1. Introduction

Optical coherent tomography (OCT) is a recently developed technology capable of imaging tissue structures and blood flow at micron scale resolution [1]. For OCT Doppler flow imaging, birefringence imaging and en face structural imaging [2–4], a sufficient Doppler frequency shift is needed to perform highly sensitive heterodyne demodulation. In those applications, the Doppler shift is commonly achieved by using an electro-optic (EO) phase modulator or an acousto-optic (AO) frequency modulator in the reference arm [2, 5–7]. However, the crystal in the modulator introduces large dispersion, which degrades the OCT axial resolution if not compensated [8, 9]. Up to date, the dispersion is generally compensated to the second order by hardware (i.e., up to the group velocity dispersion or GVD), except in cases where a crystal of the same material and the same length as in the modulator is placed in the sample arm [10]. The latter approach for dispersion compensation may be suboptimal since it is generally expensive and often introduces an extra attenuation for the signal from the sample, in addition to the potential inconvenience to place the matching crystal due to the limited space in the sample arm of a fiber-optic OCT system. Recently, elegant digital signal processing algorithms have been developed to compensate the dispersion mismatch based upon a priori information about the sample [11–13]. More recently, a real-time numerical dispersion compensation method has been developed for cases in which the dispersion between the two arms in the OCT interferometer is approximately matched initially [14]. This numerical dispersion compensation can be made automatic by using an iterative procedure that optimizes the image sharpness; thus this method is particularly suitable for images having good contrast and sharp, fine features (such as retinal images). However, for a heterodyne OCT system having a modulator in the reference arm, the dispersion mismatch between the two arms due to the modulator (in the reference arm) is fairly large, and the large unmatched dispersion can significantly compromise the signal-to-noise ratio of the OCT interference signal as well as the image contrast before digitization. Thus it would be very challenging (if not impossible) to use a numerical post-processing method for dispersion compensation once the interference signal is digitized. In this paper, we report a method to compensate the large dispersion mismatch to the third order, resulting in an axial resolution close to the ideal value predicted by the spectral bandwidth of the light source. In our approach, we combine a grating-lens based optical phase delay line in the reference arm of an OCT system and an extra length of single-mode fused silica fiber in the sample arm to fully compensate the dispersion between the sample and the reference arms up to the third order. The approach reported here can be designed to manage the dispersion over a large range and it can be applied to both time- and spectral-domain OCT. When used in time-domain OCT, our approach can also be designed for dynamically compensating the depth-dependent dispersion introduced by the sample.

 

Fig. 1. Schematic diagram of a RSOD. The beam path is: M (incident on the grating) → A (on the lens) → O (on the tilting mirror) → B (back on to the lens) → N (on the grating). θ_i is the beam incident angle on the grating (with respect to the normal of the grating); θ_λ is the diffraction angle at wavelength λ (again with respect to the normal of the grating); θ ₀ is angle between the grating normal and the optical axis of the lens; Δθ=θ ₀-θ_λ is the angle between the diffracted beam at wavelength λ and the optical axis; γ is the tilt angle of the scanning mirror. x _o is the vertical offset of the mirror pivoting axis relative to the optical axis; y _o is the vertical distance between the beam incident point on the grating (M) and the optical axis. Line MP is perpendicular to the outgoing beam and P is the point of interception; Line OO’ goes through the center of the lens and is perpendicular to Line MO’; MO’ intercepts with the extension of the Line BN at point P’. Note that Lines MA, BN and OO’ are parallel with each other. L is the distance between the grating and the lens and f is the focal length of the lens.

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2. Theory

Figure 1 shows the schematic of a grating-lens delay line which was originally designed for pulse shaping, and later developed to become a rapid-scanning optical delay line (RSOD) for real-time OCT imaging [15, 16]. In order to find the dispersion in the RSOD up to any arbitrary order, we first use ray tracing to determine the wavelength-dependent phase delay or the change in optical pathlength.

