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

62 Hee

To determine the output signal power, we propagate the interferometric signal through the detection and demodulation electronics. For simplicity, we will ignore

the degradation in signal power caused by group velocity dispersion. Let PS and PR be the time-averaged optical powers returning from a perfectly reflective sample and the reference mirror, respectively, after recombination at the beamsplitter. From Eq. (69), the peak signal current after photodetection is given by the correlation of the

where is the quantum efficiency of the photodetector, e is the electronic charge, and hv is the photon energy. The transimpedance amplifier converts the current i into a voltage v ¼ iR with a gain of R. If we assume for simplicity that the signal power is preserved through bandpass filtering and demodulation, then the signal power as defined above is

The noise variance is given by 2q NEB where q is the amplitude of the white noise input to the bandpass filter defined in Eq. (73) and NEB is the noise equivalent bandwidth defined in Eq. (78) for demodulation by mixing. In the shot noise limit, the white noise amplitude q is dominated by the shot noise from light reflected from the reference mirror. Section 2.6.1 will discuss the design considerations so that this limit can be achieved in practice. In the shot noise limit, the noise variance after filtering and demodulation is

Therefore, the signal-to-noise ratio in the shot noise limit is obtained by substituting Eqs. (80) and (81) into Eq. (68), resulting in

In the shot noise limit, the dynamic range does not depend on the reference arm

reflective sample and is inversely proportional to the noise equivalent measure of the detection bandwidth. Section 2.6.2 will discuss how Eq. (82) leads to design tradeoffs between power, image acquisition time, and dynamic range.

2.5.6Signal-to-Noise Ratio for Demodulation by Envelope Detection

Demodulation by envelope detection can be modeled as square-law detection followed by low-pass filtering as described in Section 2.3.4. Because the square-law detector is a nonlinear device, the output noise power will in general depend on both the input signal and the input noise power. Thus, a simple expression for the NEB that depends only on the input noise cannot be derived as in the case of demodulation by mixing. However, a fairly involved calculation eventually leads to a simple expression for the SNR at the output of the envelope detector in relation to the SNR at its input.

The results of this derivation can be summarized as follows. Let sðtÞ denote the signal at the output of the bandpass filter and the input to the square-law detector. We assume that sðtÞ has the form of a randomly amplitude modulated sine wave:

where pðtÞ is the modulating envelope (or axial reflectivity profile as a function of depth), !D is the electronic carrier frequency determined by the Doppler shift of the scanning reference mirror, and is a uniformly distributed random phase. In this case, with a low-pass filter bandwidth that is approximately twice the signal bandwidth, the SNR at the output of the square-law detector and low-pass filter can be written as

SNRout ¼ ðSNRinÞ2 1 þ 2SNRin

where

Efp4ðtÞg¼ E2fp2ðtÞg

and SNRin is the SNR at the input of the square-law detector. Note that is a function of only the form of the probability distribution of the signal envelope. For example, if pðtÞ can be modeled as a Gaussian random variable, then ¼ 3 exactly.

Equation (84) demonstrates that for an input SNR that is substantially greater than 1 the output SNR is linearly dependent on the input and is given by

Aside from the factor =2, demodulation by square-law detection is similar to demodulation by mixing except that only the NEB of the bandpass filter and not that of the final low-pass filter appears in the expression for the SNR. Although the derivation for Eq. (84) does assume that low-pass filter bandwidth is approximately twice the signal bandwidth [because the bandwidth of s2ðtÞ is twice the bandwidth of sðtÞ], the exact form of the final low-pass filter is less important in the SNR for square-law detection than it is for demodulation by mixing. In addition, for square-law detection, the carrier cosð!Dt þ Þ is not required to have a constant phase (i.e., is a random variable). This property is advantageous when the velocity of the reference mirror is not perfectly linear. In this case, square-law detection is relatively insensitive to small variations in the phase of the Doppler shift or carrier frequency, which would otherwise cause problems for demodulation by mixing.

2.6DESIGN ISSUES

2.6.1Design for Shot Noise Limited Sensitivity

Equation (73) shows that the noise after photodetection and amplification can be represented as a zero-mean, white, WSS stochastic process that contains contribu-

tions from shot noise, thermal noise, and RIN and/or ASE noise. Quantum-limited operation is attained only when the shot noise dominates the other sources of noise. In the shot noise limit, described by Eq. (82), the detection sensitivity is approximately two photons per resolution element.

It is possible to design the system for shot noise limited sensitivity by using a spectrum analyzer to measure the noise vnðtÞ at the output of the transimpedance amplifier. Because the spectrum analyzer combines positive and negative frequencies, it effectively measures [from Eq. (73)] the positive frequency power spectral density

which contains, from left to right, the white noise contributions from shot noise, RIN and/or ASE noise, and thermal noise.

