## Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting

Optics Express, Vol. 12, Issue 20, pp. 4822-4828 (2004)

http://dx.doi.org/10.1364/OPEX.12.004822

Acrobat PDF (563 KB)

### Abstract

A novel technique using an acousto-optic frequency shifter in optical frequency domain imaging (OFDI) is presented. The frequency shift eliminates the ambiguity between positive and negative differential delays, effectively doubling the interferometric ranging depth while avoiding image cross-talk. A signal processing algorithm is demonstrated to accommodate nonlinearity in the tuning slope of the wavelength-swept OFDI laser source.

© 2004 Optical Society of America

## 1. Introduction

^{22. B. Golubovic, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Optical frequency-domain reflectometry using rapid wavelength tuning of a Cr4+:forsterite laser,” Opt. Lett. 22, 1704–1706 (1997). [CrossRef] }Furthermore, the inability to distinguish between a positive and negative electrical frequency in a conventional interferometry leads to the ambiguity between positive and negative depths. To avoid the superposition or folding of the positive-delay image upon the negative-delay image, the reference delay of the interferometer can be adjusted to be outside of the sample. This, however, further limits the ranging depth for a given coherence length of the source. To avoid this limitation, researchers have measured quadrature interference signals based on active or passive phase biasing using a piezoelectric actuator

^{99. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, Opt. Lett. 27, 1415–1417 (2002). [CrossRef] }, birefringence plate

^{1010. Y. H. Zhao, Z. P. Chen, Z. H. Ding, H. W. Ren, and J. S. Nelson, Opt. Lett. 27, 98–100 (2002). [CrossRef] }or 3×3 coupler

^{1111. M. A. Choma, C. Yang, and J. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003). [CrossRef] [PubMed] }. These techniques could unfold otherwise overlapping images associated with positive and negative depths, but tended to leave residual artifacts due to the difficulty of producing stable quadrature signals.

^{55. H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optics systems”, J. Lightwave Technol. 7, 3–10, (1989). [CrossRef] }or a recirculating frequency shifting loop.

^{66. X Zhou, K. Iiyama, and K. Hayashi, “Extended-range FMCW reflectometry using an optical loop with a frequency shifter,” IEEE Photon. Technol. Lett. 8, 248–250 (1996). [CrossRef] }In this work we use an acousto-optic frequency shifter and apply the technique to high-speed OFDI with several orders of magnitude faster ranging speed. Furthermore, we demonstrate a signal processing algorithm to accommodate nonlinear tuning in the wavelength-swept OFDI source.

## 2. Principle

### 2.1 Frequency shift

*f*in the reference arm and an interferometer pathlength difference (or depth) of

*z*, the detector signal can be expressed as

*η*denotes the quantum efficiency of the detector,

*P*(

_{r}*t*) and

*P*(

_{s}*t*) the optical powers of the reference and sample arm light, respectively,

*R*(

*z*) the reflectivity profile of the sample,

*G*(|

*z*|) the coherence function corresponding to the fringe visibility, c the speed of light,

*ν*(

*t*) the optical frequency, and

*ϕ*(

*z*) the phase of backscattering. For linear tuning, i.e.

*ν*(

*t*) =

*ν*

_{0}-

*ν*

_{1}

*t*, the frequency of the detector signal is given by

*z*=

*c*Δ

*f*/(2

*ν*

_{1}). Therefore, by choosing Δ

*f*and

*ν*

_{1}to have opposite signs, the zero signal-frequency can be made to point to a negative depth.

*z*

_{c}indicates the depth where the visibility drops to 0.5 and thereby the SNR drops by 6 dB. One may arguably define the effective ranging depth as the maximum depth span where the SNR penalty is less than 6 dB. In Fig. 2(a), with no frequency shift, only one side of the coherence range (hatched region) can be used due to the sign ambiguity of the signal frequency. In contrast, with an appropriate frequency shift, both sides of the coherence range from -

*z*

_{c}to

*z*

_{c}can be utilized without any image crosstalk between the negative and positive depths.

### 2.2 Nonlinear tuning

^{55. H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optics systems”, J. Lightwave Technol. 7, 3–10, (1989). [CrossRef] ,77. J. P. von der Weid, R. Passy, G. Mussi, and N. Gisin, “On the characterization of optical fiber network components with optical frequency domain reflectometry,” J. Lightwave Technol. 15, 1131–1141 (1997). [CrossRef] }As a solution to this problem, the detector signal may be sampled with nonlinear time intervals compensating for the frequency chirping. Alternatively, the detector signal can be sampled with a constant time interval if the sampled data is re-mapped to a uniform

*ν*-space by interpolation prior to discrete Fourier transform (DFT).

^{1212. C. Dorrer, N. Belabas, J-P Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000). [CrossRef] }Both methods have been demonstrated to yield a transform-limited axial resolution given by the tuning spectral range of the source.

