## Abstract

A new formulation describing the interference term of a two beam interferometer with unequal Gaussian spectra propagating in different dispersive media is provided by defining a composite standard deviation and a composite center frequency of the interfering spectra. This formulation is generalized to arbitrary spectra by decomposing each spectrum into a linear composition of Gaussian distributions. The effective phase and group delays indicate the effect of the unequal spectral distributions and the dispersive media. An effective coherence length is derived, different than the coherence lengths of the interfering fields. The accuracy of the new formulation is proven experimentally by using optical coherence tomography systems.

© 2006 Optical Society of America

## 1. Introduction

Since Young’s double-slit experiment and Michelson’s interferometer carried out in 1807 and in 1881[1] respectively, coherence measuring technology has been broadly developed and applied to high resolution measurements[2–8]. Since the first optical coherence tomography (OCT) experiments based on the Michelson interferometer for biological tissue imaging in 1991[9], OCT technology has shown excellent resolution of around a few micrometers. OCT for noninvasive three-dimensional imaging in the biomedical sciences has been extensively investigated [10–14].

In an OCT system, axial resolution primarily depends on the coherence length of the light source, while the lateral resolution depends on the numerical aperture of the optics system[15]. Following this basic principle, researchers are expending considerable effort to improve the axial resolution of OCT by pursuing improved light sources with broader bandwidths to produce greater axial resolution[16–19]. However, the value of this is only fully realized if the spectra from the two reflection arms of the interferometer are identical and not significantly altered by interaction with the optical system or the sample. The experimental results from an actual interferometer equipped with a broadband source (such as those reported in this letter) may show axial resolution significantly different from theoretical expectations in which identical spectra[15] are assumed.

In a standard OCT system, the broad-band optical beam illuminates two separate pathways; the reference arm or delay line, and the sample arm. Unlike many interferometers, such as in gravitational wave detection where the two arms of the interferometer are typically designed identically [20], the two arms of an OCT interferometer are usually different. This is especially true for applications in biomedical imaging. For instance, the scanning optics in the sample arms for bench-top OCT and for endoscopic OCT contain very different optical components both different from the optical delay line in the reference arm. In practice, each optical component, including biological tissue, will alter the spectrum of the beam illuminating it, resulting in differing spectra in each arm of the interferometer. The spectra can differ in center wavelength, bandwidth, amplitude and phase. In this letter, we provide a general model to describe the interferogram of a two beam interferometer in which the spectra returning from the two interferometer arms have experienced different filtering functions resulting in different center wavelengths, bandwidths, amplitudes and phase.

## 2. Theory derivation

Superluminescent diodes are commonly used as a light source in low coherence interferometers and usually operate at room temperature. According to basic optical gain analysis, the fundamental linewidth of this kind of light source is inhomogeneously broadened by crystalline defects, which is generally modeled by a Gaussian distribution [21]. However, the reality is that no real broadband source spectrum and certainly no such spectrum passively shaped by optical components or biological tissues, conforms very closely to a Gaussian profile. State of the art extremely broadband sources based on supercontinuum generation are especially divergent from the Gaussian distribution. However, it is practical to assume that the local distribution of such a non-Gaussian spectrum can be described in terms of a Gaussian function. Therefore, in this more general approximation, a non-Gaussian distribution can be decomposed into a linear combination of Gaussian distributions [22]. This method has been used to describe inhomogeneous velocity distributions [23, 24]. Based on this assumption, we write a general plane wave as a sum of Gaussian functions,

And the intensity of the wave in Eq. (1.1) is written as,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.6em}{0ex}}=\sum _{i=j}^{n}{a}_{i}^{2}{e}^{-\frac{{\left(\omega -{\omega}_{i}\right)}^{2}}{2{\sigma}_{i}^{2}}}+\sum _{i\ne j}^{n}{a}_{i}{a}_{j}{e}^{-\frac{{\left(\omega -{\omega}_{i}\right)}^{2}}{{4{\sigma}_{i}}^{2}}-\frac{{\left(\omega -{\omega}_{j}\right)}^{2}}{{4{\sigma}_{j}}^{2}}}{e}^{-\Delta {\varphi}_{\mathit{ij}}}.$$

Where, *a _{i}*,

*σ*,

_{i}*ω*and

_{i}*Δϕ*are respectively the amplitude ratio, standard deviation, the center frequency of

_{ij}*i*Gaussian distribution and the phase mismatch between the

^{th}*i*and

^{th}*j*components. The subscript

^{th}*j*represents the

*j*component of the complex conjugate wave. The first and second summations in Eq. (1.2) refer to the DC and interference parts respectively. The following analysis is derived using two Gaussian distributions, but can be generalized utilizing this assumption.

