## Abstract

We examine the effects of dispersion and absorption in ultrahigh-resolution optical coherence tomography (OCT), particularly the necessity to compensate for high dispersion orders in order to narrow the axial point-spread function envelope. We present a numerical expansion in which the impact of the various dispersion orders is quantified; absorption effects are evaluated numerically. Assuming a Gaussian source spectrum (in the optical frequency domain), we focus on imaging through water as a first approximation to biological materials. Both dispersion and absorption are found to be most significant for wavelengths above ~1*µ*m, so that optimizing the system effective resolution (ER) requires choosing an operating wavelength below this limit. As an example, for 1-*µ*m source resolution (FWHM), and propagation through a 1-mm water cell, if up to third-order dispersion compensation is applied, then the optimal center wavelength is 0.8*µ*m, which generates an ER of 1.5*µ*m (in air). The incorporation of additional bandwidth yields no ER improvement, due to uncompensated high-order dispersion and long-wavelength absorption.

©2005 Optical Society of America

## 1. Introduction

Optical coherence tomography [1] (OCT) has rapidly established itself as an *in-vivo* imaging technique, capable of producing cross-sectional images of living tissue to depths of up to 2mm. Conventional OCT has been limited to axial resolutions of ~10*µ*m. However, the recent application of broadband pulsed Ti:Al_{2}O_{3} lasers and other sources [2] has allowed this figure to approach 1*µ*m, thereby ushering in a new generation of ultrahigh-resolution OCT devices [3], [4]. The exceedingly wide source spectral bandwidths required to achieve these resolutions has increased the susceptibility of the modality to the deleterious effects of frequency-dependent sample absorption and dispersion. The resulting distortion of the axial point-spread function (interferogram) envelope severely degrades both the system resolution and dynamic range. A slowly-decaying tail introduced by high-order dispersion may exacerbate the problem of blindness, the invisibility of a weak signal due to a strong signal in its vicinity. Signal chirping may also introduce misleading image artifacts due to a beating effect [5]. Sample dispersion can theoretically be compensated for, but this process is complicated by the numerous orders that must be considered.

At conventional resolutions, only second-order (group-velocity) dispersion generally has a significant impact on the signal, and it can be ameliorated by inserting a dispersion compensating element into the reference arm [6]. If multiple dispersion orders must be compensated for, each constitutes a distinct parameter which must be fitted separately, critically restricting the choice of compensating medium. Furthermore, the dispersion correction only applies to one lateral plane, and the additional absorption introduced into the reference arm must be taken into account. Recent experimental results [7] showed that the presence of unmatched reference arm dispersion (due to a 2-cm water cell) induced broadening of the interferogram envelope by a factor of ~20. The center wavelength of the Fourier domain OCT system was 850nm, and the system resolution (in the absence of dispersion) was 2.7*µ*m.

Novel dispersion-compensation techniques such as utilizing grating tilt in a rapid scanning frequency-domain optical delay line [8], [9], [10] suffer from the same limitation, that multiple dispersion orders cannot be compensated simultaneously: fixed third-order dispersion is introduced for given second-order compensation. Recently, the combination of a delay line and an additional length of single-mode fiber in the sample arm has been used to compensate for both the second-and third-order dispersion introduced when an electro-optic phase modulator or an acousto-optic frequency modulator is incorporated into the reference arm [11]. (Note that the limited bandwidth of grating-based delay lines may also limit the system resolution.)

Modern numerical post-processing dispersion compensation techniques have been employed to some advantage, utilizing phase-sensitive digital spectral filtering [12], applying depth-dependent correlation kernels to the signal [13], or digital resampling algorithms in the optical frequency domain [14]. Most recently, a Fourier-domain approach was used to directly fit dispersion compensation parameters to the spectral OCT signal [7], a fine-tuning method for optimizing image sharpness which did not, therefore, rely entirely on *a priori* knowledge of the sample. These approaches have been demonstrated to provide compensation against both second-and third-order dispersion (but no higher).

