OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 7040–7045
« Show journal navigation

Comparison of length measurements provided by a femtosecond optical frequency comb

Dong Wei and Masato Aketagawa  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 7040-7045 (2014)
http://dx.doi.org/10.1364/OE.22.007040


View Full Text Article

Acrobat PDF (770 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

This paper presents a comparison of length measurements between the wavelength and the adjacent pulse repetition interval length (APRIL) provided by a femtosecond optical frequency comb. A theoretical estimation of the frequency stability for stabilizing the wavelength and APRIL, the frequency parameters that affect the stability of the APRIL in air, and the ambiguity in the length measurement by the APRIL are investigated. We find that the APRIL can be used as a low-cost measurement for the absolute length over a range of hundreds of meters in laboratory conditions.

© 2014 Optical Society of America

1. Introduction

The meter is defined by the distance over which light propagates in vacuum. The wavelength of an iodine-stabilized He–Ne laser has been used as an achievement means of the meter. The frequency of a He–Ne laser is stable. Because the wavelength is inversely proportional to the frequency, the wavelength is also stable; therefore, the wavelength can be used to measure the length.

The national standard of length in Japan changed from the iodine-stabilized He–Ne laser to a femtosecond optical frequency comb (FOFC) in 2009. The characteristics of an FOFC and He–Ne laser are different. A He–Ne laser is a single-frequency source; however, an FOFC is a pulsed laser source with a coherent combination of several hundreds of thousands of ultra-stable wavelengths. Because of this difference, it is possible to measure length with an FOFC in two ways. One is by using the wavelength, and another is by using the adjacent pulse repetition interval length (APRIL), which is a coherent representation of individual wavelengths.

Several methods for generating a single frequency from an FOFC or multiple FOFCs have been proposed. For example, an individual wavelength can be extracted from an FOFC to measure length [1

1. N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, and R. Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006). [CrossRef] [PubMed]

, 2

2. S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 095302 (2009). [CrossRef]

]. In addition, two FOFCs can be used together to generate a beat signal [3

3. T. Yasui, Y. Kabetani, Y. Ohgi, S. Yokoyama, and T. Araki, “Absolute distance measurement of optically rough objects using asynchronous-optical-sampling terahertz impulse ranging,” Appl. Opt. 49(28), 5262–5270 (2010). [CrossRef] [PubMed]

, 4

4. S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, and T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express 17(20), 17324–17337 (2009). [CrossRef] [PubMed]

]. Further, the possibility of measuring length with an APRIL has also been explored. Yamaoka et al. [5

5. Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41(21), 4318–4324 (2002). [CrossRef] [PubMed]

] first tested the possibility of measuring length by an APRIL. Later, Ye [6

6. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29(10), 1153–1155 (2004). [CrossRef] [PubMed]

] independently proposed using the integer time of an APRIL for the absolute length measurement. In addition, various experiments [7

7. M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Europ. Opt. Soc. Rap. Public. 3, 08003 (2008).

12

12. X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013). [CrossRef]

] were performed for distance detection. We examined the stability of an APRIL in air [13

13. D. Wei, K. Takamasu, and H. Matsumoto, “A study of the possibility of using an adjacent pulse repetition interval length as a scale using a Helium–Neon interferometer,” Precis. Eng. 37(3), 694–698 (2013). [CrossRef]

] and the group refractive-index characteristic of an APRIL in air [14

14. D. Wei and M. Aketagawa, “Characteristics of an adjacent pulse repetition interval length as a scale for length,” Opt. Eng. 53(5), 051502 (2014). [CrossRef]

].

In this work, we compare the wavelength and APRIL of an FOFC to test their characteristics for length measurement. This paper is organized as follows. First, a theoretical estimation of the frequency parameter required to stabilize the wavelength and APRIL is presented in Section 2. Next, the stability of the APRIL in air is described in Section 3 when only the repetition frequency is stabilized. In Section 4, the ambiguity in the length measurement that uses an APRIL is examined. Finally, the main conclusions are summarized in Section 5.