All the parameters to be considered are shown in Fig. 1 and explained in detail in the figure caption. The pivoting point of the scanning mirror is placed on the focal plane of the lens. The wavelength-dependent optical pathlength following the beam path (M→A→O→B→N→P) is then calculated as:

Here we have |OO′=Lcos(Δθ)-y ₀sin(Δθ)+f cos/(Δθ)+[f tan (Δθ)-x _o] tanγ, |NP|=|MN|sinθ_i and |NP′|=|MN|sinθ_λ. L is the distance between the grating and the lens, f is the focal length of the lens, x _o and y _o are the vertical offsets of the mirror scanning axis and the incident beam on the grating relative to the optical axis, respectively. θ _i is the incident angle on the grating and θ_λ is the diffraction angle at wavelength λ. θ ₀ is the angle between the normal of the grating and the optical axis of the lens, and Δθ=θ ₀-θ_λ is the angle between the diffracted beam and the optical axis. Utilizing the diffraction grating equation, sinθ_λ-sinθ _i=m(λ/d) (where m is the diffraction order, d is the grating line-spacing), and considering a phase correction term 2πm|MN|d due to the grating [17], we find from Eq. (1) that the phase delay in a RSOD with a double-pass configuration is:

where ω=2πc/λ is the angular frequency of light.

Equation (2) is the general expression for the phase delay. Clearly, the phase delay and thus the dispersion of the RSOD can be tuned by setting the separation between the grating and the lens L (relative to the focal length f), choosing the beam incident angle on the grating θ_i and by choosing the orientation angle θ ₀ of the grating with respect to the optical axis of the lens. When taking L=f,y _o=0, and to the first order of Δθ, Eq. (2) will then be consistent with the results reported in References [16, 18]. Straightforward calculations indicate the dispersion produced by the RSOD would not be sufficient to compensate a 70-mm long crystal in an EO phase modulator if we choose, L=f. Therefore in the following analyses, we start with the general expression for the phase delay (Eq. (2)), which is a function of L for a given focal length f. For simplicity, we choose zero vertical offset for the incident beam on the grating (i.e., y _o=0), and we first consider a non-scanning configuration where γ=0. Notice θ ₀ is a constant and thus we have ∂ΔΔ/∂ λ=-∂θ_λ/∂λ where θ_λ is related to the wavelength λ by the grating equation. To the first order of Δθ, we find the second-order dispersion (GVD) and the third-order dispersion (TOD) are respectively given by:

From these two equations, we see that both the GVD and TOD depend on the diffraction angle θ_λ through the term 1/cos θ_λ, indicating the GVD and TOD of the RSOD can be tuned by choosing the incident angle θ _i which relates to the diffraction angle θ_λ through the diffraction grating equation. Equations (3) and (4) also tell that the GVD only depends on (L-f); however the TOD has two terms with one term depending on (L-f) and the other term on the focal length f. In practice Δθ is small; consequently the second term in TOD (Eq. (4)) would be small. We notice that the dependence of the GVD and TOD on the (L-f) term has opposite signs. It can be shown ϕ″ϕ‴<0 is generally valid using the fact 1≥sinθ_λ≥λ/d -1 (which is given by the first-order diffraction grating equation). This implies it is not possible to simultaneously achieve a large negative GVD and a large negative TOD using a RSOD.

Table 1. Second and Third Order Dispersions at 1.29 µm

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Fig. 2. Schematic of a fiber-optic OCT system in which an EO phase modulator (or an AO frequency modulator) and a RSOD are implemented in the reference arm. An extra length of single-mode fiber (indicated by “Extra SMF”) in the sample arm is introduced in conjunction with the RSOD in the reference arm to fully compensate the dispersion in the OCT system up to the third order. Details are described in Sections 3 and 4.