^{1313. J. G. Proakis and D. G. Manolakis, Digital signal processing, 3rd Ed. (Prentice Hall, New Jersey, 1996), Chap. 5.}was used to achieve nearly transform-limited axial resolution over the entire ranging depth. The algorithm is as follows:

- 1. Obtain N samples of the signal with uniform time interval during each wavelength sweep of the source.
- 2. Calculate DFT of N data points in the electrical frequency domain.
- 3. Separate two frequency bands below and above Δ
*f*corresponding to negative and positive depths, respectively. - 4. Shift each frequency band such that the zero depth is aligned to the zero electrical frequency.
- 5. Apply zero-padding to each frequency band and calculate inverse DFT resulting in an array of increased number of samples in the time domain with smaller time interval for each frequency band.
- 6. Interpolate each array in the time domain into a uniform
*ν*space using a mapping function calibrated to the nonlinearity of the source with linear interpolation. - 7. Calculate DFT of each interpolated array.
- 8. Combine the two arrays (images) by shifting the array index.

## 3. Experiment

### 3.1 OFDI System

^{st}-order diffraction). The two frequency shifters were driven with voltage controlled oscillators to produce a net shift of Δ

*f*= FS2 - FS1. The use of two frequency shifters balanced the material dispersion of the acousto-optic crystals between the interferometer arms and suppressed image crosstalk due to finite frequency sidebands of the frequency shifters. The insertion loss of each device including fiber coupling was less than 2.5 dB. A swept laser was constructed to provide a tuning range of 108 nm from 1271 nm to 1379 nm (

*ν*

_{1}= 135 GHz/μs). To generate an optical trigger signal, a portion of the laser output was tapped from the reference arm and transmitted through a narrowband filter. The photodetector then detected a train of short pulses generated when the output spectrum of the laser swept through the pass band of the filter. From the photodetector output, TTL pulses were generated with adjustable phase delay and used as trigger and gating pulses in a data acquisition board (National instruments Inc., PCI 6115). Although a repetition rate up to 36 kHz could be achieved, the laser was operated at a reduced rate of 7 kHz so that 1300 samples could be acquired during a 91% of each wavelength sweep with the available digitizer board operating at 10 Ms/s. The depth span in the image was 5.8 mm corresponding to the Nyquist frequency of 5 MHz. A low-pass electrical filter (order 6 elliptic function) with a 3dB cutoff at 5.7 MHz was used as an anti-aliasing filter. The probe comprising a galvanometer mirror and an imaging lens produced a 30 μm 1/e

^{2}diameter focal spot on the sample with a confocal parameter of 1.1 mm.

### 3.2 Point Spread Functions

*f*= 0 (FS1 = -25 MHz, FS2 = -25 MHz). With a calibrated partial reflector at various depths in the sample arm, the detector signal was acquired for individual frequency sweeps of the source. For comparison, the sampled data acquired at each depth were processed with and without the mapping algorithm described in Section 2.2. Figure 4 shows the results, where the y-axis represents the square of the DFT amplitudes normalized to the value at zero frequency, and the bottom and top x-axes represent the signal frequency and the depth

*z*, respectively. Without mapping, the point spread function suffers from significant broadening and large degradation of the peak power as the depth increases, because of the tuning nonlinearity of our swept laser [Fig. 4(a)]. With the mapping algorithm, however, the point spread function exhibits nearly transform-limited axial resolution at all depths, as shown in Fig. 4(b). The finite coherence length of the laser accounts for the decrease of the signal power with depth. Over the entire depth span of 5.8 mm, the SNR is reduced by more than 11 dB. According to the criterion for the effective ranging depth introduced in Section 2.1, the ranging depth without frequency shifting was only 2.9 mm.

*f*= -2.5 MHz (FS1 = -22.5 MHz, FS2 = -25 MHz). The magnitude of the frequency shift corresponded to a half of the Nyquist frequency. Figures 5(a) and (b) show the point spread functions measured with and without the mapping process, respectively. In this case, the peak signal power occurs at the depth corresponding to the net acousto-optic frequency shift of 2.5 MHz. The measured axial resolution, defined as the full-width-at-half-maximum of the point spread function, was 14.5 – 15.5 μm over the entire depth range. This compares with the theoretical value of 14 μm calculated from the Fourier transform of the integrated optical spectrum of the source. The nearly transform-limited axial resolution observed in Fig. 5(b) validates the mapping algorithm described in Section 2.2. The reduction in signal power with depth is less than 5 dB over the entire depth span of 5.8 mm, demonstrating the benefit of the frequency shifting technique in terms of extending the ranging depth. The measured system sensitivity was approximately 110 dB at zero delay and 105 dB at the maximum depth. No indication of image crosstalk between positive and negative depths was observed up to a -70 dB power level limited by the finite signal to noise ratio.

