Optics Communications 286 (2013) 46–50

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Optics Communications

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Influence analysis of the refractive index of air on the cross-correlationpatterns between femtosecond pulses

Yan Xu a,b,c, Weihu Zhou d, Deming Liu a,c,n

a National Engineering Laboratory for Next Generation Internet Access System, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, Hubei, PR Chinab College of Applied Science, Jiangxi University of Science and Technology, Keja Ave. 156, Ganzhou 341000, Jiangxi, PR Chinac School of Optical and Electronic Information, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, Hubei, PR Chinad Academy of Opto-electronics, Chinese Academy of Sciences, Dengzhuang South Road 9, Beijing 100094, PR China

a r t i c l e i n f o

Article history:

Received 15 February 2012

Received in revised form

1 September 2012

Accepted 4 September 2012Available online 17 September 2012

Keywords:

Laser optics

Femtosecond pulse

Cross-correlation

Refractive index of air

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.09.006

esponding author at: National Engineering La

Access System, Huazhong University of Sci

37, Wuhan 430074, Hubei, PR China. Tel.: þ

6 27 8754 3055.

ail address: dmliu@mail.hust.edu.cn (D. Liu).

a b s t r a c t

The environmental parameters of refractive index of air and their effect on the cross-correlation model

between femtosecond pulses propagation in air were discussed. Using the dispersion pulse propagation

theory, the equation of cross-correlation between femtosecond pulses was obtained and the relation

between the cross-correlation with the environmental parameters was established. Results show that

the change of the atmospheric conditions gives rise to the change of the group refractive index, and

then contributes to the shift of the correlations patterns without any extra linear broadening or chirp.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Recent advances in the field of femtosecond pulses have givenbirth to the development of credible sources of carrier-envelope-phase stabilized femtosecond pulses [1,2]. The unique propertiesof femtosecond pulse make the frequency comb to be applied as aversatile tool, not only in frequency and time metrology [3,4], butalso in laser noise description [5], and high-precision spectro-scopy [6]. Moreover, the phase relationship between femtosecondpulses emitted by the optical frequency comb has led to a newpath for large range high accuracy absolute distance measure-ment [7–11]. An arbitrary plane wave pulse would propagateunchanged in shape at the phase velocity of the wave field in thenon-dispersive medium. Thus, the analysis of the resultingcorrelation patterns is uncomplicated. However, the analysis ofcorrelation patterns becomes complex for femtosecond pulsepropagation in a dispersive medium such as air as a result ofthe difference between the group velocity and phase velocity[7–10]. Therefore, the spatial and temporal coherence betweenthe femtosecond pulses has a highly important research value. Tothe best of our knowledge, few studies have been carried out

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boratory for Next Generation

ence and Technology, Luoyu

86 27 8755 6188;

regarding the influence analysis of refractive index of air ontemporal coherence function between the femtosecond pulsespropagation in dispersive media, though such influence analysisof refractive index of air is likely to be useful for absolute distancemeasurement applications.

In this paper, according to the dispersion pulse propagationtheory, the numerical model of the cross-correlation between thefemtosecond pulses propagating in air is obtained based onGaussian pulse model. The refractive index of air of widespectrum of the femtosecond laser is calculated by the Ciddorequation [12] and the Edlen equation [13], and the influence ofthe refractive index of air on the cross-correlation patterns isnumerically simulated. The results have a certain theoreticalsignificance for large-scale high-precision distance measurementusing the cross-correlation method.

2. Theoretical analysis

2.1. Cross-correlation

Fig. 1 shows the schematic representation of this model, whichis a Michelson type interferometry with optical interferencebetween individual pulses [12]. The pulse train from the femto-second pulse laser is divided into two beams which are recom-bined after having passed through various optical delays. Theintensity of the recombined beam is recorded as a function of thevariable delay, i.e. piezo-element positions[7]. In dispersive media

Fig. 1. The schematic representation.

Y. Xu et al. / Optics Communications 286 (2013) 46–50 47

like air, a maximum coherence is obtained when the path lengthdifference between both arms is equal to an integer q multiple ofthe effective laser cavity length or the pulse-to-pulse distancelpp ¼ c=ðnge � f repÞ . Here c is the speed of light in vacuum, f rep isthe repetition frequency of the laser, and nge is the effective grouprefractive index of the medium.

The cross-correlation is readily measured by placing a slowdetector at the output of the Michelson interferometer. Assumingthe electric field from the reference arm is eðx1,tÞ, the electric fieldfrom the measuring arm is eðx2,tÞ. If the time response of thedetector is much larger than the time duration of the signal eðtÞ,the intensity detected by a slow detector is calculated as theinterference of two electric fields which have traveled thedistances x1 and x2, respectively [7].

The frequency spectrum emitted by a mode-locked lasercontains a comb of regularly spaced frequencies,

om ¼morepþo0 ð1Þ

where o0 is the common offset frequency, m is a positive integer,and orep is the repetition frequency f rep expressed in angularnotation that is explicitly given as orep ¼ 2pf rep. The offsetfrequency o0 is caused by the difference between the groupvelocity and the phase velocity inside the laser cavity.