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3. Method 1 — Dispersion compensation for an EO phase modulator

The overall schematic of a fiber-optic OCT system involving an EO-phase modulator (or an AO frequency modulator) is shown in Fig. 2 where the modulator and a RSOD are placed in the reference arm. As shown in Table 1, the crystal LiNbO₃ in an EO phase modulator has a large GVD and TOD, which were calculated using the optical properties of the crystal given in a Handbook [19]. We notice that both the GVD and TOD of the EO crystal are positive around the wavelength λ _o=1.29 µm. We first consider the case where only the GVD of the crystal is compensated. As indicated by Eqs. (3) and (4), when $\Delta {\theta }_{{\lambda }_o}=0$ (i.e., the diffracted beam at the source center wavelength 1.29 µm is along the optical axis), the GVD and TOD of the RSOD are both simply proportional to (L-f) but with opposite signs, and the numerical values are given in Table 1 for per millimeter of (L-f). Obviously when choosing L>f, the RSOD produces a negative GVD and can compensate the positive GVD of the phase modulator. However, the RSOD also produces a positive TOD when L>f, which will add to the positive TOD of the crystal. This leaves the TOD uncompensated, resulting in a non-ideal OCT axial resolution. To investigate the effect of the uncompensated TOD on the OCT axial resolution, we adopted a conventional fiber-optic OCT system with the sample and reference arms of an equal length of single-mode fibers (SMF28) and air spaces (e.g., similar to the one shown in Fig. 2 but without the “Extra SMF”). A super luminescence diode with a 31-nm bandwidth at a 1.29-µm center wavelength was used. The line-spacing of the grating is d=6.67µm and the first order diffraction (m=1) was chosen. Notice that the GVD and TOD of the RSOD change during depth scanning when the angle γ is varied, and the rate of change depends on the beam incident angle θ _i (more discussions on this topic are given at the end of this section). This dispersion change in the RSOD during depth scanning can be potentially used for dynamically compensating the dispersion caused by the dispersive sample [18]. In our case, we chose θ _i=5.5° (and the corresponding diffraction angle θλ ₀=16.8°) taking into account the need for: (1) compensating the dispersion of the EO crystal; (2) avoiding the obstruction of the beam by the optical components in the RSOD; and (3) enabling the depth-dependent dispersion compensation of water (which will be discussed at the end of this section). Given the chosen parameters (θ _i=5.5°,θλ ₀=1.68°,m=1 and d=6.67µm), we found, for a lens of f=100mm, the GVD of the LiNbO₃ crystal could be compensated by the RSOD. The theoretical GVD (ϕ′) and the TOD (ϕ‴) of the OCT system are shown in Fig. 3A. Although the GVD is zero at 1.29 µm, the slope of the GVD curve at 1.29 µm is nonzero, indicating a nonzero TOD. Experimentally, the OCT interference signal was measured (Fig. 3B). The wavelength-dependent phase was then obtained by taking the Fourier transform of the measured interference signal, from which the GVD D (ϕ″) and TOD (ϕ‴) were further calculated (shown in Figs. 3C & 3D). Note that the experimental GVD (Fig. 3C) crosses zero at the source center wavelength 1.29 µm. The measured OCT axial resolution given by the FWHM of the fringe envelope was ~28.5 µm (Fig. 3B). This represents about 21% degradation from the optimal resolution offered by the light source. The degradation is mainly caused by the uncompensated TOD. When using a source with a broader spectrum, the effect of the uncompensated TOD on the axial resolution can be worse.

 

Fig. 3. (A) The theoretical net GVD (ϕ″) and TOD (ϕ‴) of the OCT system in which the sample and reference arms have an equal length of single-mode fibers (SMF28) and the GVD of the LiNbO3 is compensated by the RSOD. The net GVD (TOD) is defined as the GVD (TOD) of the sample arm minus the GVD (TOD) of the reference arm; (B) The measured OCT interference signal; (C) and (D) Experimental GVD (ϕ″) and TOD (ϕ‴) of the OCT system calculated from the Fourier transform of the measured OCT interference signal.

