( τ meas ) dτ meas = 2 π ( G rf ( τ meas ) · ν 1,2 + O rf ( τ meas ) ) .
I τ , meas ( τ meas ) = u ( φ 2 ( τ meas ) − φ 1 ( τ meas ) 2 π ( ν 2 − ν 1 ) ) · e i · ν 2 · φ 1 ( τ meas ) − ν 1 · φ 2 ( τ meas ) ν 2 − ν 1 · { e iϕ ∣ f r ∣ · Rect } ·
τ opt ( τ meas ) = φ 2 ( τ meas ) − φ 1 ( τ meas ) 2 π ( ν 2 − ν 1 ) and θ ( τ meas ) = ν 2 · φ 1 ( τ meas ) − ν 1 · φ 2 ( τ meas ) ν 2 − ν 1 .
I num ( τ opt ) = I τ , meas ( τ meas ) · e − i φ 1 ( τ meas ) = u ∣ ( τ opt ( τ meas ) ) · e − i 2 π · ν 1 · τ opt ( τ meas ) · { e iϕ ∣ f r ∣ · Rect } ·
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