## Optical frequency domain reflectometry based on real-time Fourier transformation

Optics Express, Vol. 15, Issue 8, pp. 4597-4616 (2007)

http://dx.doi.org/10.1364/OE.15.004597

Acrobat PDF (627 KB)

### Abstract

We propose and demonstrate an ultrahigh-speed optical frequency domain reflectometry (OFDR) system based on optical *frequency-to-time* conversion by pulse time stretching with a linearly chirped fiber Bragg grating (LCFG). This method will be referred to as OFDR based on real-time Fourier transformation (OFDR-RTFT). In this approach the frequency domain interference pattern, from which the desired axial depth profile is reconstructed, can be captured directly in the time-domain over the duration of a single stretched pulse, which translates into unprecedented axial line acquisition rates (as high as the input pulse repetition rate). We provide here a comprehensive, rigorous mathematical analysis of this new OFDR approach. In particular, we derive the main design equations of an OFDR-RTFT system in terms of its key performance parameters. Our analysis reveals the detrimental influence of nonlinear phase variations in the input optical pulse (including higher-order dispersion terms and group delay ripples introduced by the LCFG stretcher) on the system performance, e.g. achievable resolution. A simple and powerful method based on Hilbert transformation is successfully demonstrated to compensate for these detrimental phase distortions. We show that besides its potential to provide ultrahigh acquisition speeds (in the MHz range), LCFG-based OFDR-RTFT also offers the potential for performance advantages in terms of axial resolution, depth range and sensitivity. All these features make this approach particularly attractive for imaging applications based on optical coherence tomography (OCT). In our experiments, single-reflection depth profiles with nearly transform-limited ≈ 92.8 μm (average) axial resolutions over a remarkable 18 mm depth range have been obtained from OFDR-RTFT interferograms, each one measured over a time window of ≈50 ns at 20 MHz repetition rate. Improved sensitivities up to -61 dB have been achieved without using any balanced detection scheme.

© 2007 Optical Society of America

## 1. Introduction

1. D. Uttam and B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” IEEE J. Lightwave Technol. **3**, 971–977 (1985) [CrossRef]

4. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. **117**, 43–48 (1995) [CrossRef]

5. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express **11**, 889–894 (2003) [CrossRef] [PubMed]

8. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftima, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**, 2953–2963 (2003) [CrossRef] [PubMed]

11. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and application for optical coherence tomography,” Opt. Express **14**, 3225–3237 (2006) [CrossRef] [PubMed]

12. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. **31**, 2975–2977 (2006) [CrossRef] [PubMed]

14. S. Moon and D. Y. Kim, “Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source,” Opt. Express **14**, 11575–11584 (2006) [CrossRef] [PubMed]

*real time*. This frequency-to-time mapping is achieved by simply inducing a large amount of chromatic dispersion over the input optical pulse (so-called real-time Fourier transformation, RTFT [16–18

16. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. **24**, 1–3 (1999) [CrossRef]

14. S. Moon and D. Y. Kim, “Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source,” Opt. Express **14**, 11575–11584 (2006) [CrossRef] [PubMed]

8. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftima, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**, 2953–2963 (2003) [CrossRef] [PubMed]

14. S. Moon and D. Y. Kim, “Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source,” Opt. Express **14**, 11575–11584 (2006) [CrossRef] [PubMed]

8. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftima, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**, 2953–2963 (2003) [CrossRef] [PubMed]

*amplitude*mapping [14

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

19. T. -J. Ahn, J. Y. Lee, and D. Y. Kim, “Suppression of nonlinear frequency sweep in an optical frequency-domain reflectometer by use of Hilbert transformation,” Appl. Opt. **44**, 7630–7634 (2005) [CrossRef] [PubMed]

*single*interferogram measurement. The application of the HTCM is fundamental to optimize the performance of the OFDR technique, e.g. in terms of achievable axial resolution and axial depth range. We note that a somehow similar approach has been recently reported for nonlinear frequency sweep compensation in OCT systems [20

20. Y. Yasuno, V. D. Madjarova, S. Makita, M. Akiba, A. Morosawa, C. Chong, T. Sakai, K.-P. Chan, M. Itoh, and T. Yatagai, “Three-dimensional and high-speed swept-source optical coherence tomography for in vivo investigation of human anterior eye segments,” Opt. Express **13**, 10652–10664 (2005) [CrossRef] [PubMed]