The effect of absorption in OCT is to frequency-shift, distort, and globally attenuate the source power spectrum [15], with consequent reduction in both signal strength and resolution. Since, in general, the effects of absorption cannot be reversed by compensation, care should be taken to select operating wavelengths so that they are minimized.

In this paper, we provide an expansion that quantifies the extent to which the axial point-spread function envelope is broadened due to the distinct dispersion orders, explicitly determining the dependence on source resolution (SR). Additionally, we simulate the dispersive and absorptive effects of water upon an OCT signal at various resolutions, confirming the necessity to compensate for multiple-order dispersion terms in order to maximize signal quality. We determine necessary conditions, within the limitations of our model, to optimize system effective resolution (ER), in terms of source resolution, center wavelength, and dispersion compensation requirements.

## 2. Theory

A typical time-domain OCT (TdOCT) schematic, based on the Michelson interferometer, is shown in Fig. 1. One input arm is illuminated by a light source with high spatial coherence, but broad optical bandwidth (low temporal coherence). Interference between the back-reflected (or back-scattered) signals from the sample and reference arms is only observed when their optical pathlengths are matched to within about one coherence length of the source. Optical sectioning in the sample can thus be performed by recording the modulation pattern as a reference reflector is scanned over a range of distances. OCT can also be performed in the Fourier domain (FdOCT) [7], where the interference pattern between the two optical signals is measured as a function of optical frequency, and no time-delay scanning is required. The detected interference signals from each approach are related by a Fourier transform, and therefore, the theory presented below is equally valid for both TdOCT and FdOCT.

The effects of the reference and sample arms may be described by transfer functions in the optical frequency domain, *H*_{R}
(*v*)=|*H*_{R}
(v)|exp[*jf*_{R}
(*v*)] and *H*_{S}
(*v*)=|*H*_{S}
(*v*)|exp[*jf*_{S}
(*v*)], respectively, where *j*=p-1, and f
_{R}
and f
_{S}
represent the phase responses of their respective transfer functions. These functions take into account relative delays due to scanning, and frequency-dependent dispersion and signal attenuation. (The interfering waves are assumed to have the same polarization state, so that scalar wave optics may be used to describe the scenario.) The interferometric component of the detected response can be represented in terms of the zero*th* moment of the cross-spectral density of the returning optical signals [16, Eq. (21)],

where *G*
_{0}(*v*) represents the power spectral density of the source, ℜ denotes the “real part” operator, and the asterisk (*) denotes complex conjugation. Assuming that the only contributions to the magnitude and phase of the product *H*_{S}
(*v*)*H**
_{R}
(*v*) are due to the effects of propagation through the sample medium, then:

where *z* represents the single-pass propagation distance through the sample medium, *α* the (power) attenuation coefficient and *β* the propagation constant of the medium, which is related to refractive index *n*(*ν*) via the equation *β*=2*πνn*/*c*, where *c* is the speed of light in vacuo. (The factor of two in the exponent is due to the double pass associated with reflection-mode imaging, and the factor of one-half preceding the *α* term is due to the fact that it is the returning reference field (not the intensity) which contributes to the interferometric term.)

We assume that the source has a Gaussian power spectral density, so that *G*
_{0}(*v*)=exp{-4ln2[(*ν*-*v*
*ν̄*)/Δ*v*]^{2}}, where *v*ν is the center frequency and ν*v* the full-width-at-half-maximum (FWHM) spectral bandwidth of the source. The effect of spectral deviations from this mathematically convenient approximation will be discussed later. Additionally, we assume at this stage that *α* may be ignored. (This allows us to quantify the dispersion effects; absorption will be simulated in the next section.) Then the product 2*βz* can be Taylor-series expanded:

where β_{k}=(2*z*/*k*!)(d
^{k}
*β*(*ν*)/d*ν*
^{k}
)|_{ν=ν}̄. The parameters *β*
_{0} and *β*
_{1} are related to the phase and group delays associated with propagation through the medium, via the expressions τ
_{p}
=*β*
_{0}/(2*πν̄*) and τ_{g}=*β*
_{1}/(2*π*). For values of *k* greater than or equal to 2, the terms *β*_{k}
represent the various orders of chromatic dispersion. Following the convention in OCT, we identify the term *β*_{k}
with *k*th-order dispersion. The following identities relate the second-order term *β*
_{2} to alternative quantities found in the literature, specifically: the group dispersion parameter GD [6], [13]; the dispersion coefficient *D*_{λ}
[16], [17, Eq. (5.6-21)] (also known as the group-velocity dispersion *D* [18]); dispersion coefficient *D*_{v}
[17, Eq. (5.6-20)] (group-delay dispersion [16]); and group-delay dispersion *D*_{w}
[9] (group-velocity dispersion [11]).

The width of the interferogram envelope is determined entirely by the argument of the ℜ function, which we define:

where *u*=*ν*-*ν̄*, *g*(*u*)=exp[-*Ku*
^{2}+*j*${\sum}_{k=2}^{\infty}$
*β*_{k}*u*^{k}
], and *K*=4ln2/(Δ*ν*)^{2}. The Fourier transform operation *𝓕*, acting between the *u* and τ_{g} domains, may be defined:

The expression of the interferogram in the τ_{g} domain is possible since group delay τ_{g} is proportional to sample optical pathlength *ℓ̃*. We can measure the width of the interferogram as a function of *ℓ̃*, and hence also of τ_{g}.

If *β*_{k}
=0 for all *k*≥3 (there is no third-or greater-order dispersion), the interferometric signal retains a Gaussian envelope *E*(τ_{g})=|*Ĩ*
_{C}
(τ_{g})|=|*G*(τ_{g})|. (At this point, and hereafter, we ignore the factor of 2 in Eq. (1).) We define its root-mean-square (RMS) width to be:

and find, for this case,

where ${\sigma}_{0}=\sqrt{k\u2044\left(2{\pi}^{2}\right)}$ is the RMS envelope width in the absence of dispersion.

Now, if *β*_{k}
≠0 for some *k*≥3, then we cannot derive a simple closed-form expression for the Fourier transform of *g*. Instead, we resort to determining the width of the interferogram envelope *E*(τ
_{g}
) using the moments of the Fourier transform of some function of *g*(*u*). The determination of an expression for *E* as the Fourier transform of a function of *g*(*u*) would be sufficient to determine its RMS width. Such an expression is not easily attained, but the *square* of *E can* be expressed as the Fourier transform of a function of *g*(*u*), using the autocorrelation theorem:

We define *N*(*u*)=∫^{∞}-_{∞}
*g*(*w*)*g**(*w*-*u*)d*w*, and consider the Moment Generating Theorem, i.e., $\U0001d4d5\left\{{\left(\frac{j}{2\pi}\right)}^{n}{m}^{\left(n\right)}\left(u\right)\right\}={\tau}_{g}^{n}M\left({\tau}_{g}\right)$ and the Central Ordinate Theorem, i.e., ∫^{∞}-_{∞}
*M*(τ_{g})dτ_{g}=*m*0), where *m* and *M* are arbitrary functions in the *u* and τ_{g} domains, respectively, related by the Fourier transform operation, and *m*
^{(n)}(u) is the *n*-th derivative of *m* with respect to *u*. These may be applied directly to Eq. (7), after *E* is replaced by *E*
^{2}, to obtain the RMS width ${\sigma}_{{\tau}_{g}}^{\prime}$ of the squared envelope (the Fourier transform of *N*),

where the prime (′) notation on the right-hand side is used to indicate derivatives taken with respect to *u*.

The derivatives may be calculated by reversing the order of integration and differentiation in the definition of *N*(*u*); some calculation yields the result:

where ${\sigma}_{{\tau}_{g},\mathrm{ND}}^{\prime}=\sqrt{K}\u2044\left(2\pi \right)$ is the RMS (squared) envelope width in the absence of dispersion. We define the factor (1+^{∑}∞_{t=2}${\sum}_{s=t}^{\infty}$
*C*
_{s,t}
*β*
_{s}
*β*_{t}
)^{1/2} to be the envelope broadening factor (EBF), in which the coefficients *C*
_{s,t} are given by:

where *δ*
_{s,t} represents the Kronecker delta function. Note that the coefficients *C*
_{s,t} are entirely dependent on the source spectrum, whereas the parameters *β*_{k}
are also dependent on the properties of the medium.