The transimpedance gain R and the reference arm power hvi are chosen such that the shot noise term is greater than the RIN and thermal noise. To ensure that shot noise eclipses the thermal noise, we require 2ehviR > 4kTR, or

This limit defines the absolute minimum reference arm power. Thus, in the absence of RIN or ASE noise, whenever the DC output of the reference arm power is greater than 50 mV, the system will be shot noise limited. The gain R must be chosen so that the shot noise dominates the RIN, or

The upper limit of R is determined by the transimpedance amplifier stability and rolloff frequency in Eq. (47). Ideally the gain R should be as large as possible and the reference arm power hvi should be as small as possible to minimize the RIN and ASE noise (Section 2.4.3).

In practice, to construct the detector circuit, R is chosen before hvi. Because the RIN parameter is usually unknown, given R, the optimal value of hvi must be determined experimentally. The procedure is to successively attenuate the reference arm power hvi and examine the noise spectral density on the spectrum analyzer until the white component equals the predicted shot noise value 2ehviR. Attenuation of the reference arm intensity reduces the RIN as the square hvi2 of the intensity but affects the shot noise component only linearly with hvi. If hvi needs to be decreased below the thermal noise limit of 0.05 V, then the amplifier gain R should be increased and the procedure repeated. If increasing R places the transimpedance amplifier roll-off frequency below the signal bandwidth, then other methods of reducing the RIN, such as dual-balanced detection, need to be considered.

2.6.2Trade-Off Between Resolution, Power, Speed, and Sensitivity

The four fundamental design issues for OCT are the optical power, acquisition speed, signal-to-noise ratio, and axial resolution. Equation (82) establishes the linear relationship between the signal-to-noise, optical power incident on the sample, and noise equivalent bandwidth of the electronics:

The NEB, however, is essentially equal to the electronic bandwidth f~ used for detection of the interferometric signal. The electrical bandwidth is linearly related to the reference mirror scanning velocity vs and the wavelength bandwidth of the light source by Eq. (44). Therefore, we have

The axial resolution l is inversely proportional to the spectral bandwidth of the light source, as seen in Eqs. (21) and (22), so we also may write

By combining Eqs. (91) and (93), we obtain the fundamental relationship between the four design parameters:

Equation (94), which is linear in all its parameters, expresses the basic idea that ‘‘you can’t get something for nothing’’ and allows the major design issues to be easily considered in relation to one another.

The limitations expressed by Eq. (94) may be intuitively understood with an alternative interpretation of Eq. (82). The minimum detectable reflectivity Rmin is equal to the reciprocal of the SNR. Equation (82) can then be rewritten as

In Eq. (95), the quantity PsRmin is equal to the power reflected from a minimally detectable point within the sample. The quantity 1/NEB is the inverse detection bandwidth or, equivalently, the time spent observing each resolution element within the sample. From Eq. (93) it is evident that this observation time is proportional to the longitudinal resolution l divided by the scanning velocity vs. The right-hand- side numerator PsRmin=NEB is therefore equal to the optical energy that returns from a minimally detectable resolution element during the observation period. By dividing this quantity by the photon energy hv, we obtain the number of photons returning from the resolution element. Equation (95) shows that the number of returning photons must be equal to or greater than 2= in order to be detected. This requirement ensures that a minimum of one photon is detected by the photodetector, because on average, only half of the reflected photons return to the detector through the beamsplitter and the probability that each returning photon is converted into a unit of electric charge is :

The preceding discussion shows that the shot noise limit is directly related to the quantization of light. In summary, at least one (but actually 2=) photons need to be received for each resolution element in the OCT image in order for that resolution element to have a detectable reflectivity. Sensitivity can be increased only by increas-

ing the number of incident photons, increasing the size of the resolution element, or increasing the time spent observing each element.

REFERENCE

1.Hee MR. Optical coherence tomography of the eye. PhD Thesis. Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, February 1997.

ever, presents a nearly monotonic decrease with increasing wavelength over the visible and near-infrared spectral region. Maximizing OCT imaging depth of penetration therefore requires the use of a center wavelength that balances these two influences. Although no comprehensive study of the wavelength dependence of imaging contrast and depth of penetration in OCT has been published, theoretical treatments [1,2] and investigations of tissue optical properties [3–5] suggest that optimal image depth of penetration should occur near 1:3 m and near 1:65 m.

Axial resolution in OCT imaging is determined by the bandwidth of the light source through the equation

!

where is the source center wavelength and is the full width at half-maximum (FWHM) spectral width. This equation determines the FWHM of the axial point spread function, L, assuming a Gaussian spectral distribution. For a given source wavelength, an increase in bandwidth gives rise to a proportional increase in resolution. Although in frequency space relation (1) gives an inverse proportionality between bandwidth and resolution, the square law scaling in wavelength space points out a fundamental difficulty in using long-wavelength sources. For example, to achieve 10 m resolution at 800 nm requires a bandwidth of 28 nm, yet this same resolution requires a 1:6 m wavelength source to provide 113 nm of spectrum.