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

16. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. **24**, 1–3 (1999) [CrossRef]

17. J. Azaña and M. A. Muriel, “Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings,” IEEE J. Quantum Electron. **36**, 517–526 (2000) [CrossRef]

*linear*pulse stretching rate is essential to achieve a

*uniform*sensitivity and resolution over a longer depth range. An LCFG can provide the desired nearly linear group delay over a very broad bandwidth. Specifically, LCFG technology has evolved to the point that several meters long, high-quality gratings can be readily fabricated. This should easily allow scaling the technique for operation over input pulse bandwidths > 100 nm [21]. In contrast, it is extremely difficult to obtain a linear group delay over such broad bandwidths using a long section of conventional SMF or dispersion-shifted optical fiber.

## 2. Basic operation principle of OFDR-RTFT

16. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. **24**, 1–3 (1999) [CrossRef]

17. J. Azaña and M. A. Muriel, “Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings,” IEEE J. Quantum Electron. **36**, 517–526 (2000) [CrossRef]

2. U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol. **11**, 1377–1384 (1993) [CrossRef]

19. T. -J. Ahn, J. Y. Lee, and D. Y. Kim, “Suppression of nonlinear frequency sweep in an optical frequency-domain reflectometer by use of Hilbert transformation,” Appl. Opt. **44**, 7630–7634 (2005) [CrossRef] [PubMed]

13. M. Wojtkowski, V. J. Srinivasan, T. J. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express **12**, 2404–2422 (2004) [CrossRef] [PubMed]

22. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006) [CrossRef]

## 3. Theoretical analysis of OFDR-RTFT

### 3.1. OFDR-RTFT

*ω*

_{0}as an electromagnetic wave,

*a*

_{0}, with a complex slowly varying envelope,

*â*

_{0}as shown in Fig.2(a) i.e.

*a*

_{0}(

*t*) =

*â*

_{0}(

*t*) ∙ exp(

*jω*

_{0}

*t*). In practice, an LCFG operated in reflection can provide a flat spectral response and a (nearly) linear group delay over the input pulse spectral bandwidth [23

23. K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, “Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion,” Opt. Lett. **19**, 1324–1326 (1994) [CrossRef]

*ω*

_{0}and can be mathematically described using the Taylor series expansion at

*ω*

_{0}, [17

17. J. Azaña and M. A. Muriel, “Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings,” IEEE J. Quantum Electron. **36**, 517–526 (2000) [CrossRef]

_{0}=Φ(

*ω*

_{0}) is a phase constant, Φ

_{0}= [

*∂*Φ(ω)/

*∂ω*)

_{ω=ω0}is the group delay,

_{0}= [

*∂*

^{2}Φ(

*ω*)/

*∂ω*

^{2}]

_{ω=ω0}is the first-order dispersion coefficient and

*δ*Φ is the phase deviation (including higher-order dispersion terms and group-delay ripples of the LCFG). An LCFG is predominantly a first-order dispersive element. The spectral phase of an ideal LCFG is shown in Fig. 2(b). As a result, the phase deviation

*δ*Φ is expected to be much smaller than the phase component of the main transfer function

*Ĥ*́(

*ω*) (transfer function of an ideal first-order dispersive element). Notice that in the above notation

*ω*=

*ω*-

_{opt}*ω*

_{0}, where

*ω*is the optical frequency variable and ω is the baseband frequency variable. For the sake of simplicity, the amplitude ripples on the spectral transmission of the LCFG has not been considered in the theoretical analysis. It is well known that the impulse response corresponding to the transfer function

_{opt}*Ĥ*́(

*ω*) can be written as [17

**36**, 517–526 (2000) [CrossRef]

*Â*

_{1}(

*ω*) =

*Â*

_{0}(

*ω*) ∙ exp(

*jδ*Φ) = |

*Â*

_{0}(

*ω*)| ∙

*exp*[

*j*(

*ϕ*+

*δ*Φ)],

*ϕ*(

*ω*) being the spectral phase profile of the input optical pulse

*â*

_{0}. The corresponding time-domain expression of Eq. (3) is as follows:

*â*

_{2}(

*t*) =

_{R}*â*

_{1}(

*t*)*

_{R}*h*̂(

*t*), where

_{R}*â*

_{1}and

*â*

_{2}are the inverse Fourier transforms of

*Â*

_{1}and

*Â*

_{2}, respectively. Introducing Eq. (2) into this last expression, the reflected pulse can be written in the following integral form:

*t*is the time window limiting the duration of

*â*

_{1}(

*t*). Assuming that the LCFG phase deviation

*δ*Φ is sufficiently small, the time duration of

*â*

_{1}(

*t*) should be similar to that of the input optical pulse

*â*

_{0}. In fact, if this time duration is sufficiently short such that the following condition is satisfied [17

**36**, 517–526 (2000) [CrossRef]

*ω*́ =

*t*/

_{R}_{0}is the transformed frequency variable, which is scaled by the first-order dispersion term (frequency-to-time conversion ratio). Thus, Eq. (6) indicates that under the conditions of Eq. (5), the amplitude spectrum of the input optical pulse is efficiently mapped into the time domain (i.e. the output pulse time intensity is directly proportional to the input pulse power spectrum, |

*â*

_{2}

*(t*|

_{R})^{2}∝ |

*Â*

_{1}

*(*ω ´
)|

^{2}∝ |

*Â*

_{1}

*(*ω ´
)|

^{2}) as shown in Fig 2(b). This operation is usually referred to as RTFT and can be interpreted as the time-domain equivalent of Fraunhofer spatial diffraction [16

**24**, 1–3 (1999) [CrossRef]

**36**, 517–526 (2000) [CrossRef]

*â*

_{2}(

*t*) from a sample is assumed to be approximated by the convolution of this pulse with the characteristic time-flying impulse response,

*f*(

*t*), of the sample in the moving time frame,

_{R}*t*. Notice that the function

_{R}*f*(

*t*) is proportional to the desired sample (amplitude and phase) depth profile with a finite frequency bandwidth restricted by the bandwidth of the input light source. In the frequency domain, this relation can be written as

_{R}*F*̂ is the spectral transfer function associated with the sample impulse response. A similar approach was previously used to model the problem of light scattering in conventional OFDR [4

4. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. **117**, 43–48 (1995) [CrossRef]

*replicated*along the duration of the stretched pulse when this pulse is scattered and/or reflected from the sample. In other words, the sample spectral transfer function (in amplitude) can be directly captured in the temporal domain by simply measuring the time intensity envelope of the reflected/scattered pulse using a high-speed photodetector. This is so because the time intensity envelope of the reflected/scattered pulse is directly proportional to its optical power spectrum, i.e. |

*â*|

_{s}(t_{R})^{2}∝ ∙|

*Â*

_{1}

*(*ω ´
)|

^{2}∙|

*F*̂

*(*ω ´
)|

^{2}. However, it is well known that this measurement only allows one to recover the sample power spectrum, i.e. it provides information only about the autocorrelation of the sample impulse response but does not allow one to fully reconstruct the sample depth profile (

*f*̂(

*t*)). As previously discussed in detail (see for instance Ref. [2

_{R}2. U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol. **11**, 1377–1384 (1993) [CrossRef]

4. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. **117**, 43–48 (1995) [CrossRef]

*â*

_{2}(

*t*- 2

_{R}*δt*), where 2

*δt*is the relative time delay in a Michelson interferometer (see Fig. 1). Using Eq. (6) and Eq. (8), the signal resulting from the interference of

*â*(

_{s}*t*) and

_{R}*â*

_{2}(

*t*- 2

_{R}*δt*) can be written as

*δt*/

_{0}is the frequency shearing induced by the time delay, 2

*δt*, in the frequency-to-time converted coordinate. In this derivation, the same Fraunhofer approximation as that in Eq. (5) has been used, i.e.

*δt*

^{2}/(2

_{0})≪1. Thus, the corresponding temporal intensity (measured with a high-speed photodetector) is given by the following expression:

*ϕ*represents the spectral phase profile of the input optical pulse

*â*

_{0}. The first term,

*I*, on the right-hand-side of the above equation is considered as a DC term composed by two contributions, namely the power spectrum of the optical pulse reflected/scattered from the sample and the spectrally sheared reference spectrum. The second and the third terms,

_{DC}*I*and

_{AC}*I*

_{AC}^{*}are the interference terms from which one can directly reconstruct the complex impulse response of the sample under test (i.e. depth complex profile) by use of the DFT.