The pulse field emitted by the femtosecond laser, propagatingin the direction of positive x, at x¼0 can be written as

eð0,tÞ ¼X1

m ¼ 0

Amcos½ðmorepþo0Þtþfm� ð2Þ

where Am is a real amplitude, and fm is the phase.The cross-correlation of the electric field between the refer-

ence arm and measuring arm is given by

GðXÞ ¼ 2X1

m ¼ 0

9am92

1þcos ðmorepþo0Þnðmorepþo0ÞX

c

� �� �ð3Þ

Where amj j2 is the power spectral density of both pulses,

nðmorepþo0Þ is the refractive index nðoÞ for all frequencies ofthe comb, and X ¼ x2�x1. It shows that the first-order cross-correlation function depends on the knowledge of the sourcespectrum, rather than the pulsed electric field.

2.2. Group refractive index

After traveling through different paths in the Michelson inter-ferometer, the broadband pulses emitted by the femtosecondlaser superpose and form interference fringes. For absolute dis-tance measurement, the brightest fringe position is used toanalyze [8]. Accomplishing absolute distance measurement inair requires the knowledge of the dispersion relation. The phase

velocity is

upðoÞ ¼o=kðoÞ ¼ c=nðoÞ ð4Þ

where kðoÞ is the wave number, and nðoÞ is the refractive indexgiven by the Ciddor’s equation or the Edlen’s equation. After acertain propagation distance, the velocity of the pulse envelope,the group velocity is equal to ug ¼ do=dkðoÞ ¼ c=nge. Here nge isthe group refractive index defined at the pulse carrier frequency(oc), conventionally accepted as the frequency with the max-imum intensity in the spectrum, and given by

nge ¼ nðocÞþocdðnðoÞÞ

do

� �oc

ð5Þ

The position of the maximum of the cross-correlation patternbetween the femtosecond pulses propagation in air would changeas the refractive index of air changes. For absolute distancemeasurement based on cross-correlation principle, the maximumfringe is reached at a given distance L. L¼ qUlpp ¼ qUc=f rep invacuum, while Ln ¼ qUlpp ¼ qUc=ðngeUf repÞ in a dispersive medium.

3. Results and discussions

The refractive index of air on wide spectrum of the femtose-cond laser can be calculated according to the Ciddor’s equation orthe Edlen’s equation. Fig. 2(a) and (b) shows the Gaussianspectrum distribution with the repetition frequency of 1 GHz,spectral width of 10 THz, and central wavelength of 800 nm and1560 nm, respectively. The refractive index of air on the entirespectral range of 800 nm and 1560 nm Gaussian spectrum isillustrated in Fig. 2(c) and (d), respectively. The refractive index ofair is calculated by the Ciddor’s equation or the Edlen’s equation,and the atmospheric parameters are 101.325 kPa, 20 1C, 50%humidity, and 450 ppm of CO2. Environmental fluctuation wouldgive rise to the change of the refractive index of air. In order tosimplify the uncertainty estimation, we have calculated thesensitivity coefficients of the mean group refractive index of airon a variation of atmospheric conditions for central wavelength of800 nm and 1560 nm, respectively, nearby 101.325 kPa, 20 1C,50% humidity, 450 ppm of CO2, as indicated in Table 1.

A small variation of environmental parameters will make animportant shift for the cross-correlation pattern [8] as indicatedin Fig. 3. Fig. 3(a) and (b) depicts the cross-correlation pattern ofthe Gaussian spectrum distribution with central wavelength of800 nm and 1560 nm, respectively, after propagating for a dis-tance difference of 30 m at 101.325 kPa, 20 1C, 50% humidity,450 ppm of CO2. The peak of the cross-correlation pattern iscentered on zero. From Fig. 3(c) and (d), we can see that for atemperature variation of 0.5 1C, the position of the peak is shifted

Fig. 2. The spectra and dispersion of the femtosecond pulse. (a) Spectra, lc¼800 nm, (b) Spectra, lc¼1560 nm, (c) Dispersion, lc¼800 nm and (d) Dispersion, lc¼1560 nm.

Table 1Sensitivity coefficients of mean group refractive index on environment parameters.

nge Sensitivity coefficient (ppb)

800 nm 1560 nm

Ciddor’s Edlen’s Ciddor’s Edlen’s

Pressure 2.7167 2.7168 2.6638 2.6639 per Pa

Temperature �966.13 �966.09 �948.86 �948.73 per 1C

Humidity �8.27 �8.29 �8.63 �8.61 per %

CO2 content 0.1452 – 0.1424 – per ppm

Fig. 3. The effect of temperature and pressure on the cross-correlation model with the path length difference of 30 m. (a) lc¼800 nm, T¼20 1C, P¼101.325 kPa, (b)

lc¼1560 nm, T¼20 1C, P¼101.325 kPa, (c) lc¼800 nm, DT¼0.5 1C, (d) lc¼1560 nm, DT¼0.5 1C, (e) lc¼800 nm, DP¼100 Pa and (f) lc¼1560 nm, DP¼100 Pa.

