Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

Figure 2 The Michelson interferometer.

from the reference and sample mirrors is recombined at the same beamsplitter and incident on a detector. The reference and sample mirrors are positioned at distances lR and lS, respectively, from the beamsplitter. If the light source is perfectly coherent (i.e., monochromatic), then reflection from the reference and sample mirrors produces a sum of two monochromatic electric field components ES and ER at the detector. These fields may be expressed by the phasors

ER ¼ AR exp ½ jð2 RlR !tÞ& and ES ¼ AS exp ½ jð2 SlS !tÞ& ð1Þ

where ! is the optical frequency of the light source and is the propagation constant. The factors of 2 multiplying the propagation constants R and S arise from the round-trip propagation of light to and from the reference and sample mirrors.

Note that ER and ES represent the field components in the interferometer after reflection and recombination at the beamsplitter, which eliminates the necessity of considering the split ratio and phase shift of the beam splitter. In general, the timeaveraged photocurrent I at the detector is given by

where is the detector quantum efficiency, e is the electronic charge, h is the photon energy, and 0 is the intrinsic impedance of free space. For monochromatic fields, Eq. (2) can be written

describes the variation of the photocurrent with the positions of the reference and sample mirrors. In free space, the propagation constants are equal for the reference

where l ¼ lS lR is the mismatch in distance between the reference and sample beam paths. Equation (5) shows that the photocurrent contains a sinusoidally varying term representing the interference between the reference and sample fields. The interference has a period of =2 relative to the length mismatch l.

Because all of the interference information is contained in the real part of the cross-spectral term ESER, subsequent analysis of interferometry in the case of low coherence light source will focus on calculating this term only.

2.2.2Interferometer with Low Coherence Light

A low coherence light source consists of a finite bandwidth of frequencies rather than just a single frequency. The analysis in Section 2.2.1 can be extended to account for partially coherent light by integrating the cross-spectral term ESER over the harmonic content of the light source. We allow the reference and sample fields to be functions of frequency:

ERð!Þ ¼ ARð!Þ expfj½2 Rð!ÞlR !t&g

ð6Þ ESð!Þ ¼ ASð!Þ exp j½2 Sð!ÞlS !t&

The interference signal at the photodetector is proportional to the sum of the interference due to each monochromatic plane wave component, according to

If the sample and reference arm fields have the same spectral components as the light source (i.e., if the reflectors in each arm and the fiber beamsplitter are spectrally uniform), then Sð!Þ is essentially equivalent to the power spectrum of the light source. ð!Þ expresses the phase mismatch at the detector of each frequency component. We will show that Eq. (7) expresses a fundamental inverse Fourier transform relation between the source spectrum and the detector photocurrent.

2.2.3Nondispersive Medium

Consider the case where the sample and reference arms consist of a uniform, linear, nondispersive material. Let the spectrum of the light source Sð! !0Þ be bandlimited with a center frequency of !0. We assume that the propagation constants

in each arm are the same and rewrite them as first-order Taylor expansions around the center frequency:

Then the phase mismatch ð!Þ in Eq. (9) is determined solely by the length mismatch l ¼ lS lR between the reference and sample arms through

where the phase delay mismatch p and the group delay mismatch g are defined as

Thus, vp ¼ !0= ð!0Þ corresponds to the center frequency phase velocity and vg ¼ 1= 0ð!0Þ is the group velocity. Equation (12) is basically a statement of the familiar Wiener–Khintchin theorem: The autocorrelation function is equal to the inverse Fourier transform of the power spectral density. Equations (12)–(14) show that the interferometric term of the photocurrent consists of a carrier and an envelope. The carrier oscillates with increasing path length mismatch 2 l at a spatial frequency of ð!0Þ. The envelope, which determines the axial point spread function of the interferometer, is essentially the inverse Fourier transform of the source power spectrum Sð! !0).

Gaussian Power Spectrum and Nondispersive Medium

Assume that the light source has a Gaussian power spectral density given by

and where !0 is defined as the center frequency and 2 ! is the standard deviation power spectral bandwidth (radians per second). Substitution of this power spectrum into Eq. (12) gives the interferometric photocurrent.

"#

where the real f g notation has been discarded for convenience. The photocurrent contains a Gaussian envelope with a characteristic standard deviation temporal width 2 (seconds) that is inversely proportional to the power spectral bandwidth:

Note that ! ¼ 1 (i.e., the time–frequency uncertainty relation is minimized for a Gaussian waveform). The envelope falls off quickly with increasing group delay mismatch g and is modulated by interference fringes that oscillate with increasing phase delay mismatch p. Thus, Eq.(17) defines the axial resolving properties of the OCT system. The detector sees interference fringes only when the reference and sample arm lengths are matched, so the group delay mismatch falls within the Gaussian envelope:

The standard deviation axial resolution, or width of the axial point spread function (i.e., the width of the Gaussian envelope in units of length mismatch l) is, from Eqs. (14), (18), and (19),

For propagation in free space, both the phase velocity and group velocity equal the speed of light c, and Eq. (20) reduces to

showing that the axial resolution is inversely proportional to the bandwidth.

For measurements of bandwidth and resolution, a full-width at half-maximum

(FWHM) criterion is more convenient than a standard deviation measure. For a p

Gaussian with standard deviation , the FWHM equals 2 2 ln 2. Thus, for the interferometer in free space, the FWHM resolution lFWHM is related to the FWHM wavelength bandwidth for a Gaussian source by

!

where 0 is the center wavelength.