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As indicated by Eqs. (3) and (4), the GVD and TOD of a RSOD have opposite signs (when L-f≠0). In comparison, the GVD and TOD of the EO crystal both have a positive sign. Therefore a RSOD alone cannot fully compensate the dispersion of the LiNbO₃ crystal up to the third order. Fortunately, the standard fused silica fiber has a small GVD but a moderate positive TOD around 1.29 µm, of which the values are given in Table 1. By introducing an extra length of SMF fiber in the sample arm, the TOD of the modulator and the RSOD in the reference arm can then be fully compensated. The optical pathlengths in the two OCT arms remain equal with a longer air space in the reference arm due to the RSOD. Using Eqs. (3) and (4), we found that both the GVD and TOD can be completely compensated when choosing L-f=77 mm with a 671-mm-long extra SMF in the sample arm. The calculated net GVD and TOD of the system are plotted in Fig. 4A, showing that both the GVD and the TOD are zero at 1.29 µm. The OCT interference signal was experimentally measured (Fig. 4B), and the phase of the interference signal was computed by taking the Fourier transform of the interference signal. The calculated GVD is shown in Fig. 4C, which is zero at the center wavelength of the source (1.29 µm) and flat over the source spectral range (1.27–1.32 µm). A flat GVD implies the TOD is zero as shown in Fig. 4D. A nearly ideal OCT axial resolution of 23.5 µm is recovered as given by the FWHM of the fringe envelope (Fig. 4B).

 

Fig. 4. (A) The theoretical net GVD and TOD of the OCT system with a 671-mm-long extra fiber in the sample arm and L-f=77 mm; (B) The measured OCT interference signal; and (C) and (D) Experimental GVD (ϕ″) and TOD (ϕ‴) of the OCT system calculated from the Fourier transform of the interference signal.

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In addition to providing dispersion compensation, the RSOD can also perform depth scanning by varying the mirror angle γ. However, as indicated in Eq. (2), the phase and consequently the GVD and TOD of the RSOD change as the mirror angle γ varies, and the γ-dependent GVD and TOD can be found as:

Clearly the change of GVD and TOD versus the angle γ depends on the incident θ _i (through the variable Δθ=θ ₀-θ_λ), the diffraction angles θ_λ, the focal length f and the grating line-spacing d. For the configuration chosen above (e.g., θ _i=5.5°, θ _λ0=16.8°, f=100mm, d=6.67µm and m=1) and a few millimeters pathlength scan (e.g., cϕ′_γ=c∂ϕ/∂ω~a few millimeters), we experimentally found that the variation of the axial resolution was less than 3% within a 3-mm depth scan in air. As shown by Eq. (5), the depth-dependent GVD and TOD in the RSOD have the same signs as the GVD and TOD of water around 1.29 µm (see Table 2). Considering water is the major constituent of soft tissues, the changes in GVD and TOD of the RSOD can then help partially and dynamically compensate the depth-dependent tissue dispersion during depth scanning. Again, for the above RSOD configuration, our experimental results showed that the variation of the axial resolution was reduced from the previous 3% when the sample was air to about 2% when the sample was water during a 3-mm optical pathlength scan (or equivalent to a 2.25-mm depth scan in water).

Table 2. GVD and TOD Change versus Pathlength Scanning

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Note: ϕ″ cϕ′ and ϕ″ cϕ′ represent respectively the change in GVD and TOD during 1-millimeter pathlength scan. The GVD and TOD of water were calculated using the results from Reference [20]. The change of the GVD and TOD in the RSOD (e.g., the last two columns) are achieved by scanning the tilting angle γ while all other parameters remain unchanged. For a light source at 1.29 µm, the RSOD configuration is: θ _i=5.5, θ _λ0=16.8°, f=100 mm, L-f=77 mm, d=6.67 µm and m=1; for a light source at 825 nm, the RSOD configuration is: θ_i=17.0°, θ _λ0=2.57°, f=100 L-f=-12 mm, d=3.33 m=-1.