This definition for the EBF can be extended to the case where absorption has an impact on the signal. We define it to be the ratio between the RMS width of the squared broadened interferogram envelope (in the presence of dispersion and absorption) and that of the squared unbroadened envelope (in their absence). This is clearly consistent with the series expansion provided in Eq. (11).

If *β*_{k}
=0 for all *k*≥3, then the expressions under the radical signs in both Eqs. (8) and (11) are equal. This is due to the fact that if only second-order dispersion impacts upon the signal, then the envelope retains its Gaussian functional form. Furthermore, for any two curves which have the same functional form (but where one is possibly dilated with respect to the other), the ratio between their widths will be independent of the precise manner in which the functional “width” is defined. Consequently, the envelope RMS ratio and the *squared* envelope RMS ratio are identical. However, it is known that third-and higher-order dispersion terms distort the interferogram envelope [19], and so, in general, the precise definition of envelope width will have some bearing on the EBF. A comparison between the broadening factors associated with various different definitions of envelope width is provided in the next section.

The EBF is defined in the t_{g} domain, but it equivalently describes the axial point-spread function broadening. This is because t_{g} is linearly related to the optical pathlength via the equation τ_{g}=2*ℓ̃*/*v*
_{g}, where *v*_{g}
is the group velocity of the propagating wave. The full-width-at-half-maximum (FWHM) resolution of the (unsquared) interferogram envelope can be expressed in terms of the quantity *K* via the relation [16, Eq. (18)]: *ℓ̃*_{FWHM}=(*c*
^{2}
*K*ln2/*π*
^{2})^{1/2}, where the tilde denotes optical length. Hereafter, we refer to this quantity as the source resolution (SR). Therefore, the EBF coefficients *C*
_{s,t} may be expressed in terms of the SR instead of *K*.

The analysis of this section is directly analogous to that of the propagation of unchirped Gaussian pulses in single-mode fibers [20]. In that context, the squared magnitude of the pulse has a natural physical interpretation: it is the power signal. In OCT, the EBF has previously been reported up to the second-[5] and third-[16] order.

## 3. Results and discussion

#### 3.1. Absorption and refractive index data

The propagation of radiation through biological tissue can be modeled as the propagation of light through water, a major component of tissue constituents. This is the recommendation of Fercher *et al.* [13], due to the wide availability of published wavelength-dependent refractive index and absorption data for water (which are in good agreement), and we follow it here. (Wang *et al.* [18] have previously examined second-order dispersion effects in water.) The data used in our calculations were obtained by applying a cubic spline interpolation to tabulated data due to Segelstein [21]. It should be noted that direct multiple numerical differentiation of such data is virtually impossible, since high-frequency noise rapidly dominates the signal, and so specific high-order dispersion parameters cannot be obtained directly from such data. (At-tempts to directly measure water dispersion parameters up to the third order do exist, however [22].) Empirical expressions approximating the refractive index of water have been published, which can easily be differentiated multiple times. Harvey *et al.* [23] present one such expression, valid over a wide range of input parameters (temperature, density, and wavelength), based on approximating the Lorentz-Lorenz function for water. However, despite its convenience, it was not sufficiently accurate over the large wavelength range associated with broad-bandwidth sources. This is particularly true of its derivatives. Additionally, the use of a consistent source for both refractive index and absorption data ensures that the Kramers-Kronig relationship between the two quantities is satisfied. Figure 2 shows the curves utilized in this analysis (over an appropriate wavelength range). Note that the characteristic resonance structure of the refractive index curve shows the dominance of the single absorption peak at ~3*µ*m in determining the dispersion properties in its wavelength neighborhood (from ~1*µ*m to ~5*µ*m).

