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

the envelope, Risð lgÞ, is a function of group delay, whereas the carrier is a function of phase delay [4]. If dispersion in the sample and reference arms is not matched, then it is necessary to distinguish between group and phase pathlength difference. This is the case, for example, if the group delay and the phase delay effected by the delay line are not equal, as with the Fourier domain rapid scanning ODL to be described in Section 5.5. The complex envelope can be expressed as the convolution of the autocorrelation function of the optical source, Riið lgÞ and the amplitude backscatter profile of the sample, rsð lg), which can be thought of as a set of impulses with various amplitudes representing discrete reflection or scattering locations in the sample [3]:

An OCT system with a perfect mirror in the sample arm measures the interferometric autocorrelation of the source, R~ii, which can be expressed similarly to Eq. (2):

4.1.2Phase Delay Scanning

When the pathlength difference is swept by a scanning delay line in the reference arm, the photodetector response is a time domain signal related to the interferometric autocorrelation by the speed of the scan of the delay. The center frequency of the detector response signal is related to the carrier of the autocorrelation by the phase delay scan speed, and hence the center frequency f0 can be written in terms of the center of the optical source spectrum:

Here, V is the scan speed of the phase delay, defined as the time derivative of the phase delay, V ¼ d l ðtÞ=dt, and 0 and 0 are the center frequency and the center wavelength, respectively, of the optical source. The center frequency corresponds to the Doppler shift frequency of the center wavelength component of the reference arm light and equivalently to the beat frequency of the optical heterodyne detector response. In the case of a simple translating retroreflecting mirror, V ¼ 2s, where s is the velocity of the mirror, and Eq. (6) becomes

f0 ¼ 2s=0

which is the familiar Doppler shift equation. Note also that if the scan is linear (the translation speed is constant), then V will be constant, and therefore f0 will be constant. In the case of a nonlinearly scanning delay line, V is a time-varying function; therefore, f0 will also change with time correspondingly.

4.1.3Group Delay Scanning

The frequency components of the detector response signal, expressed as offset from the carrier frequency f 0 ¼ ðf f0Þ, are related to the complex envelope of the autocorrelation by the scan speed of the group delay. They can thus be written in terms of the offset frequency 0 ¼ ð 0Þ or wavelength components of the optical source:

Differentiating Eq. (7) gives the expression for the bandwidth of the detector response signal in terms of the optical source frequency bandwidth or wavelength bandwidth :

Note that the result expressed in Eq. (8) is independent of the optical source spectral shape [1]. In the case of a simple translating retroreflecting mirror, Vg ¼ V ¼ 2s, where s is the velocity of the mirror. When V ¼ Vg, the familiar expression f =f0 ¼=0 holds true, which follows from Eqs. (6) and (8). Note also, as with phase delay, that if the scan is linear (the translation speed is constant), then Vg will be constant, and therefore f will be constant. In the case of a nonlinearly scanning delay line, Vg is a time-varying function; therefore, f will also change with time correspondingly. For most of the ODLs to be discussed here (except in Section 5.5), the phase delay scan speed and the group delay scan speed will be assumed to be identical.

4.1.4 Characteristics of Scanning Optical Delay Lines

Working Pathlength Scan Range

The working pathlength scan range (in meters per scan) is the portion of the delay sweep during which measurements can be made. In other words, this parameter refers to the depth range that will be imaged by the OCT system. It is important to note that group delay (as opposed to phase delay) sets the location of the coherence gate in OCT; therefore, this parameter specifically refers to the working scan range of the group delay. Typically 3 mm ( 10 ps) is a sufficient depth scan range for OCT in turbid samples. The desired range depends on the application, however, and is determined by the scale of the features to be imaged and tissue attenuation. It is useful if this parameter can be varied continuously and easily.

Pathlength Scan Velocity

Pathlength scan velocity (in meters per second) is a parameter introduced in Sections 4.1.2 and 4.1.3. The scan velocity of the phase delay, V , determines the Doppler shift of the reference light and therefore the center frequency of the detected signal. The scan velocity of the group delay, Vg, determines the bandwidth of the detected signal and relates to the imaging sweep. In the dispersionless case, which may be assumed for the first three categories of delay lines listed in the Introduction, V and Vg are identical. The distinction becomes important only for category 4. This parameter also goes to the fundamental trade-off between imaging speed and sensitivity in OCT. Imaging speed obviously scales with pathlength scan velocity, and from Eq. (8) we see that the bandwidth f of the detected signal also scales with scan velocity. The detection bandwidth B must be broad enough to accept f . The signal-to-noise

ratio (SNR) of an OCT system is inversely proportional to the detection bandwidth. Therefore, for a fixed image size and light intensity, SNR is inversely proportional to imaging speed.