### 3.2. Numerical time-to-frequency conversion: HTCM

*ϕ*(

*ω*́), by simply acquiring the interference pattern |

*â*(

_{d}*t*)|

_{R}^{2}in Eq. (10) from a single reflection point, i.e. when

*F*̂(

*ω*́) =

*cons tan t*, and applying the so-called Hilbert transformation compensation method (HTCM) [19

19. T. -J. Ahn, J. Y. Lee, and D. Y. Kim, “Suppression of nonlinear frequency sweep in an optical frequency-domain reflectometer by use of Hilbert transformation,” Appl. Opt. **44**, 7630–7634 (2005) [CrossRef] [PubMed]

*I*, can be obtained from the measured interference pattern using the following procedure. First, the real part of the AC interference component can be directly extracted from the measured interference pattern,

_{AC}*I*+

_{AC}*I*

_{AC}^{*}=

*real*(

*I*), by numerically subtracting the DC component. In practice, this subtraction can be easily performed by numerically filtering the Fourier transform of the measured temporal interference pattern with a narrow bandpass filter.

_{AC}^{-1}denotes the inverse Fourier transformation, H denotes the Hilbert transformation, Θ is a suitable window function (used to screen out the term

*real*(

*I*) by filtering in the Fourier domain) and the variable

_{AC}*ξ*, is the Fourier transform counterpart of the variable of

*ω*’. From the calculated differential phase factor ΔΦ, the numerical time-to-frequency mapping to be applied over the measured interference patterns can be precisely recalibrated as follows:

*single*interferogram measurement (using a single reflection point); in other words, this recalibration process does not need to be performed for each interferogram acquisition. This is due to the fact that (i) the used pulse stretching system is almost an entirely passive, linear process (only negligible nonlinear effects may be induced by the LCFG), and (ii) the SPM-broadened input optical pulse is highly stable with very low RIN noise. It should be mentioned that a similar Hilbert transform – based technique has been previously used for compensating the interferometer dispersion imbalance and dispersion induced by the sample under test in a Fourier-domain OCT system [13

13. M. Wojtkowski, V. J. Srinivasan, T. J. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express **12**, 2404–2422 (2004) [CrossRef] [PubMed]

## 4. Performance evaluation of OFDR-RTFT

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

7. M. A. Choma, M. V. Sarunic, C. Y. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express **11**, 2183–2189 (2003) [CrossRef] [PubMed]

**11**, 2953–2963 (2003) [CrossRef] [PubMed]

11. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and application for optical coherence tomography,” Opt. Express **14**, 3225–3237 (2006) [CrossRef] [PubMed]

12. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. **31**, 2975–2977 (2006) [CrossRef] [PubMed]

### 4.1. Axial resolution

26. B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. **20**, 1486-(1995) [CrossRef] [PubMed]

*λ*[nm] of the input pulse (e.g. SPM-broadened pulse in our specific implementation) is narrower than the LCFG reflection bandwidth, the optimal (transform-limited) axial resolution can be estimated as [26

_{S}26. B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. **20**, 1486-(1995) [CrossRef] [PubMed]

*λ*

_{0}is the center wavelength of the pulse source and here, the pulse spectrum is assumed to be Gaussian-like.

*T*(to avoid overlapping among the individual optical pulses); mathematically, |

_{R}_{λ}|∙Δ

*λ*<

_{S}*T*, where

_{R}_{λ}= -

*(2πc/λ*

_{0}^{2})_{0}is the LCFG first-order dispersion in [ps/nm] units. We recall that an LCFG is advantageous in that it can be specifically designed to achieve a desired dispersion over a prescribed bandwidth. For instance, the LCFG reflection bandwidth Δ

*λ*can be designed to ensure that |

_{LCFG}_{λ}|∙Δ

*λ*<

_{LCFG}*T*, thus avoiding temporal overlapping of the stretched optical pulses regardless of the input optical bandwidth. In fact, if the input optical bandwidth is larger than the LCFG reflection bandwidth, then the system axial resolution will be limited by the LCFG bandwidth. A more accurate estimation of the corresponding axial resolution in this case is given by the following expression [27,28

_{R}28. J. M. Schmitt, “Optical Coherence Tomography (OCT):A Review,” IEEE J. Select. Topics Quantum Electron. **5**, 1205–1215 (1999) [CrossRef]