### 3.3 Image

*ex vivo*was conducted with the OFDI system. Figure 6 depicts two images, A and B, obtained under identical experimental conditions except that Δ

*f*= 0 for A and Δ

*f*= -2.5 MHz for B. Each image was obtained using the mapping algorithm described earlier and plotted in logarithmic grayscale over a dynamic range of 55 dB in reflectivity. The surface of the tissue was placed at an angle with respect to the probe beam axis, and the reference mirror was positioned such that the signal was present at both positive and negative depths in the image. In A, the tissue image is contained within the effective ranging depth of 2.8 mm, i.e. the top half of the total depth span. However, the slope of the sample surface resulted in a folding of the negative-depth portions of the sample onto the positive-depth portions. In contrast, in B the entire positive and negative depths could be displayed without ambiguity, taking advantage of the ranging depth increase to 5.8 mm by the frequency shifting technique.

## . Conclusion

## Acknowledgement

## References and links

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

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

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

4. | W. Eickhoff and R. Ulrich, “Optical frequency domain reflectrometry in single-mode fiber”, Appl. Phys. Lett. |

5. | H. Barfuss and E. Brinkmeyer, “Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optics systems”, J. Lightwave Technol. |

6. | X Zhou, K. Iiyama, and K. Hayashi, “Extended-range FMCW reflectometry using an optical loop with a frequency shifter,” IEEE Photon. Technol. Lett. |

7. | J. P. von der Weid, R. Passy, G. Mussi, and N. Gisin, “On the characterization of optical fiber network components with optical frequency domain reflectometry,” J. Lightwave Technol. |

8. | S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. |

9. | M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, Opt. Lett. |

10. | Y. H. Zhao, Z. P. Chen, Z. H. Ding, H. W. Ren, and J. S. Nelson, Opt. Lett. |

11. | M. A. Choma, C. Yang, and J. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. |

12. | C. Dorrer, N. Belabas, J-P Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B |

13. | J. G. Proakis and D. G. Manolakis, |

**OCIS Codes**

(110.1650) Imaging systems : Coherence imaging

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 18, 2004

Revised Manuscript: September 21, 2004

Published: October 4, 2004

**Citation**

S. Yun, G. Tearney, J. de Boer, and B. Bouma, "Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting," Opt. Express **12**, 4822-4828 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4822

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### References

- A.F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, �??Measurement of intraocular distances by backscattering spectral interferometry,�?? Opt. Comm. 117, 43-48 (1995). [CrossRef]
- B. Golubovic, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, �??Optical frequency-domain reflectometry using rapid wavelength tuning of a Cr4+:forsterite laser,�?? Opt. Lett. 22, 1704-1706 (1997). [CrossRef]
- S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, �??High-speed optical frequency-domain imaging,�?? Opt. Express 11, 2953-2963 (2003) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2953">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2953</a> [CrossRef] [PubMed]
- W. Eickhoff and R. Ulrich, �??Optical frequency domain reflectrometry in single-mode fiber,�?? Appl. Phys. Lett. 39, 693-695 (1981). [CrossRef]
- H. Barfuss and E. Brinkmeyer, �??Modified optical frequency domain reflectometry with high spatial resolution for components of integrated optics systems,�?? J. Lightwave Technol. 7, 3-10, (1989). [CrossRef]
- X. Zhou, K. Iiyama, and K. Hayashi, �??Extended-range FMCW reflectometry using an optical loop with a frequency shifter,�?? IEEE Photon. Technol. Lett. 8, 248-250 (1996). [CrossRef]
- J. P. von der Weid, R. Passy, G. Mussi, and N. Gisin, �??On the characterization of optical fiber network components with optical frequency domain reflectometry,�?? J. Lightwave Technol. 15, 1131-1141 (1997). [CrossRef]
- S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, �??High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,�?? Opt. Lett. 28, 1981-1983 (2003). [CrossRef] [PubMed]
- M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, Opt. Lett. 27, 1415-1417 (2002). [CrossRef]
- Y. H. Zhao, Z. P. Chen, Z. H. Ding, H. W. Ren, and J. S. Nelson, Opt. Lett. 27, 98-100 (2002). [CrossRef]
- M. A. Choma, C. Yang, and J. Izatt, �??Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,�?? Opt. Lett. 28, 2162-2164 (2003). [CrossRef] [PubMed]
- C. Dorrer, N. Belabas, J-P Likforman, and M. Joffre, �??Spectral resolution and sampling issues in Fourier-transform spectral interferometry,�?? J. Opt. Soc. Am. B 17, 1795-1802 (2000). [CrossRef]
- J. G. Proakis and D. G. Manolakis, Digital signal processing, 3rd Ed. (Prentice Hall, New Jersey, 1996), Chap. 5.

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