## 3.2. Effect of dispersion compensation on the envelope broadening factor

Figure 3 presents plots of the EBF vs. source center wavelength (defined as *c*/*ν̄*, and subsequently referred to as center wavelength) for two distinct cases: when absorption is taken into account, and when it is ignored. The plots were generated at three distinct SRs, under four different conditions: no dispersion compensation; second-order dispersion compensation; both second-and third-order dispersion compensation; and full dispersion compensation. The intermediate conditions were realized by subtracting, respectively, a quadratic and cubic polynomial (as a function of ν) from the expression for *β*, choosing the coefficients (via a numerical fitting algorithm) so that the EBF was minimized. This method showed good immunity to the discrete noise and experimental error associated with the specific *β* vs. *ν* data set used. The plots were obtained by performing the integration of Eq. (2) numerically, substituting in the absorption and refractive index data from the previous subsection. A Gaussian source spectrum, as described in Section 2, was assumed. This spectrum was truncated at 2.5×10^{10}Hz, so that lower frequencies were excluded. (This choice had no bearing on the results in this paper, since the effect of very low source frequencies is entirely suppressed by water absorption, and in plots where water absorption was ignored, the simulated Gaussian sources had negligible power in this extreme region.) In each case, the single-pass propagation distance *z*=1mm.

The EBF rapidly increases as SR is improved, a consequence of Eq. (12). Specifically, since SR is proportional to √*K*, then coefficient *C*
_{s,t} is inversely proportional to ${\stackrel{\mathit{~}}{\mathit{\ell}}}_{\text{FWHM}}^{s+t}$. Therefore, the individual terms in the double-sum of Eq. (11) depend, at minimum, on the fourth power of inverse SR. (Some, of course, may be negative if *s*≠*t*.) This expansion demonstrates that high-order dispersion terms, which are suppressed at high values of SR, contribute significantly to the EBF as the bandwidth is increased. If both second-and third-order dispersion are eliminated, then the most significant term corresponds to (*s*, *t*)=(4,4), which is (in general) entirely negligible at 10-*µ*m SR, but rapidly increases in significance at lower values of SR. The expansion of Eq. (11) does not, of course, strictly apply when the effects of absorption are significant. Nonetheless, it remains useful in determining the relative magnitude of the various dispersion-order effects.

Surprisingly, it is not possible to adequately compensate for dispersion using only second-and third-order correction for either the 3-*µ*m or 10-*µ*m SR cases, if the center wavelength is greater than ~1*µ*m. This demonstrates the dominance of high-order dispersion terms in this region. The fact that effective low-order dispersion compensation *is* possible at 1-*µ*m SR is seemingly in conflict with the claim in the previous paragraph that these orders should have minimal relative impact on the EBF as SR is improved. However, large-scale structure in the *β* vs. *v* function does significantly impact upon the EBF at broad bandwidths, and when it can be approximated by a low-order polynomial, it is possible to see significant EBF reduction by introducing low-order dispersion correction. (The ripples that are clearly visible in the 10-*µ*m SR curves are, in general, described by low-order dispersion terms, and demonstrate the ability to resolve small-scale variations in *β* when narrow bandwidths are used.) In Eq. (11), this effect may be explained in terms of the “coupling” between all the even-order terms, and between all the odd-order terms. For example, merely compensating for second-order dispersion (by effectively setting *β*
_{2} to zero and leaving all other parameters unchanged) will eliminate an infinite number of terms in the EBF expansion (those corresponding to *t*=2). However, for maximum EBF reduction, *β*
_{2} should not be set to zero, but instead chosen so that the net sum of these particular terms is negative: that is, full compensation of *β*
_{2} might be traded off in favor of greater partial compensation of higher even-order dispersion. This is equivalent to approximating large-scale *β*(*v*) structure with a quadratic polynomial.