2. Frequency stability for stabilizing the wavelength and APRIL

First, we examine the frequency stability required to obtain a stabilized wavelength and an APRIL. For convenience, we summarize the features of an FOFC, the details of which can be found elsewhere [15

15. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).

]. In the frequency domain, a mode-locked laser generates equidistant frequency comb lines with the pulse repetition frequency frep, and the entire equidistant frequency comb is shifted by the offset frequency fCEO from the zero frequency. In the time domain, when the electric-field packet repeats at the pulse repetition period TR=1/frep, the carrier phase changes according to Δφce=2πfCEO/frep to the carrier-envelope phase.

In vacuum, the relation λvac=cvac/fholds between the wavelength λvac and the frequency f. Here, cvac is the speed of light in vacuum. The uncertainty in the wavelength is given by u(λvac)/λvac=u(f)/f, where u(x) is the uncertainty of the variable x.

One of the frequencies of an FOFC fP is expressed as
fP=(Q+P)×frep=fCEO+P×frep,
(1)
where P is the number of comb lines on the order of 106 and 0Q<1. First, we consider the stability of the pulse repetition frequency frep. We note that fCEOP×frep; thus, we have fPP×frep. Then, the stability estimate for the pulse repetition frequency frep is
u(frep)/frep=u(fP)/fP.
(2)
Next, we consider the stability of the offset frequency fCEO by considering the uncertainty of Eq. (1):
u(fP)/fP=[u(fCEO)+P×u(frep)]/(fCEO+P×frep).
(3)
After dividing the denominator and numerator on the right-hand side of Eq. (3) by fCEO, we obtain u(fP)/fP=[u(fCEO)/fCEO+P×u(frep)/fCEO]/(1+P×frep/fCEO). Here, we note that frep>fCEO(fCEO is the fractional part of the frequency f), and recall that P is on the order of 106. Further, we obtain 1P×frep/fCEO.Then, we have
u(fP)/fP[u(fCEO)/fCEO+P×u(frep)/fCEO]/(P×frep/fCEO).
(4)
Because the accuracy of u(fP)is limited by whichever is the larger value between u(fCEO) or P×u(frep), we assume that u(fCEO)P×u(frep). In the case where u(fCEO)P×u(frep), an improvement in the accuracy of u(fP)cannot be realized by improving the accuracy of u(fCEO) and vice versa. We obtain
u(fCEO)/fCEO+P×u(frep)/fCEO2×u(fCEO)/fCEO.
(5)
By substituting Eq. (5) into Eq. (4), and moving u(fCEO)/fCEO to the left-hand side, we obtain

u(fCEO)/fCEO12(u(fP)/fP)×(P×frep/fCEO).
(6)

In vacuum, the following relation holds between an APRIL δvac and the pulse repetition frequency frep:
δvac=cvac/frep.
(7)
The uncertainty of the APRIL is given by
u(δvac)/δvac=u(frep)/frep.
(8)
To stabilize a wavelength, both fCEO and frep need to stabilized in Eq. (6) and Eq. (2), respectively. In the case of an APRIL, only frep needs to be stabilized in Eq. (8), leading to a cost reduction, as fCEO does not need to be stabilized.

Without the stability requirement for fCEO, the frequencies (wavelengths) of an FOFC change in time. In the following, we examine how this change influences an APRIL in air.

3. Error in the APRIL in air caused by a change in the central frequency of an FOFC

The wavelength in air λair is a function of λvac and the phase refractive index of air np(λvac) according to λair=λvac/np(λvac). The phase refractive index of air can be derived from the temperature, atmospheric pressure, etc., by using empirical equations [16

16. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996). [CrossRef] [PubMed]

19

19. J. A. Stone and J. H. Zimmerman, “Index of refraction of air,” Available in: http://emtoolbox.nist.gov/Wavelength/Edlen.asp.

].

The APRIL in air δair is a function of δvac and the group refractive index of air ng(λcen_vac) according to
δair=δvac/ng(λcen_vac),
(9)
where λcen_vac is the central frequency of the FOFC. The group refractive index of air is estimated using [20

20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).