2.2.4Group Velocity Dispersion

Group velocity dispersion (GVD) causes different frequencies to propagate with nonlinearly related velocities. A short pulse in dispersive media will broaden if significant dispersion is present. In analogy to the short-pulse case, the interferometric autocorrelation (i.e., the axial PSF) will also broaden if there is significant GVD mismatch between reference and sample arms. Both the fiber optics and the sample may have significant GVD.

To include GVD in the analysis, the propagation constants R and S are Taylor expanded to second order around the center frequency !0:

We assume that a GVD mismatch exists in a length L of the sample and reference paths. The frequency-dependent phase mismatch from Eq. (9) is then

where l is defined as before and 00ð!0Þ ¼ S00ð!0Þ R00ð!0Þ is the GVD mismatch between the reference and sample paths. Note that only the difference in GVD between the two interferometer arms enters Eq. (24). Thus, the deleterious effects of dispersion may be decreased by equalizing the GVD in the two interferometer arms.

Inserting ð!Þ into the propagation equation Eq. (7), gives the photocurrent

where the phase delay mismatch p and group delay mismatch g have been defined in Eqs. (13) and (14). Again the real f g notation has been dropped for convenience. The GVD mismatch multiplies the source power spectral density Sð! !0Þ in a frequency-dependent quadratic phase term. The interferometric signal looks like a short pulse, with Fourier transform Sð! !0Þ, which propagates through a length L of dispersive medium with second-order dispersion equal to the difference in GVD between the interferometer arms. Thus, just as a short pulse broadens and chirps after propagation through a dispersive medium, the interferometric signal should also broaden and chirp due to GVD mismatch in the two interferometer arms.

Gaussian Power Spectrum and Dispersive Medium

To establish the analogy further, assume that the source has a Gaussian power spectral density according to Eq. (15). Then, after substitution into Eq. (25), we obtain a modulated interferometric signal with a complex Gaussian envelope described by

"#

where is the standard deviation half-width from Eq. (18). The characteristic width of the axial point spread function in the presence of dispersion, ð2LÞ, is a complex parameter that depends on both the round-trip length of GVD mismatch 2L and via

The real and imaginary components of 1= ð2LÞ2 describe the broadening and chirping, respectively, of the interferometric signal and are

48 Hee

The broadening factor becomes appreciable when the magnitude of the dispersion

parameter critical becomes greater than the unbroadened standard deviation temporal width . For a typical fused silica fiber at 800 nm, 00 ¼ 350 fs2=cm. For a

typical resolution achieved by OCT, lFWHM ¼ 10 m, implying ¼ 28 fs. Thus, dispersive broadening becomes a factor if the fiber arm lengths are mismatched by at least L 1 cm.

The chirping of the interferometric signal with increasing path length mismatchl can be described by differentiating the phase in the exponent of Eq. (26), leading to

where k describes the spatial frequency of the interference fringes versus the distance measured l. For example, in the positive dispersion mismatch regime 00ð!0Þ > 0, as the reference arm path length is increased, l decreases, the wavenumber k increases, and interference fringes occur at the detector more often. It is important to note that GVD changes the phase but not the bandwidth of the interference signal.

Dispersion mismatch also degrades the peak height of the interferometric envelope, which reduces the system dynamic range. The degradation in the photocurrent amplitude is described by the multiplicative factor

The reduction of the signal amplitude peak scales as the square root of the broadening. Assuming that the dynamic range is measured in terms of reflected optical power ,which is proportional to photocurrent power, then the loss in dynamic range scales linearly with the broadening.

2.2.5Non-Gaussian Light Sources

If the light source has a non-Gaussian spectrum, then several abnormalities in the axial point spread function may occur that can be predicted using the inverse Fourier transform relation in Eq. (12).

Spectral Modulation

Modulation or ripples in the light source spectrum can cause echoes to appear in the axial point spread function. Assume that we may write the modulated spectrum Sm ð!Þ as a function of an ideal, unmodulated spectrum Sð!Þ in the form

where 0 is the center wavelength, M is the ratio of the peak-to-peak modulation height to the average spectral height, and m is the approximate modulation period measured in the wavelength domain. Then by the inverse Fourier transform relation in Eq. (12) for a nondispersive medium, the photocurrent will consist of a main peak situation at l ¼ 0 and two side peaks or echoes with a height of M=r relative to the main peak situated at

Equation (34) reduces to l ¼ 20=2 m in free space.

Blindness

Blindness occurs if the tails of the axial point spread function are sufficiently broad to prevent the observation of a weak reflection placed next to a strong reflection. The effect is a function of the light source spectrum according to the inverse Fourier transform relationship in Eq. (12). If the source has a Gaussian spectrum, then the axial point spread function is also Gaussian and its tails decay exponentially with the squared path length mismatch between reference and sample arms. However, if the source is non-Gaussian, then significant tails in the axial point spread function may occur, preventing the detection of a weaker reflector nearby. A Lorentzian line shape, for example, will only decay exponentially with the linear path length mismatch.

2.3DETECTION ELECTRONICS

Section 2.2 showed that the interferometric photocurrent consists of an axial point spread function envelope superimposed on a carrier frequency. An electric circuit is used to extract the envelope of the detected signal in a manner similar to the demodulation of AM radio signals. In the actual OCT system, the reference mirror oscillates rapidly and travels at a velocity vs during scanning. Therefore, the axial point spread function, which is a function of the path length mismatch l, is mapped into a function of time by the change of variables l ¼ vst (where time t ¼ 0 occurs when the two path lengths are matched, and t increases as the reference mirror is translated farther from the beamsplitter). From Eq. (13), the resulting electric signal has a carrier frequency of

where !0 is, as before, the center frequency of the source spectrum, and the tilde denotes the electronic frequency counterpart of the corresponding optical frequency. If free space propagation is assumed, then Eq. (35) simplifies to