At both 1-*µ*m and 3-*µ*m SR, whether or not absorption is included, the minimum value of the EBF occurs in the vicinity of the 1-*µ*m wavelength. This point has been identified as the group velocity dispersion zero of water [18]. (It does not appear to have any significance at 10-*µ*m SR, indicating that this attribution is based upon large-scale structure in the *̂* vs. *v* curve, as described above.) Accordingly, it is in this vicinity that second-order dispersion compensation alone has little or no effect upon the EBF. In the absence of any dispersion compensation, the EBF at this point (for 1-*µ*m SR, when absorption is taken into account) is ~6, a figure which is large enough to demonstrate that some form of dispersion compensation is warranted even at the dispersion minimum.

The only case in which third-order dispersion compensation seems to give any significant improvement in the system resolution over merely second-order compensation is at 1-*µ*m SR, as evidenced by the difference between the black and the red curves. For 3-*µ*m SR, the difference is marginal, and for 10-*µ*m SR, it is entirely negligible. This is consistent with the observation that second-order dispersion effects dominate over third-order effects at high values of SR (even granting dispersion-order coupling arguments).

It is possible for absorption effects to actually help mitigate the effects of dispersion, by effectively cutting off the effect of high wavelengths for which the refractive index varies most significantly. In general, when full dispersion compensation is applied, an unattenuated spectral bandwidth provides the best resolution. However, even this is not exclusively true. In the upper-right panel of the figure, we observe that it is possible for absorption effects to distort the source spectrum in such a way that that the EBF decreases below 1, near a center wavelength of ~1.15*µ*m. The near-center wavelengths of the source are attenuated in such a way that the effective bandwidth increases. Although interferogram side lobes are produced, these are accounted for in the RMS definition of EBF. The resolution gain (of ~2%) is negligible, and comes at the expense of a significant reduction in signal power.

## 3.3. Comparison between broadening-factor definitions

As indicated earlier, the EBF is dependent upon the specific definition of the width of the interferogram, particularly when the functional form of the envelope deviates significantly from a Gaussian. The RMS width of the square of the envelope was chosen for mathematical convenience; other suitable choices would be: the RMS width of the (unsquared) envelope; the envelope FWHM; and the inverse-squared fringe visibility of the envelope, where the visibility of the envelope is defined to be its maximum magnitude. (The inverse-square operation arises from power considerations [6].) Figure 4 compares the EBFs due to the other three definitions with that of the previously defined EBF, corresponding to both 1-*µ*m and 3-*µ*m SR, in the presence of absorption. (The blue curves in Fig. 4 are identical to the respective blue curves in the top row of Fig. 3, for the same SRs.)

If the envelope remained Gaussian after propagation, all four of these curves would be identical. This would be the case when second-order dispersion is dominant (and absorption negligible). The greatest deviation between the curves occurs near the aforementioned group-velocity dispersion zero of water (~1–1.1*µ*m), for which much of the interferogram power is concentrated into a long tail, and at higher center wavelengths, for which the distorted spectrum (following absorption) remains centered at around 1.1*µ*m. At 1-*µ*m SR, the presence of the tail tends to magnify both RMS-width EBFs but has less bearing on the other two EBFs. (This effect may be exploited; Hsu *et al.* [24] deliberately introduced dispersion into the reference arm of their system, in order to improve the FWHM resolution.)

## 3.4. Plots of interferograms

In Fig. 5, plots of interferogram envelopes corresponding to some of the cases in the left-hand column of Fig. 3 are presented, both including and ignoring the effects of absorption. The dispersion compensation methods invoked are identical to those described in Subsection 3.2. A number of observations may be drawn from the plots: As previously demonstrated in Fig. 3, even third-order dispersion correction is insufficient to completely mitigate the dispersion broadening effects. Additionally, when merely second-order dispersion correction is applied, the structure of the envelope is characteristic of the (now predominant) odd-order dispersion [16] -a slowly-decaying tail with prominent fringes. Finally, the ability of absorption to actually mitigate the effects of dispersion is clearly evidenced in the right columns of the figure.