4. Method 2 — Dispersion compensation for an AO frequency modulator

The dispersion compensation approach discussed above can also be applied to an OCT system where an AO frequency modulator is employed. An AO modulator is a valuable substitute to an EO phase modulator for introducing a frequency shift since a linear and broadband EO phase modulator is generally not available for wavelengths not within the 1.3-µm range. To demonstrate the need for compensating both the GVD and TOD caused by an AO modulator, we used a broadband light source with an approximately 125-nm bandwidth at a center wavelength 825 nm. Similar to the EO phase modulator, the crystal in the AO modulator (PbMoO₄) has a large positive GVD and TOD. However, unlike the single-mode fiber near 1.3-µm which has a negligible GVD but a large TOD, a single-mode fused silica fiber at 825 nm exhibits a large GVD and a large TOD. Using the data from the Handbook [19], the ratio of the SMF TOD to GVD at 825 nm is calculated and the value (0.831 fs) is close to that of the crystal (0.740 fs for o-beam and 0.634 fs for e-beam in the crystal). Therefore we can use an extra length of SMF in the sample arm to compensate most of the GVD and TOD of the AO crystal, and the fine-tuning of the dispersion compensation can be realized by using a RSOD. Different from the 1.3-µm system, it is not obvious whether the separation between the grating and the lens L should be larger or smaller than the focal length f. To systematically analyze this issue, we start with the following equations that balance the GVD and TOD between the reference and sample arms of an OCT system (shown in Fig. 2):

Here C _f2 and C _f3 are respectively the GVD and TOD of 1-mm SMF at 825 nm; ϕ″_M and ϕ‴_M are the GVD and TOD of the AO modulator; C _R2 and C _R3 are the GVD and TOD of the RSOD per millimeter of (L-f) (which can be found from Eqs. (3) and (4), i.e.,

C _R2=-16π ² cm ²/ω ³ d ² cos² θ_λ),

C _R3=48m ² π ² c[1+2πmc tan θ_λ/(ωd cos θ_λ)] (d ² ω ⁴ cos² θ_λ)] (d ² ω ⁴ cos² θ_λ),

where the condition $\Delta {{\theta }_{\lambda }}_o=0$ has been applied (e.g., the diffraction beam at γ ₀ is along the optical axis of the lens). From Eqs. (7) and (8), we find the condition for full dispersion compensation up to the third order is:

Notice that C _R2 is negative and C _R3 is positive. Since C _f2 and C _f2 are both positive (e.g., the GVD and TOD of the SMF around 825 nm are positive), the denominators on the righth-and-sides of Eqs. (9) and (10) are negative. Thus we will have the flowing two scenarios for the grating-lens separation L relative to the focal length f according to Eq. (9): (1) L-f<0 if ϕ‴_M/ϕ″_M>C _f3/C _f2 ; and (2) L-f >0 if ϕ‴_M/ϕ″_M<C _f3/C _f2. We also notice that we will always have l>0 according to Eq. (10) regardless the ratio of the TOD to GVD in the AO modulator (ϕ‴_M/ϕ″_M) relative to the ratio in the single-mode fiber (C _f3/C _f2). This indicates it would be legitimate to use an extra length of single-mode fiber in the sample arm to compensate the dispersion of the AO modulator and the RSOD up to the third order.

In principle, we can consider either the e-beam or o-beam incident on the AO modulator. Without losing generality, we chose the e-beam for our RSOD setup. For e-beam, the GVD and TOD of the AO crystal are 489.4 fs² and 310.44 fs³, respectively, which were calculated using the Sellmeier coefficients from the Handbook. The ratio of the AO crystal’s TOD to GVD is thus 0.634 fs, smaller than that of the single-mode fiber (0.83 fs), i.e., ϕ‴_M/ϕ″_M<C _f3/C _f2. According to above analysis, we should expect the second scenario for the RSOD configuration, whereL-f > 0. However, our experimental results suggested the dispersion can only be fully compensated up to the third order when we have the first scenario whereL-f > 0. This could be potentially caused by the inaccuracy in the Sellmeier coefficients in the near-infrared region for the AO crystal used in calculating the high-order dispersion.