As described in Chapter 2, the power required from an optical source is constrained by the relation

where SNR is the signal-to-noise ratio (or sensitivity), vs is proportional to the axial scanning acquisition rate, and Ps is the maximum source optical power that can reach the detector by way of the interferometer sample arm. This relation implies that to increase image acquisition rate and/or resolution while preserving sensitivity to weak sample reflections requires an increased source power. The derivation of this equation assumed that the source emitted all of its power into a single transverse mode. Spatially incoherent sources are difficult to use for OCT because the signal that is detected arises from interference between light returning from the sample and light returning from the interferometer reference arm. Unless both the sample and reference electric fields are returned to the detector with identical wave fronts, the interference will be washed out upon integration. In typical interferometers, the wave front transformations of the two pathways are not identical. When a single-trans- verse-mode source is used, spatial filters can be employed to remove any light that is scattered from the lowest order mode into higher order modes and thus preserve interference fringe contrast. The most common interferometer used in current OCT systems is based on single-mode optical fiber. In this case the fiber itself acts both to guide the light with minimal scattering into higher spatial modes and to spatially filter any higher order modes generated by the sample or other optics external to the fiber.

The derivation of Eq. (2) assumed shot noise limited detection. This is fairly straightforward to achieve using heterodyne detection and refers to a quantum-

limited detection sensitivity. It is interesting to note that shot noise limited detection can be achieved using relatively low optical power ( 10 W) in the local oscillator (interferometer reference arm). Increasing the local oscillator power at the detectors above this level does not increase sensitivity. The most common interferometer design used for OCT, however, is a Michelson interferometer with a single beam splitter that is double-passed. In this case, 25% of the source power travels through the reference arm to the detector and 25% travels through the sample arm. Fully 50% of the source light is thrown away. In early OCT systems using superluminescent diodes, the available source power was only on the order of 100 W so that the local oscillator power at the detectors was near the optimal value for shot noise limited detection. With the higher power sources currently available, the use of a Michelson interferometer has become extremely inefficient, because, in addition to the base 50% loss, nearly all of the reference arm light is superfluous. Recently a Mach–Zehnder interferometer with optical circulators was demonstrated that overcomes this inefficiency by biasing most of the source light into the sample arm of the interferometer [6]. This advancement will allow the efficient utilization of new higher power sources.

A final general criterion for evaluating optical sources for OCT is suitability to the application environment. Early studies demonstrating OCT imaging were performed in the laboratory, where the complexity of the system or source is not a critical issue. Many studies are now moving toward in situ and even in vivo imaging, for which a simple, compact, and robust system is essential. Unfortunately, the best sources in terms of resolution and image acquisition rate (femtosecond solid-state lasers) are the worst in terms of complexity, size, and environmental stability. Femtosecond lasers can already provide resolution near 1m and are available at several near-infrared wavelengths; the most exciting future advancements for OCT sources are likely to be reductions in the size, cost, and complexity of these sources.

3.1.2Spectral Shape

In addition to the primary source criteria of wavelength, bandwidth, and power, it is important to recognize the significance of the specific shape or distribution of the source power spectrum. As was derived in Chapter 2, an OCT system free of unbalanced dispersion and wavelength-dependent loss provides an axial point spread function given by the Fourier transform of the source power spectrum:

where the delay coordinate is given by . This relationship provides a simple understanding of the effects of nonuniform spectral distributions. This section will describe some of the general implications of Eq. (3).

Because one of the primary distinguishing features of OCT is its high dynamic range, most applications under investigation and biomedical imaging in particular require an axial point spread function that has not only a narrow FWHM but also a well-behaved far-field dropoff. A spectral distribution that meets this criterion is the Gaussian

70 Bouma and Tearney

Another spectral distribution of interest is the hyperbolic secant:

In both Eqs. (4) and (5) represents wavelength and is the FWHM of the spectral distribution. To compare these distributions, Fig. 1 shows both the spectra and resulting point spread functions on linear and logarithmic scales. Also shown in this figure is a hyper-Gaussian distribution, a realistic approximation of a rectangle function. While the Gaussian (dots) and hyperbolic secant (solid) spectra give rise to rapidly extinguishing wings in the point spread function, the sharp vertical edges in the hyper-Gaussian spectrum (gray) gives rise to significant far-field ringing. The implications of this ringing for OCT imaging are most noticed near sharp discontinuities in the axial reflectivity or scattering profile, such as at the sample surface or cell or nuclear membranes. In these cases, the far-field wings can mask more weakly scattering adjacent structures. Examples of this artifact will be discussed in the sections on individual sources.

Figure 1 Power spectra (a,b) and autocorrelations (c,d) on linear (a,c) and logarithmic (b,d) scales for Gaussian, hyperbolic secant, and super-Gaussian functions.