Scan Repetition Rate

The scan repetition rate (in scans per second or hertz) is the number of pathlengths scans performed every second, or the inverse of the total scan period. This parameter depends on scan length and scan velocity and is important for determining the imaging frame rate (images per second) of the OCT system. A high scan repetition rate is necessary for real-time imaging and for eliminating motion artifact when living samples are being imaged. Motion artifact is the misalignment of adjacent depth scans in an OCT image due to motion in the sample that occurs during the course of the depth scan. Motion artifact can seriously degrade OCT images, requiring postprocessing to register or realign the adjacent depth scans, but it can be eliminated simply by imaging with a scan period much shorter than the time constant of the motion in the sample.

Pathlength Scan Duty Cycle

The pathlength scan duty cycle is defined as the usable fraction of the total scan period. This parameter can be calculated from the preceding three parameters as

duty cycle ¼ scan length (m/scan) rep rate(scans/s) scan velocity (m/s)

This parameter may not seem important at first, but it directly scales the power budget of the OCT system and therefore the SNR. In other words, if an OCT system uses an ODL with a 50% duty cycle, then it is imaging only half of the time, so it is effectively using only 50% of the light power provided by the optical source. An ideal ODL has a 100% duty cycle. Most of the ODLs reviewed here are actuated by ‘‘forward and back’’ motion. In other words, the delay is scanned in one direction, then the direction is reversed. Usually, imaging is performed on the forward stroke and the return stroke is minimized. We will refer to this technique as single-sided scanning. Alternatively, the actuator motion can be symmetrical on the forward and return strokes and images can be recorded during both. We will refer to this technique as double-sided scanning. Double-sided scanning requires that alternating data lines be reversed for display, because they were recorded during the ‘‘backward’’ scan stroke. If the ‘‘forward and back’’ actuator is driven at a frequency well below its mechanical resonance, a nearly 100% dutycycle can be realized. If the drive frequency is increased in order to increase the scan repetition rate, the duty cycle is compromised due to nonlinearity around the direction transitions. The extreme is the sinusoidal motion of a resonant scanner. A resonant scanner can move much faster than an equivalent linear scanner, but the duty cycle is decreased significantly. The trade-off must obviously be governed by the design requirements.

Linearity of Pathlength Scan

As discussed in Sections 4.1.1–4.1.3, the phase delay scan rate directly relates the measured interferogram center frequency to the carrier of the optical source autocorrelation, and the group delay scan rate directly relates the measured interferogram envelope to the envelope of the optical source autocorrelation. Therefore, any

nonlinearity in the delay scan rate will result in variations in the measured interferogram center frequency f0 and bandwidth f .

Several sources of scan nonlinearity are associated with various ODLs. One type of nonlinearity is quantization of the scan motion. This can occur if the ODL actuator is a stepper motor, for example. In this case, the motion of the ODL is in discrete steps, approximating a constant motion. The ODL may also move in discrete steps if the control waveform driving the actuator is synthesized with too few bits of resolution. Another source of scan nonlinearity is the jitter exhibited to some degree by many scanning devices used to actuate ODLs. Resonant devices can also be used to actuate some ODLs. In this case, instead of scanning at a constant rate, the ODL scans with a sinusoidally varying rate. This situation causes special problems, but resonant devices are sometimes desirable for fast scanning. If f0 andf are not constant, then the bandwidth of the detection electronics must be broadened in order to capture all of the image information, decreasing the SNR. In the case of a resonant ODL, it is possible to avoid broadening the detection bandwidth by implementing a bandpass filter that tracks the variation of f0. If the group delay (which determines depth of the coherence gate in the sample) is not scanned at a constant rate, the image may be warped if the data are sampled at a constant rate.

Insertion Loss

Insertion loss is defined as the fraction of optical power incident on the delay line that is lost and not returned to the interferometer. The typical implementation of OCT uses a Michelson interferometer with a single detector. In this configuration, the reference light is attenuated to improve the SNR by reducing excess photon noise. In this situation, optical power loss in the ODL is tolerable. If, however, a power-conserving interferometer configuration is used (as discussed in Chapter 6), then optical power loss in the ODL is undesirable. In any case, it is important that the ODL maintain a reasonably constant optical power throughput throughout the course of the scan in order to maintain a constant SNR.

Polarization Effects

Only those polarization components of the sample and reference light that are matched will interfere. The loss of fringe contrast due to polarization mismatch is called polarization fading. If the ODL changes the state of polarization (SOP) of the reference light in a way that does not change during the course of the delay scan or drift over the imaging time scale, then a simple polarization control can be used to realign the polarization state with that of the sample light. If the ODL causes a dynamic change in the SOP of the reference light, then further steps must be taken to reduce or reverse the change and avoid polarization fading.

Dispersion Effects

If dispersion in the reference arm of the interferometer is not matched with that of the sample arm, then the cross-correlation of the reference and sample light will be broadened, reducing the axial resolution of the OCT system. Even the dispersion mismatch due to a small difference in optical fiber length can significantly affect the resolution; therefore, it is good design practice to carefully account for dispersion in all elements of the sample and reference arms, including the ODL. If the ODL has dispersion that changes over the course of the delay scan, then it is very difficult to

match the dispersion in the sample arm. Some ODLs allow dispersion to be adjusted easily, which is a useful feature.