^{th}It is often misleading, especially in high-resolution OCT, to assume that the spectra returning from the sample and reference arms of an interferometer are identical to the spectrum of the light source. To model the interferogram and derive an expression for an effective coherence length, we assume that the altered interfering fields have Gaussian spectral distributions, and the center frequencies of the reference and sample fields are ω_{1} and ω_{2} respectively. The standard deviations of the fields are σ_{1} and σ_{2}, respectively. Based on this asymmetrical Gaussian distribution assumption, the correlation function in the interference term should be a cross correlation function rather than an autocorrelation function. The interference term of the two beam interferometer can be written in the frequency domain as,

Where A and Δ*ϕ* are the scale constant and the phase mismatch, while the *S*(*ω*, *Δϕ*) is the cross spectral density.

The correlation function described above can be experimentally measured. However, an analytical formulation describing the phenomenon is useful for understanding the mechanisms and predicting the effects. In this study, we introduce a modified cross correlation function, which will clearly indicate the effect of spectral filtering. By defining a composite center frequency and a composite standard deviation, Eq. (2) can be rewritten in the following,

Where the composite standard deviation *$\overline{\sigma}$* and the composite center frequency *$\overline{\omega}$* are respectively:

From these expressions, it can be noted that the Hadamard-product of two Gaussian functions is another Gaussian function. Thus, it can be shown that for the case in which the two interfering spectra are linear combinations of Gaussian distributions, the interference function will also be a linear combination of Gaussian distributions. The composite standard deviation *$\overline{\sigma}$* only depends on the two initial standard deviations and is independent of the separation of the two spectra, while the composite center frequency *$\overline{\omega}$* depends not only on the separation of the two spectra, but also on their standard deviations. Importantly, the first exponential factor in Eq. (3) describes the attenuation of the correlation function due to difference between the two spectra.

The phase mismatch Δϕ in Eq. (3) contains a group delay time and a phase delay time, which determine the envelope of the interferogram and the frequency of the fringes. A continuation of the approach used above may be used to provide a clear, while more general, view of the group delay, phase delay, and effective coherence length. Assuming the fields propagate in two different dispersive media and the propagation constants are expanded at the different center frequencies [25], the phases, *ϕ*
_{1}(*ω*) and *ϕ*
_{2}(*ω*), accumulated in double-pass as a function of frequency for the reference and sample fields, and the phase mismatch Δ*ϕ*(*ω*) can be expressed as,

Where *l*
_{1} and *l*
_{2} are the propagation distances in the reference and sample paths, respectively, and *L* is the distance over which second order dispersion is encountered (assumed to be equal for both). For generality, we keep the dispersion coefficients β_{1}, β_{1}′ and β_{1}″ for the first beam distinct from β_{2}, β_{2}′ and β_{2}″ for the second beam. To derive group and phase delays, we substitute Eq. (4) and Eq. (5) into Eq. (7) and then rewrite it as,

$$+L\left\{{\beta}_{1}^{\u2033}\left({\omega}_{1}\right){\left(\omega -{\omega}_{1}\right)}^{2}-{\beta}_{2}^{\u2033}\left({\omega}_{2}\right){\left(\omega -{\omega}_{2}\right)}^{2}\right\}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}$$

Where Δ*ω*=*ω*
_{1}-*ω*
_{2}, the phase delay Δ*τ _{p}* and the group delay Δ

*τ*are defined as,

_{g}It should be noted that these equations simplify to the familiar formulae used for OCT calculations in the case that *β*
_{1}(*ω*
_{1})=*β*
_{2}(*ω*
_{2}) and *β*′_{1}(*ω*
_{1})=*β*′_{1}(*ω*
_{2}) [26]. Through advantageous grouping and redefinition, Eq. (8) may be written in the more convenient form,

Where Δ*β*″=*β*″_{1}(*ω*
_{1})-*β*″_{2}(*ω*
_{2}) is the mismatch of the group velocity dispersion (GVD), and the effective phase delay Δ*τ*′_{p} and the effective group delay Δ*τ*′_{g} are defined by,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.4em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-\frac{L}{\varpi}\left[\Delta {\beta}^{\u2033}{\varpi}^{2}-{\beta}_{1}^{\u2033}\left({\omega}_{1}\right){\omega}_{1}^{2}+{\beta}_{2}^{\u2033}\left({\omega}_{2}\right){\omega}_{2}^{2}\right]+\frac{\Delta \omega}{\varpi}{\overline{\sigma}}^{2}(\frac{{\beta}_{1}^{\prime}\left({\omega}_{1}\right){l}_{1}}{{\sigma}_{2}^{2}}-\frac{{\beta}_{2}^{\prime}\left({\omega}_{2}\right){l}_{2}}{{\sigma}_{1}^{2}}),$$