### 4.2. Axial depth range

**11**, 2953–2963 (2003) [CrossRef] [PubMed]

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

*δλ*, i.e. these two features are assumed to be spectrally separated by exactly

*δλ*. The FWHM time width of the transform-limited Gaussian pulse corresponding to each of these features is given by [29]

*T*; mathematically |

_{FWHM}_{λ}|∙

*δλ*≥

*T*. The minimum spectral line-width

_{FWHM}*δλ*that can be resolved in the time domain by the implemented RTFT process can be obtained by introducing Eq. (15) into this last inequality and is given by the following expression:

**11**, 2953–2963 (2003) [CrossRef] [PubMed]

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

*BW*, where

*BW*is the detector bandwidth in Hertz) may be longer than the time width corresponding to the minimum resolvable spectral-line. In this case, the free-space depth range will be limited by the photodetector bandwidth, according to the following approximate expression:

11. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and application for optical coherence tomography,” Opt. Express **14**, 3225–3237 (2006) [CrossRef] [PubMed]

_{λ}≈ 2 ns/nm at 1545 nm.

### 4.3. Sensitivity

**11**, 2953–2963 (2003) [CrossRef] [PubMed]

**11**, 2953–2963 (2003) [CrossRef] [PubMed]

**11**, 2953–2963 (2003) [CrossRef] [PubMed]

*BW*≈ 0.4

*N*/(

_{s}_{λ}Δ

*λ*), where

_{LCFG}*N*is the sampling number, i.e. the optimum bandwidth is inversely proportional to the product dispersion-bandwidth of the LCFG, which in turns is inversely proportional to the maximum tuning rate in the OFDR system. In other words, as expected for a wavelength-swept OCT system, the optimum bandwidth is directly proportional to the wavelength tuning rate. Notice that this expression is derived from a direct application of the Nyquist criterion, assuming that the sampling bandwidth is 2.5 times wider than the detector bandwidth. From the detector

_{s}*BW*estimation given above, it can be also inferred that the sensitivity variation as a function of the axial depth should follow the detector frequency response (responsivity) because the modulation frequency at the detector is linearly proportional to the axial depth, according to the stretched pulse temporal chirp (time-to-frequency mapping). Moreover, a constant dispersion factor is required over the operation bandwidth in order to ensure uniform sensitivity over the axial depth range (in other words, variations in the group-delay slope of the dispersive device along the operation bandwidth will translate into sensitivity variations along the depth range). While in a conventional optical fiber it is very difficult to ensure a constant dispersion factor (group delay slope) over a broad bandwidth, we emphasize that this uniformity can be easily achieved in a LCFG. As a result, the use of a LCFG in the OFDR-RTFT system allows achieving a very uniform performance of the OFDR system along the whole depth range in terms of sensitivity and resolution. These issues are experimentally investigated in the next section.

## 5. Experiments and discussion

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

*single shot*using the proper sampling mode of the oscilloscope, see Fig. 4(b). This interferogram consisted of 4000 data points over a 50 ns time window (12.5 ps sampling resolution). The close-up view of the interferogram, shown in the inset of this figure, exhibits a clear sinusoidal modulation associated with single reflection interference.

*real*(

*I*) in Eq. (10) is numerically extracted from the measured interferogram (shown in the top-right inset of Fig. 5) by application of the HTCM described in Eq. (11). The reconstructed pulse instantaneous frequency was predominantly linear and is shown in Fig. 5. The higher-order chirp of this function, which was obtained by subtracting the predominant linear term from the total instantaneous frequency, is shown in the bottom-left inset of Fig. 5. The frequency tuning rate of the OFDR-RTFT was calculated to be 62.3 THz/μs from the slope of the recovered instantaneous chirp in Fig. 5; this tuning rate is approximately 10 times faster than that of the Fourier domain mode locking swept laser source demonstrated in Ref. [11

_{AC}**14**, 3225–3237 (2006) [CrossRef] [PubMed]

1. D. Uttam and B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” IEEE J. Lightwave Technol. **3**, 971–977 (1985) [CrossRef]

2. U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol. **11**, 1377–1384 (1993) [CrossRef]

*I*in Eq. (10) is also shown in the inset of Fig. 6. The FWHM width of this profile is measured to be 86 μm.