]
ng(λcen_vac)=np(λcen_vac)λcen_vac×(dnp(λvac)/dλvac)λcen_vac,
(10)
where (dnp(λvac)/dλvac)λcen_vac is the derivative of the function y=np(λvac) at λvac=λcen_vac.

In the above, we showed that the stability of fCEO is not required to stabilize an APRIL. Without the stability of fCEO, we know from Eq. (1) that the central wavelength of the FOFC will change. This change introduces an error into the group refractive index of air according to Eq. (10), and subsequently, δair in Eq. (9) will change. In the following, we consider this error caused by the change in fCEO.

We assume that frep = 100 MHz; then, we obtain fCEO[0,100) MHz. When λcen_vac_1 = 1560 nm, f1=cvac/λcen_vac_1 = 192,174.6526 GHz. In order to simplify the calculations and obtain the maximum error caused by the change in fCEO, we assume fCEO100 MHz and obtain f2=f1+fCEO = 192,274.6525 GHz. λcen_vac_2=c/f2 = 1559.18866 nm. Using Eq. (10), we obtain ng(λcen_vac_1) = 1.00026689 ± 0.00000003 and ng(λcen_vac_2) = 1.00026689 ± 0.00000003. The uncertainty of ±30×109 is the empirical value.

This result shows that the change of fCEO100 MHz introduces no error into the value of the group refractive index of air (and the value of the APRIL in air). Generally, fCEO and frep are controlled separately. (The change in fCEO will not affect the value of frep and δvac=c/frep.) Fig. 1
Fig. 1 Change in the group refractive index according to the shift in the offset frequency.
shows the calculated values of the group index for different changes in fCEO. We find that the influence of the APRIL due to the deviation in the offset frequency can be ignored up to fCEO=498 MHz.

4. Ambiguity problem by using an ARPIL

A wavelength-based interferometer suffers from the 2π ambiguity problem. We consider the ambiguity problem in the length measurement by using an APRIL. We rewrite Eq. (9) as
δair(t)=δvac(t)/ng(λcen_vac(t),T(t),P(t),H(t)).
(11)
where T is the temperature, P is the pressure, H is the relative humidity, and t is the time parameter. δair(t)is a spatial distance between the two adjacent pulses that may vary with time. An arbitrary length is expressed as
Lair(t)=(M+N)×δair(t),
(12)
where M and N are the integral and fractional parts, respectively. After substituting Eqs. (11) and (7) into Eq. (12), we obtain Lair(t)=(M+N)×cvac/[frep(t)×ng(λcen_vac(t),T(t),P(t),H(t))].The values of Lair(t) at times t1 and t2 are Lair(t1)=(M+N)×cvac/[frep(t1)×ng(λcen_vac(t1),T(t1),P(t1),H(t1))] and Lair(t2)=(M+N)×cvac/[frep(t2)×ng(λcen_vac(t2),T(t2),P(t2),H(t2))]. The change in the length ΔLair(t1,t2)=Lair(t2)Lair(t1) due to the time difference is
ΔLair(t1,t2)=(M+N)×cvac×[1/[frep(t2)×ng(λcen_vac(t2),T(t2),P(t2),H(t2))]1/[frep(t1)×ng(λcen_vac(t1),T(t1),P(t1),H(t1))]].
(13)
We assume that frep(t1)=frep(t2). After dividing both sides of Eq. (13) by cvac/frep(t1), we have
ΔLair(t1,t2)/[cvac/frep(t1)]=(M+N)×[Δng1(t1,t2)],
(14)
where Δng1(t1,t2)=1/ng(λcen_vac(t2),T(t2),P(t2),H(t2))1/ng(λcen_vac(t1),T(t1),P(t1),H(t1)).

ΔLair(t1,t2) is the length difference caused by the changes in the environmental parameters T, P, and H. cvac/frep(t1)=δvac(t1) is the length of the APRIL. When ΔLair(t1,t2)<δvac(t1), there is no ambiguity in the APRIL because Lair(t2) and Lair(t1) have the same integral part.
ΔLair(t1,t2)/[cvac/frep(t1)]<1
(15)
When Eq. (14) is substituted into Eq. (15), we have
(M+N)×Δng1(t1,t2)<1.
(16)
With the maximum change in the value of Δng1(t1,t2), we can calculate the target measurement range in which there is no ambiguity in the APRIL between two measurements.