Equations (12) and (13) indicate that both the phase and group delays are affected by not only the unequal spectral distributions, but also the mismatch of the GVD. It can be seen that Eq. (12) and Eq. (13) degenerate under simplifying assumptions, for instance, Δ*τ*′_{p}=Δ*τ _{p}* and Δ

*τ*′

_{g}=Δ

*τ*when

_{g}*ω*

_{1}=

*ω*

_{2}and Δ

*β*″=0. Since in general, the media in the two paths are different, Δ

*β*″ is not necessarily zero for

*ω*

_{1}=

*ω*

_{2}.

Finally, the cross correlation function of an OCT interferometer interrogating dispersive media can be written in a more detailed form. Substituting Eq. (11) into Eq. (3) and assuming scale constant *A*=1, we have the final formula of the cross spectral density,

The form of Eq. (14) is similar to the cross spectral density of two waves with identical spectral distributions described, for example, in Eq. (25) of Chapter 2 of Ref. [26]. The difference here is the attenuation term at the beginning, and the use of effective phase and group delay, as defined above.

Thus, the definition of the composite central frequency and effective phase and group delays generalizes previous expressions for two-beam interference in dispersive media [25, 26]. In the case of time domain OCT, a single point detector is used to collect the interference signal, so the signal current *I* should be the contributions of all the wavelengths under the interfering spectra. The integral is only performed over the last two factors in Eq. (14) because the first two factors are constants. After integrating for Eq. (14) over frequency, we obtain the modified expression for interferogram intensity:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\u2022{e}^{-\frac{{{\Delta {\tau}^{\prime}}_{g}}^{2}}{2[\frac{1}{{\overline{\sigma}}^{2}}+{\overline{\sigma}}^{2}{\left(\Delta {\beta}^{\u2033}2L\right)}^{2}]}}{e}^{-j\left\{{\varpi \Delta {\tau}^{\prime}}_{p}+\frac{1}{2}{\mathrm{tan}}^{-1}\left({\overline{\sigma}}^{2}\Delta {\beta}^{\u2033}2L\right)-\frac{{{\Delta {\tau}^{\prime}}_{g}}^{2}{\overline{\sigma}}^{2}\left(\Delta {\beta}^{\u2033}2L\right)}{2[\frac{1}{{\overline{\sigma}}^{2}}+{\overline{\sigma}}^{2}{\left(\Delta {\beta}^{\u2033}2L\right)}^{2}]}\right\}}\}.$$

The third factor in Eq. (15) represents the envelope of the interferrogram. From here, the value of the effective group delay corresponding to the position of the half of the maximum of the envelope, can be found to be:

Eq. (16) indicates that second order dispersion mismatch Δ*β*″ extends the envelope.

Assuming that GVD is matched in the two arms of the interferometer, i.e. Δ*β*″=0, we can significantly simplify Eq. (15) to:

Equations (15) and (17) are similar in form to previous descriptions of interferometric signal, e.g. [26], but are more general. Equations (15) and (17) reveal the attenuation of fringe visibility due to the difference between the spectral distributions of the two interfering fields and the effect on the group and phase delays of the two fields propagating in different dispersive media. Further, substituting Eq. (4) and Eq. (5) into Eq. (16) and assuming Δ*β*″=0, we obtain the effective coherence length *l*′* _{c}* as a function of the two individual field distributions, or of the coherence lengths

*l*

_{c1}and

*l*

_{c2}of the reference and sample fields,

$$=\sqrt{\frac{{\left({l}_{c1}\right)}^{2}+{\left({l}_{c2}\right)}^{2}}{2}}.\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.6em}{0ex}}$$

The coherence length will be longer if Δ*β*″≠0 according to Eq. (16).