_{DC}**14**, 3225–3237 (2006) [CrossRef] [PubMed]

*fixed*amplitude ripple patterns. This is better illustrated by the results presented in Fig. 8(c)–(d), where the recovered depth profiles of a single reflection at 3 mm are shown for the case (c) when no LCFG amplitude ripple correction is used (red line) and (d) when the LCFG amplitude ripple correction is applied (blue curve). In particular, the amplitude ripple correction translates into a fairly apparent decrease of the noise floor level of ≈-2 dB, and a significant side-lobes suppression by ≈-6.5 dB for the primary side-lobe and ≈-10.6 dB for the secondary side-lobe.

12. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. **31**, 2975–2977 (2006) [CrossRef] [PubMed]

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

## 6. Conclusions

*frequency-to-time*conversion of the spectral interferograms using a passive, linear optical pulse stretcher (highly-dispersive medium). In this way, each frequency-domain interference pattern, from which the desired axial depth profile is reconstructed, can be captured directly in the time-domain over the duration of a single stretched pulse; as a result the A-line acquisition rate can be as high as the pulse repetition rate from the input pulsed source. An unprecedented A-line acquisition speed of 5-MHz has been previously demonstrated using this technique [14

**14**, 11575–11584 (2006) [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | D. Uttam and B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” IEEE J. Lightwave Technol. |

2. | U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol. |

3. | R. Passy, N. Gisin, J. P.von der Weid, and H. H. Gilgen, “Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources,” J. Lightwave Technol. |

4. | A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. |

5. | R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express |

6. | J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. |

7. | M. A. Choma, M. V. Sarunic, C. Y. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express |

8. | S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftima, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express |

9. | B. Golubovic, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Optical frequency-domain reflectometry using rapid wavelength tuning of a Cr |

10. | S. H. Yun, G. J. Tearney, J. F. de Boer, and B. E. Bouma, “Motion artefacts in optical coherence tomography with frequency-domain ranging,” Opt. Express |

11. | R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and application for optical coherence tomography,” Opt. Express |

12. | R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. |

13. | M. Wojtkowski, V. J. Srinivasan, T. J. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express |

14. | S. Moon and D. Y. Kim, “Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source,” Opt. Express |

15. | Y. Park, T. -J. Ahn, J.-C. Kieffer, and J. Azaña, “Real-Time Optical Frequency-Domain Reflectometry,” to be presented in Conf. Lasers and Electro-Optics (CLEO/IQEC), CTuT1 (2007) |

16. | M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. |

17. | J. Azaña and M. A. Muriel, “Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings,” IEEE J. Quantum Electron. |

18. | Y. C. Tong, L.Y. Chan, and H.K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett, |

19. | T. -J. Ahn, J. Y. Lee, and D. Y. Kim, “Suppression of nonlinear frequency sweep in an optical frequency-domain reflectometer by use of Hilbert transformation,” Appl. Opt. |

20. | Y. Yasuno, V. D. Madjarova, S. Makita, M. Akiba, A. Morosawa, C. Chong, T. Sakai, K.-P. Chan, M. Itoh, and T. Yatagai, “Three-dimensional and high-speed swept-source optical coherence tomography for in vivo investigation of human anterior eye segments,” Opt. Express |

21. | |

22. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

23. | K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, “Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion,” Opt. Lett. |

24. | R. Kashyap, |

25. | K. Takada, “Noise in optical low-coherence reflectometry,” IEEE J. Quantum Electron. |

26. | B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, “High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. |

27. | J. W. Goodman, |

28. | J. M. Schmitt, “Optical Coherence Tomography (OCT):A Review,” IEEE J. Select. Topics Quantum Electron. |

29. | G. Agrawal, |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(110.2350) Imaging systems : Fiber optics imaging

(110.4500) Imaging systems : Optical coherence tomography

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 29, 2007

Revised Manuscript: March 26, 2007

Manuscript Accepted: March 29, 2007

Published: April 3, 2007

**Virtual Issues**

Vol. 2, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Yongwoo Park, Tae-Jung Ahn, Jean-Claude Kieffer, and José Azaña, "Optical frequency domain reflectometry based on real-time Fourier transformation," Opt. Express **15**, 4597-4616 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-8-4597

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### References

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