In our experiments, the RSOD was set to have a positive GVD but a negative TOD by choosing L<f (see Eqs. (3) and (4)). Consequently, the positive GVD of the AO crystal and the positive GVD of the RSOD in the reference arm were compensated by the positive GVD of the extra length of fiber in the sample arm, while the positive TOD of the crystal plus the negative TOD of the RSOD (in the reference arm) was compensated by the positive TOD of the extra length fiber. To fully compensate the GVD and TOD of the 37-mm-long AO crystal in the reference arm, we introduced an extra fiber length of 584 mm in the sample arm and set L-f=-12mm in the RSOD. The RSOD had an 800-nm grating of 300 lines/mm with a beam incident angle θ _i=17° of and a diffraction angle of θ _λ0=2.57° (for m=-1) at the 825-nm center wavelength. With these parameters, the optimal axial resolution (~2.8 µm) provided by the light source with a 125-nm bandwidth at an 825-nm center wavelength, was experimentally recovered. Figs. 5A and 5B show the GVD and TOD calculated from the OCT fringe signal and the fringe signal (Fig. 5C) was experimentally measured when both the GVD and TOD in the two arms were matched. An axial resolution of 2.8 µm was achieved, similar to the best axial resolution obtained in an OCT system when no AO modulator or RSOD were used. We also found that, even when the GVD was fully compensated (as shown in Fig. 5D), a slightly unmatched TOD between the two arms (as shown in Fig. 5E) degraded the axial resolution to ~8 µm in addition to the appearance of severe side lobes as shown in Fig. 5F. The situation became worse when the TOD of the AO crystal was not compensated at all, which resulted in dramatic degradation of the axial resolution and reduction in the intensity of the fringe signal due to the temporal broadening of the interference peak and severe side lobes.

 

Fig. 5. (A-B) The GVD and TOD calculated from the measured OCT fringe signal (C) when both the GVD and TOD were compensated for a light source of 125-nm bandwidth at a center wavelength 825 nm. (D-E) The GVD and TOD calculated from the measured OCT fringe signal (F) when the GVD was fully compensated but the TOD was partially compensated for the same light source. The degradation of the axial resolution and the increased side lobes were evident in this case as shown in (F).

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According to the experimental results, the GVD and TOD of per millimeter AO crystal were calculated to be 422.4 fs² and 598.15 fs³, respectively, corresponding to a ratio of TOD to GVD of 1.4 fs. We notice that the experimentally determined GVD of the AO crystal (422.4 fs²) is close to the value calculated using the Handbook parameters (489.4 fs²); however the TOD deduced from the experimental data (598.15 fs³) is almost twice as large as the one calculated from the Handbook (310.44 fs³).

As previously discussed, we also took into account the possibility of dynamically compensating the depth-dependent dispersion caused by the sample during depth-scanning when choosing the parameters for the RSOD, in particular, the beam incident angle θ_i and the center wavelength diffraction angle θ _λ0. With the chosen parameters (e.g., θ_i=17.0° and θ _λ0=2.57°), the change of GVD (ϕ″_γ) in the RSOD within 1-mm group delay (Cϕ′_γ) or 1-mm pathlength scanning is equal to the GVD during 1-mm pathlength scanning in water at 825 nm. We also notice that the change of TOD in the RSOD during depth scanning holds the same sign as the TOD of water (see Table 2), helping partially compensate the depth-depth TOD change caused by water. Experimental results showed that the variation of the axial resolution was smaller than 2% within 1-mm pathlength scan in water (equivalent to a 0.75-mm depth scan in water). This demonstrates that it is feasible to incorporate the described dispersion management method with the dynamic dispersion compensation during depth scanning.

5. Conclusion

We have systematically analyzed the dispersion in a fiber-optic OCT system involving an EO phase (or AO frequency) modulator and a RSOD. The combination of a RSOD in the reference arm and an extra length of single-mode fiber in the sample arm can be used to fully compensate the dispersion induced by the modulator up to the third order. An optimal OCT axial resolution offered by the light source can then be achieved. We have shown that the described dispersion management methods can incorporate dynamic dispersion compensation for matching the sample dispersion during depth scanning. Results also demonstrate that the third order dispersion compensation for the modulator is critical for achieving the best axial resolution when using a broadband light source.