## 3. Experiments

Two experiments were designed to verify the accuracy of the above formulations using OCT imaging systems. In the first experiment a filter was placed in the reference arm of the interferometer to cause extreme spectral shaping. The light source had a center wavelength of 1292 nm with a bandwidth of 34 nm while the filter had a bandwidth of 6.4 nm at the center wavelength of 1313.6 nm. With a mirror in the sample arm, we measured the interferogram with and without the filter in place. The signal was averaged 100 times. The envelopes of the measured interferograms are shown in Fig. 1. The interferogram measured without the filter corresponds essentially to the self-coherence function, and the full-width at half-maximum (FWHM) of this function corresponds to the coherence length of the light source, 19 µm, indicated by the cross mark in Fig. 2. The interferogram measured with the filter in place is significantly broader, as expected, with a FWHM of 86.5 µm, indicated by the down triangle in Fig. 2. This effective coherence length can also be computed from Eq. (18). Substituting the data from the experiment, we predict an effective coherence length of 85.5 µm with the filter in the reference arm, while the coherence length of the light source is predicted to be 21.6 µm. This broadening of the interference function and therefore the resultant axial resolution degradation calculated from the formulae presented here closely agrees with the experimental result. Figure 2 shows the broadening of the interferogram resulting from narrowing the spectrum in one arm of the interferometer (solid line) together with the broadening resulting from narrowing of the spectrum of the light source (dashed line) as predicted by Eq. (18). The parameters used in the prediction match the conditions of the experiment presented in Fig. 1. It is notable that while narrowing the spectrum in one arm broadens the interferogram, this broadening is not as severe as that due to narrowing of the source spectrum. The amplitudes in Fig. 1 are normalized. The measured ratio of the peak signal without filter to the peak signal with filter is about 32. We can attribute this observed attenuation to two factors. The first factor is the transmission loss of the filter. The double-pass transmission of the reference beam by the filter was measured as 5.5 %. The second attenuation factor results from the separation between the center wavelengths of the filtered (reference) light and the unfiltered (sample) light. We input the two center wavelengths and the two bandwidths into the second term of Eq. (15), resulting in an attenuation factor of 0.58. Multiplying these attenuation factors results in a net attenuation factor of 0.0318, corresponding to a peak signal ratio of 31.5. This expected signal attenuation agrees closely to the experimental measurement, and is significantly greater than that predicted by the filter transmission loss alone.

In the second experiment, the effective coherence length Eq. (18) was evaluated by comparing the coherence lengths obtained in different ways. We used a commercial optical spectral analyzer (OSA) to directly measure the spectrum of an OCT light source and the spectra returned from the sample and reference arms of a Fourier-domain OCT (FD-OCT) system with a sample beam scanner[27, 28]. A non-linear single Gaussian Least-square fit was used to find the equivalent center wavelengths for the spectra measured above as shown in the first row in Table I. We estimated the individual coherence lengths at FWHM of the measured spectra by using the inverse Fourier transformation (IFFT). The different coherence lengths shown in the second row in Table I indicate slight spectral filtering. Substituting the individual coherence lengths of reference and sample arms into Eq. (18) above, the predicted effective coherence length of the interference signal was calculated and shown in the third row in Table I. It was found to be in good agreement with the results shown in the last row in Table I from the interferogram measured using FD-OCT. These results were analyzed by using Eq. (18) assuming parameters matching the conditions of the experiment presented in Table I. In Fig. 3, the curves labeled with star, plus and solid dot symbols were calculated assuming that one of the two spectral widths was fixed while the other varied, and that the two center wavelengths in Eq. (18) were not necessarily the same. The curve by the plus symbol was calculated by assuming two different center wavelengths of 1319.4 nm and 1322.3 nm and a fixed bandwidth of 56.36 nm at 1322.3 nm and a varied bandwidth at 1319.4 nm. The curve marked by the star symbol assumes the spectra are both centered at 1319.4 nm while the curve marked by the solid dot symbol assumes that both spectra are centered at 1322.3 nm. When the ratio is increased (the spectral width narrowed in one arm), the effective coherence length broadens, as indicated by the curves. For the two curves labeled with cross and triangle symbols, two identical spectra were assumed, representing the coherence length broadening due to narrowing of the source spectrum. As was shown in Fig. 2, narrowing the spectrum of one beam in the two-beam interferometer broadens the effective coherence length more slowly than the coherence length of the light source is broadened by equivalently narrowing the light source.

## 4. Conclusion

In conclusion, we have obtained a generalized expression for two-beam interference applicable to OCT imaging systems where different sample and reference paths result in unequal interfering fields. The formulation also includes the effects of unequal dispersive media on the interferogram. By defining the composite center frequency, composite standard deviation, effective phase and group delays and effective coherence length, we express the generalized formulation in the form familiar for interference of fields with identical spectral distributions. Our experiments are consistent with our theoretical predictions and demonstrate that the degradations of both the amplitude and the resolution of an OCT signal associated with the spectral filtering can be accurately described by the reported theory. In practice, the spectra from OCT reference and sample arms are generally not identical because of various factors such as differing geometry of the optics in each arm and dispersion and wavelength-dependent loss in the optics and sample. The reported results are specifically relevant to OCT interferometry, but can be widely applied to describe any two beam interferometer.

## Acknowledgments

The research was supported in part by the National Institute of Health (National Institute of Biomedical Imaging and Bioengineering) R03EB004044, and (National Cancer Institute) R01CA114276. Zhilin Hu’s email: zhilin.hu@case.edu

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