## 3.5. Dependence of envelope broadening factor on propagation distance

In each of the results presented thus far, the single-pass propagation distance z was set to 1mm. This was selected as an appropriate imaging depth for OCT, which has been regarded as an “intermediate” depth range modality, occupying a niche between the alternate modalities of confocal microscopy and ultrasound [25]. In this subsection, we present results showing the dependence of the EBF on *z*, when the SR and center wavelength are held fixed. The plots are given in Fig. 6, and the same dispersion compensation criteria have been applied as in Subsections 3.2 and 3.4. The simulated center wavelengths were 0.8,1 and 1.3*µ*m, and the SRs were 1, 3, and 10*µ*m. The effects of absorption were included in all plots.

In the absence of absorption effects, the dependence of the EBF on z would be given by Eq. (11). Since *C*
_{s,t} does not depend on *z*, and *β*_{s}
, *β*_{t}
are both proportional to *z*, then EBF=(1+*Tz*
^{2})^{1/2} for some constant *T*. Therefore, when the EBF is much greater than 1, it is approximately proportional to *z*. This characteristic is clearly observed in the first (and to a lesser extent, the second) column of Fig. 6, which correspond to center wavelengths for which water absorption is modest. In the third column, corresponding to a center wavelength of 1.3*µ*m, the impact of water absorption is observed even at very short propagation distances. For 1 and 3-*µ*m SRs, the EBF increases very rapidly with *z*, when *z* is small. In a sample, this corresponds to considerable resolution degradation at shallow penetration depths. As evidenced by the other plots in the top two rows of Fig. 6, when sample absorption is not significant, at low values of SR, the EBF is approximately linearly dependent on *z*.

## 3.6. Conditions for attaining maximum effective resolution

For any detected OCT signal which has been impacted by dispersion or absorption, we may define the effective resolution (ER) to be the product: ER=SR×EBF. (Recall that SR is defined in the FWHM sense, but EBF is defined in an RMS sense. Therefore, ER is directly comparable to SR, and represents an approximation of the FWHM resolution of the broadened interferogram, to the extent that the EBF and EBF_{FWHM} curves in Fig. 4 are identical.) The ER is defined with respect to optical distance, like SR, and therefore represents effective physical resolution in air. We seek to determine conditions under which it is minimized.

Figure 7 shows plots of the ER vs. center wavelength for multiple SRs under the four different dispersion-compensation conditions considered in previous subsections. The single-pass propagation distance is 1 mm. If full dispersion compensation is applied, 1-*µ*m ER is attainable with a 1-*µ*m SR only at center wavelengths below ~ 0.9*µ*m, and even increasing the bandwidth will not improve upon the ER at higher wavelengths. If merely second-order dispersion compensation is utilized, then the maximum attainable ER, for *any* SR, is 1.8*µ*m, at a center wavelength of 0.55*µ*m. (The required SR is 1*µ*m). If an SR greater than 1*µ*m is used (1.5*µ*m or 2*µ*m), then an ER of less than 2.5*µ*m is attainable up to center wavelengths of ~0.85*µ*m, but ultrahigh SRs of 1*µ*m or less give poor results at these center wavelengths. This situation is significantly improved if third-order dispersion compensation is incorporated into the system. Under this condition, an SR of 1*µ*m gives consistent results below 0.9-*µ*m center wavelength, attaining a minimum ER of 1.5*µ*m at center wavelength 0.8*µ*m. If no dispersion compensation is applied at all, the ER is minimized at around 1–1.1 *µ*m for all SRs (the GVD “zero” of water, identified earlier). The minimum ER (3.6 *µ*m) occurs at a center wavelength of 1.0*µ*m, for a 2-*µ*m SR. At all center wavelengths, an SR of 2*µ*m or greater is required to attain the minimum ER at that wavelength, due to the additional uncompensated dispersion introduced with broader-bandwidth sources.

## 3.7. Limitations of the Gaussian spectral density assumption

The calculations in this paper assumed the use of a source with Gaussian spectral density (in the optical frequency domain). This assumption is, of course, invariably invalid for any real source at some level. In the absence of dispersion and absorption effects, for a given RMS spectral width, the RMS axial point-spread function width is minimized if the spectral distribution is Gaussian. Therefore we might reasonably expect that under most circumstances, the results in this paper represent an optimal scenario: deviations from a Gaussian source would, in general, degrade the resolution. It may be possible to construct a light source that compensates for absorption by specifically shaping the spectrum to counter the sample absorption characteristics, or invoke numerical post-processing to achieve the same result. Resolution advantages would come at the expense of signal power or signal-to-noise ratio. Moreover, this approach would have limited effectiveness in water, since attenuated wavelengths tend to be entirely cut off.