Y. Xu et al. / Optics Communications 286 (2013) 46–5048

Table 2The shift amount of cross-correlation patterns on environment parameters.

The shift amount (mm)

800 nm 1560 nm

Ciddor’s Edlen’s Ciddor’s Edlen’s

Pressure �0.0656 �0.0642 �0.0700 �0.0700 per Pa

Temperature 24.65 24.65 23.94 23.94 per 1C

Humidity 0.260 0.255 0.260 0.260 per %

CO2 content �0.0045 – �0.0040 – per ppm

Fig. 4. The position shift amount of the maximum of the correlation pattern caused by the environmental parameters for different propagation distances in air. (a)

lc¼800 nm and (b) lc¼1560 nm

Y. Xu et al. / Optics Communications 286 (2013) 46–50 49

from 14.2 and 15.7 mm to the forth respectively. Same for thepressure, a variation of 100 Pa will be able to shift the peak8.0 and 7.9 mm to the back respectively, as can be seen inFig. 3(e) and (f). The fluctuation of environmental parameters,such as humidity and CO2 content has the similar influence oncross-correlation model. The simulations of Fig. 3 shows that thecorrelations patterns only shift without any extra linear broad-ening or chirp. Table 2 presents the position of the maximum ofthe cross-correlation pattern between the femtosecond pulsespropagation at 30 m in air for subtle changes in the four relevantparameters of atmospheric pressure, temperature, humidity, andCO2 content.

Fig. 4 shows the shift amount of the maximum of the correla-tion pattern between the femtosecond pulses with the centerwavelength of 800 nm and 1560 nm [14], respectively, afterpropagating for a distance difference from 0 m to 120 m for a0.2 1C temperature variation (from 20 1C to 20.2 1C), a 50 Papressure increasing (from 101.325 kPa to 101.375 kPa), a 10%variation in humidity (from 50% humidity to 60% humidity), and a100 ppm CO2 content increasing (from 450 ppm to 550 ppm).From Fig. 4, we find that there is a linear relationship between thechange of environmental parameters, such as pressure, tempera-ture, humidity and CO2 content and the shift amount of themaximum of the correlation pattern with the propagation dis-tance increasing. The accurate measurement of environmentalparameters is essential to improving the accuracy of absolutedistance measurement.

4. Conclusions

According to the dispersion pulse propagation theory,the cross-correlation pattern distribution is given by the

cross-correlation equation between femtosecond pulses. Thesensitivity coefficients of mean group refractive index on envir-onmental parameters are presented for two central wavelengths.The variation of the cross-correlation model between femtose-cond pulses propagating in air with the refractive index of air issummarized by numerical simulation, which is undoubtedlysignificant for conducting long-range high-precision absolutedistance measurements based on cross-correlation of femtose-cond pulses.

Acknowledgments

This work was supported by the Chinese Natural ScienceFoundation under Grant agreement 60937002, and the NaturalScience Foundation of Beijing City of China under Grant agree-ment 4112065.

References

[1] D. Wei, S. Takahashi, K. Takamasu, H. Matsumoto, Optics Express 17 (2009)7011.

[2] T. Chattopadhyay, A.K. Saw, Optics Communications 284 (2011) 4611.[3] T. Shioda, T. Mori, T. Sugimoto, Y. Tanaka, T. Kurokawa, Optics Communica-

tions 282 (2009) 2909.[4] M. Musha, Y. Tamura, K. Nakagawa, K. Ueda, Optics Communications 272

(2007) 211.[5] O. Prochnow, R. Paschotta, E. Benkler, U. Morgner, J. Neumann, D. Wandt,

D. Kracht, Optics Express 17 (2009) 15525.[6] M.J. Thorpe, J. Ye, Applied Physics B 91 (2008) 397.[7] M.G. Zeitouny, M. Cui, N. Bhattacharya, H.P. Urbach, S.A. van den Berg,

A.J.E.M. Janssen, Physical Review A 82 (2010) 023808.[8] P. Balling, P. Kren, P. Masika, S.A. van den Berg, Optics Express 17 (2009)

9300.

Y. Xu et al. / Optics Communications 286 (2013) 46–5050

[9] M. Cui, M.G. Zeitouny, N. Bhattacharya, S.A. van den Berg, H.P. Urbach,J.J.M. Braat, Optics Letters 34 (2009) 1982.

[10] M. Zeitouny, N. Bhattacharya, H.P. Urbach, S.A. van den Berge, Advancedoptical and mechanical technologies in telescopes and instrumentation. in:Proceedings of the SPIE, 7018, Amsterdam, (2008) 70183R-1.

[11] J. Ye, Optics Letters 29 (2004) 1153.[12] P.E. Ciddor, Applied Optics 35 (1996) 1566.[13] K.P. Birch, M.J. Downs, Metrologia 30 (1993) 155.[14] Y. Xu, W.H. Zhou, D.M. Liu, Chinese Journal of Lasers 38 (2011) 1108002 (in

Chinese).