4.2LINEAR TRANSLATION

4.2.1Linear Translator Mounted Retroreflector

The simplest scanning ODL is a retroreflector mounted on a linear translating stage (Fig. 1). The reference light is collimated and directed toward the retroreflector, which redirects the light back to the collimator, which recouples the light into the delivery fiber. The retroreflector is mounted on a linear translating stage in order to scan the delay. A mechanically translating stage will have more than enough translation range to provide a sufficient working pathlength scan. The major drawback of this type of ODL is a low scan velocity. Commercial linear translators can move at a speed of up to 100 mm/s (e.g., Newport model PM500-L). A linear translating ODL is very flexible, able to operate with a wide range of scan lengths and scan velocities. The scan repetition rate and duty cycle will depend on the selected scan range and velocity and on whether one-sided or two-sided scanning is used. Translating stages are commonly commercially available with several types of actuators such as dc motors, stepper motors, and linear motors. All will have some degree of nonlinearity in their motion due to jitter and/or because of motion taken in discrete steps. Care should be taken to choose a product that minimizes these nonlinearities, for reasons discussed in the preceding section. When necessary, mechanical damping can further

(a)

(b)

Figure 1 Schematic of an optical delay line consisting of a fixed delivery fiber and collimating lens plus a retroreflector mounted on a linear translating stage. A flat mirror (a) or a corner-cube retroreflector (b) can be used as the retroreflecting element. The corner-cube retroreflector is less alignment-sensitive than a flat mirror.

of the galvanometer (Fig. 6 illustrates the ¼ 0 position.) The small-angle approximation is valid for configurations appropriate for OCT. For example, with a 4 cm swing arm, a scan range of 4 mm is effected with a rotation of less than 3 . If the galvanometer is turning at a constant angular velocity !, then the translation velocity is s ¼ r! cosð!tÞ r!. The OCT signal center frequency and bandwidth are then given by Eqs. (9) and (10), respectively.

Figure 7 illustrates a scanning ODL very similar to the configuration just described, using rotational actuation but approximating linear translation [10]. This ODL is mechanically more sophisticated than the one described above but will not be discussed in detail here. This design has been commercialized (ClarkMXR Model ODL-150) and is capable of scan rates of more than 30 Hz.

4.3ANGULAR SCANNING METHODS

4.3.1Rotating Cube

The optical pathlength (OPL) that a beam traverses can be scanned rapidly by passing the beam through a glass cube that is rotating at a constant speed. As the glass cube rotates, the thickness of glass that the beam passes through changes, and therefore the OPL changes. An important motivation for this type of ODL is speed. A cube can be rotated rapidly, generating delay sweeps at repetition rates much higher than those of the ODLs described so far. Several configurations based on this principle have been implemented. The pathlength scan range depends on the size of the cube as well as the configuration (how many passes through the cube.) The pathlength scan range is easily adequate for OCT, even with a small cube, but the

Figure 7 An ODL configuration similar to that of Fig. 6. The design is detailed in Ref. 10. (Courtesy of American Institute of Physics.)

110 Rollins and Izatt

scan range is not adjustable—it is fixed for a given cube size. The scan velocity depends on the configuration, the cube size, and the cube rotation speed. The cube can be rotated very fast, resulting in a high scan velocity and repetition rate. The duty cycle depends on the configuration. The insertion loss and polarizationdependent loss depend on the surface treatment of the cubes. For most configurations, the scan is not linear. In other words, the delay is not a linear function of the rotation angle. The other drawback of this design is the pathlength-dependent dispersion introduced by the glass cube. A fixed amount of dispersion can be compensated for by introducing a matching dispersive element into the sample arm. As the cube spins, however, and the pathlength within the dispersive glass changes, the amount of dispersion also changes. Whether or not this dynamic dispersion mismatch is significant depends on the cube material, the wavelength and bandwidth of the optical source, and the scan length.

Several ODL configurations have been implemented using rotating cubes. The following have been demonstrated in optical low coherence reflectivity measurements.

Single Pass, No Internal Reflections

Figure 8 illustrates the most straightforward rotating cube configuration. The index of refraction of the cube is higher than that of air (typically 1.5), so the beam is refracted as it enters and leaves the cube. As the cube rotates, the length of the path within the cube varies, but in a nonlinear way. The single-pass pathlength variation as a function of the incidence angle of the beam on the cube is given by

where L is the dimension of the cube and n is the index of refraction of the cube [11,12]. When the cube is rotated at a fixed angular frequency !, then ¼ !t an the scan velocity s ¼ 2 @d=@t. The interferogram center frequency and bandwidth are

Figure 8 Rotating cube ODL that operates with a single pass, no internal reflections. (Adapted from Refs. 11 and 12.)