For example, in a measurement in a laboratory, we can assume that T[10,30] °C, P[80,120] kPa, and H[10,80]%. We find the maximum value max(ng(λcen_vac(t1),T(t1),P(t1),H(t1)))=max(ng(t1)) and the minimum value min(ng(λcen_vac(t2),T(t2),P(t2),H(t2)))=min(ng(t2)) of the group refractive index in air for changes in T, P, and H. Then, we calculate Δng1(t1,t2)=1/min(ng(t2))1/max(ng(t1)), which is approximately 1.3×104. For M<5×103, Eq. (16) is satisfied, and there is no ambiguity in the APRIL for the length measurement. If the APRIL is one meter [cvac/frep(t1)=δvac(t1)=1 m], and the range without ambiguity is less than 5 km (M<5×103), ΔLair(t1,t2)/[cvac/frep(t1)]=(M+N)×Δng1(t1,t2)<1, and the condition in Eq. (15) meets the requirement. In other words, for a length measurement greater than hundreds of meters (5×103) using an APRIL (namely, one meter) in the laboratory, the measured length changes caused by the changes in the environmental parameters only become changes in the fractional parts.

For the wavelength, we assume λvac=1560 nm and Δnp1(t1,t2)=1.3×104. When M<5×103, ΔLair(t1,t2)/λvac<1 is satisfied, and there is no ambiguity in the wavelength for length measurement. The range without ambiguity is less than 7.8 mm. In other words, for a length measurement greater than 8 mm using the wavelength (namely, 1560 nm) in the laboratory, the measured length changes caused by the changes in the environmental parameters may become the changes in the integral parts.

5. Conclusion

We examined the wavelength and APRIL of an FOFC used for length measurements. We conclude that an APRIL can be used as a low-cost method to measure the length compared to the wavelength because only one frequency parameter is necessary for stabilization. If the offset frequency is small enough (<498 MHz), we can disregard the influence of the group refractive index according to the changes in the offset frequency. In addition, we do not need to consider the ambiguity caused by the changes in the environmental parameters in the APRIL for a displacement measurement greater than hundreds of meters in laboratory conditions. This allows for the realization of absolute length measurements with an APRIL.

Acknowledgments

We thank Prof. Mitsuo Takeda of Utsunomiya Univ. for the informative discussions. This research work was financially supported by Grant-in-Aid for Young Scientists (B) Grant Number 25820171 and a grant from the Mazda Foundation, respectively.

References and links

1.

N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, and R. Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006). [CrossRef] [PubMed]

2.

S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 095302 (2009). [CrossRef]

3.

T. Yasui, Y. Kabetani, Y. Ohgi, S. Yokoyama, and T. Araki, “Absolute distance measurement of optically rough objects using asynchronous-optical-sampling terahertz impulse ranging,” Appl. Opt. 49(28), 5262–5270 (2010). [CrossRef] [PubMed]

4.

S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, and T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express 17(20), 17324–17337 (2009). [CrossRef] [PubMed]

5.

Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41(21), 4318–4324 (2002). [CrossRef] [PubMed]

6.

J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29(10), 1153–1155 (2004). [CrossRef] [PubMed]

7.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Europ. Opt. Soc. Rap. Public. 3, 08003 (2008).

8.

M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34(13), 1982–1984 (2009). [CrossRef] [PubMed]

9.

H. Matsumoto, X. Wang, K. Takamasu, and T. Aoto, “Absolute measurement of baselines up to 403 m using heterodyne temporal coherence interferometer with optical frequency comb,” Appl. Phys. Express 5(4), 046601 (2012). [CrossRef]

10.

C. Narin, T. Satoru, T. Kiyoshi, and M. Hirokazu, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012). [CrossRef]

11.