Indeed, some of the effects of using a non-Gaussian source can be observed by merely considering the effects of absorption upon the OCT signal, since the result of it, in general, is to eliminate the low-frequency spectral components that would be absent from the non-Gaussian source. As indicated in Fig. 3, the effects of absorption include partial suppression of the impact of dispersion-induced broadening at these low frequencies (corresponding to high wavelengths). An expansion resembling Eq. (11), with coupling between the various even-order, and odd-order terms, will remain valid provided that the source spectrum remains approximately symmetric (in the optical frequency domain); if it does not, additional coupling will take place between the dispersion components of different parities.

## 4. Conclusion

This analysis demonstrated the significance of sample dispersion and absorption in degrading axial resolution in OCT, with a particular focus on imaging in aqueous media. A multiple dispersion-order expansion was performed, in order to quantify the effects of the various dispersion coefficients, and the impact of absorption was determined by numerical simulations.

The strong absorption peak of water at 1.44*µ*m has the effect of eliminating neighboring optical frequency components which are propagated through water. The resulting amelioration of the source bandwidth results in envelope broadening, even when dispersion has been fully compensated, as is evidenced in the upper-left panel of Fig. 7. As an example, consider the maximum signal propagation distance *z* for which the ER remains less than 1.5*µ*m (an effective physical resolution of ~1*µ*m in tissue) at a common OCT operating center wavelength of 1.3*µ*m (and an SR of 1*µ*m). If up to third-order dispersion compensation is applied, *z* cannot exceed ~10*µ*m, and even for full dispersion compensation, the upper limit for *z* is ~70*µ*m. These observations suggest that in order to achieve ultrahigh-resolution OCT images over substantial depth ranges, it is necessary to choose a center wavelength of less than ~1*µ*m. For up to third-order compensation, the best ER is obtained using a 1-*µ*m SR at a center wavelength of 0.8*µ*m (dependent, but not strongly, on propagation distance). Increasing the source bandwidth is no substitute for dispersion compensation, since the compounded dispersive effects due to the additional wavelengths present yield little or no net resolution improvement. The SR should be no less than 1 *µ*m unless high-order dispersion compensation can be applied, and then only if the center wavelength is less than ~0.9*µ*m.

The choice of center wavelength is further complicated by the fact that sample scattering at short wavelengths is in general much greater than scattering at long wavelengths (in the Mie regime, there is a power law relationship between scattered intensity and inverse wavelength). That is, there is necessarily a trade-off between minimizing both dispersion/absorption effects and sample scattering effects, at least when imaging through water-based media. This trade-off has been recognized in the OCT literature [26], [27]; ideal center wavelengths in the range ~1.3–1.8*µ*m were noted to be local minima in the water absorption curve, in order to maximize penetration depth. The achievable OCT resolution may also be limited by sensor sensitivity [16, p. 517].

It may be possible to utilize the “information” provided by dispersion and absorption effects. Liu *et al.* [15] contend that the quantification of these effects through a tissue depth scan may be sufficient to provide some diagnostic capability. However, their experiments were performed using a low-resolution (~12*µ*m) source, for which only second-order dispersion was significant. The additional structure inherent in the axial point-spread function due to multiple dispersion orders at ultrahigh resolutions may provide even more discriminatory capacity for identifying tissue types, but this is not yet clear.

In conclusion, if an axial resolution target approaching 1-*µ*m is to be attained in OCT imaging, then it is necessary to compensate for multiple dispersion orders, no matter how great the source bandwidth. Moreover, the source center wavelength should not be located in the vicinity of a sample absorption peak, so that the entirety of the spectrum is utilized in generating the signal.

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