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Space position measurement using long-path heterodyne interferometer with optical frequency comb,” Opt. Express 20(3), 2725–2732 (2012). [CrossRef] [PubMed]

12.

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013). [CrossRef]

13.

D. Wei, K. Takamasu, and H. Matsumoto, “A study of the possibility of using an adjacent pulse repetition interval length as a scale using a Helium–Neon interferometer,” Precis. Eng. 37(3), 694–698 (2013). [CrossRef]

14.

D. Wei and M. Aketagawa, “Characteristics of an adjacent pulse repetition interval length as a scale for length,” Opt. Eng. 53(5), 051502 (2014). [CrossRef]

15.

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).

16.

P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996). [CrossRef] [PubMed]

17.

K. P. Birch and M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993). [CrossRef]

18.

E. Bengt, “The Refractive Index of Air,” Metrologia 2(2), 71–80 (1966). [CrossRef]

19.

J. A. Stone and J. H. Zimmerman, “Index of refraction of air,” Available in: http://emtoolbox.nist.gov/Wavelength/Edlen.asp.

20.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(140.4050) Lasers and laser optics : Mode-locked lasers

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: January 24, 2014
Revised Manuscript: March 12, 2014
Manuscript Accepted: March 12, 2014
Published: March 18, 2014

Citation
Dong Wei and Masato Aketagawa, "Comparison of length measurements provided by a femtosecond optical frequency comb," Opt. Express 22, 7040-7045 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7040


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, R. Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006). [CrossRef] [PubMed]
  2. S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 095302 (2009). [CrossRef]
  3. T. Yasui, Y. Kabetani, Y. Ohgi, S. Yokoyama, T. Araki, “Absolute distance measurement of optically rough objects using asynchronous-optical-sampling terahertz impulse ranging,” Appl. Opt. 49(28), 5262–5270 (2010). [CrossRef] [PubMed]
  4. S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express 17(20), 17324–17337 (2009). [CrossRef] [PubMed]
  5. Y. Yamaoka, K. Minoshima, H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41(21), 4318–4324 (2002). [CrossRef] [PubMed]
  6. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29(10), 1153–1155 (2004). [CrossRef] [PubMed]
  7. M. Cui, R. N. Schouten, N. Bhattacharya, S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Europ. Opt. Soc. Rap. Public. 3, 08003 (2008).
  8. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34(13), 1982–1984 (2009). [CrossRef] [PubMed]
  9. H. Matsumoto, X. Wang, K. Takamasu, T. Aoto, “Absolute measurement of baselines up to 403 m using heterodyne temporal coherence interferometer with optical frequency comb,” Appl. Phys. Express 5(4), 046601 (2012). [CrossRef]
  10. C. Narin, T. Satoru, T. Kiyoshi, M. Hirokazu, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012). [CrossRef]
  11. X. Wang, S. Takahashi, K. Takamasu, H. Matsumoto, “Space position measurement using long-path heterodyne interferometer with optical frequency comb,” Opt. Express 20(3), 2725–2732 (2012). [CrossRef] [PubMed]
  12. X. Wang, S. Takahashi, K. Takamasu, H. Matsumoto, “Spatial positioning measurements up to 150m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013). [CrossRef]
  13. D. Wei, K. Takamasu, H. Matsumoto, “A study of the possibility of using an adjacent pulse repetition interval length as a scale using a Helium–Neon interferometer,” Precis. Eng. 37(3), 694–698 (2013). [CrossRef]
  14. D. Wei, M. Aketagawa, “Characteristics of an adjacent pulse repetition interval length as a scale for length,” Opt. Eng. 53(5), 051502 (2014). [CrossRef]
  15. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).
  16. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996). [CrossRef] [PubMed]
  17. K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30(3), 155–162 (1993). [CrossRef]
  18. E. Bengt, “The Refractive Index of Air,” Metrologia 2(2), 71–80 (1966). [CrossRef]
  19. J. A. Stone and J. H. Zimmerman, “Index of refraction of air,” Available in: http://emtoolbox.nist.gov/Wavelength/Edlen.asp .
  20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited