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

Figure 10 Optical pathlength configuration with double pass, four internal reflections. L2, lens; P, right-angle prism; M, mirror; EGM, reflecting gold layers; RC, rotating cube. (#Copyright 1998 IEEE.)

[14,15]. This configuration was demonstrated using a 12 mm 12 mm 4 mm cube rotated at 310 rev/s, resulting in a scan repetition rate of 1240 Hz. A pathlength of 11 mm was scanned at 42 m/s. The resultant duty cycle is approximately 32%. In most biological samples, only about 3 mm of pathlength scanning is useful, however, so this configuration would yield an effective duty cycle of about 9%. A smaller cube could be used to improve this. The (free space) pathlength variation as a function of angular position of the cube is given by

where L is the length of the side of the cube and n is the refractive index of the cube material. This function is obviously not linear, and a scan speed variation of17% was measured. When the cube is rotated at a fixed angular frequency !, then ¼ !t, and the scan velocity is s ¼ @d=@t. The interferogram center frequency and bandwidth are then given by Eqs. (9) and (10). The autocorrelation width measured using this ODL was reported to be only 1% broader than the theoretical width (the source coherence length), so dispersion mismatch was apparently not significant. The type of glass that the cube was composed of was not specified. The optical sources used for this measurement had a center wavelength of 1310 nm and a bandwidth of 45 nm. This ODL required custom deposition of mirrors on portions of each cube face.

4.3.2Scanning Mirror

Another ODL based on the transformation of angular motion into pathlength variation is illustrated in Fig. 12 [17]. As with the rotating cubes and the galvan- ometer-mounted retroreflector, the motivation for using a rotational scanning element is speed. In addition to an angular scanning mirror, three additional mirrors and a lens are used. An examination of Figs 12a and 12b will instruct the reader as to the operation of this configuration, particularly the need to place the scanning mirror and mirror 2 one focal length from the lens. This configuration reflects the beam four times off the scanning mirror, amplifying the scan length and velocity by 4. It should be noted that 0 pathlength variation is effected by scanning the beam in a line across mirror 2. All of the pathlength variation arises from the beam being offset from the pivot of the scanning mirror so that there is an axial shift imposed on the beam in addition to the angular scan. In other words, if the beam were reflected from the pivot of the scanning mirror, then no pathlength variation would result from scanning the mirror. The pathlength variation d as a function of scan angle (referenced from the plane parallel to the lens) is given by d ¼ 4x tan 4x for small scan angles, where x is the distance that the beam is offset from the pivot of the scanning mirror. The scan velocity s is given by s 4x =t. For a constant angular velocity !, s 4x!. The interferogram center frequency and bandwidth will then be given by Eqs. (9) and (10). The pathlength variation achievable for a given configuration of this design will be limited by the need to keep the scanning mirror within the depth of focus of the lens and by the size of the lens and mirror 2. Optical power loss will be determined by the quality of the mirrors and the alignment. There should be no significant scan-dependent polarization or dispersion variation.

This ODL was demonstrated using a 1.2 kHz resonant scanner and 50 mm focal length lens. Pathlength was scanned by 3 mm at 1.2 kHz and a peak scan velocity of 11.3 m/s. Most likely the full 3 mm was not usable for imaging, because the motion of a resonant scanner is sinusoidal. The nonlinear pathlength variation with time would warp the image to an unacceptable degree near the extrema of the scan. A reasonable duty cycle for single-sided scanning using a resonant ODL is 33% (66% for double-sided scanning). In order to maintain a narrow detection bandwidth over the course of the scan (and prevent SNR degradation), the signal was bandpass filtered by a voltage-controlled oscillator that was controlled by the reso-

Figure 12 Schematic illustrations of an ODL based on the transformation of an angular mirror scan to pathlength variation. (# Copyright 1998, Taylor & Francis, Ltd.)

effect can be compensated for by introducing 45 Faraday rotators (FRs) in the air gaps of the reference and sample arms so that on a double pass through the FRs, the SOP is conjugated and the return propagation through the fiber reverses the SOP change [23]. Alternatively, polarization-maintaining fiber can be used to minimize these effects. In addition, the temperature of the actuator should be carefully controlled in order to prevent temperature-dependent drift of the delay and SOP.

4.5FOURIER DOMAIN RAPID SCANNING OPTICAL DELAY LINE

The Fourier domain rapid scanning optical delay line (RSOD) was developed for femtosecond pulse measurement [24–26] and has recently been applied to OCT [1,27,28]. The RSOD is based on Fourier transform pulse-shaping techniques, which have been demonstrated to also be capable of shaping the temporal properties of broadband incoherent light [29]. This delay line achieves scans of several millimeters at repetition rates of several kilohertz and also allows separation of group and phase delay, which provides an additional degree of control over the center frequency and bandwidth of the OCT signal.

The RSOD consists of a grating–lens pair in a folded, double-passed Fourier domain pulse-shaping configuration (Fig. 16). A flat mirror serves as a spatial phase filter, which imposes a linear phase ramp on the optical frequency spectrum. The delay line is based on the well-known property of the Fourier transform that a phase ramp in the frequency domain corresponds to a group delay in the time domain:

The angle of the incident light on the grating can be selected such that the center wavelength 0 of the diffracted beam is normal to the grating and the entire grating is in the focal plane of the lens. This is done to prevent introduction of group velocity dispersion (GVD), which varies throughout the course of the scan. If the distance from the grating to the lens is not one focal length, then GVD is introduced. Therefore, if the grating is at an angle not normal to the optical axis of the lens, then as light is laterally displaced on the grating by the scanning mirror it is also displaced from the focal plane of the lens, introducing GVD [30].

The mirror pivot can be offset from the center wavelength by an arbitrary distance x by a simple lateral translation of the scanning mirror. The phase shift ð Þ as a function of wavelength for a given mirror tilt angle can be written as (refer to Fig. 16)

where lf is the focal length of the lens, p is the pitch of the grating, and 0 is the firstorder diffraction angle of the center wavelength from the grating (measured from the grating normal). Recall that in the configuration shown in Fig. 16, 0 ¼ 0, i.e., the grating is in the focal plane of the lens. In this case cos 0 evaluates to unity, so this factor will not be propagated further in the subsequent analysis.

The function ð Þ was derived by using the grating equation and assuming that the small-angle approximation sin holds for diffraction angles between different wavelengths off the grating and for small deflection angles effected by the scan-

(a)

(b)

Figure 16 Schematics of the Fourier domain optical delay line. (a) View from above. The incident, collimated broadband light is diffracted from the grating and spectrally dispersed. The lens parallelizes the dispersed spectrum while focusing it to a line on the scanning mirror. The scanning mirror imposes a linear phase ramp on the spectrum and redirects the light back through the lens, which recollimates the beam and reconverges the spectrum onto the grating. The beam then diffracts in a reverse manner from the grating and propagates collimated and undispersed toward the double-pass mirror collinear with the incident beam. The double-pass mirror turns the light back through an identical path. Double passing is illustrated clearly in (b). (Courtesy of OSA.)

ning mirror [25]. The distance y that a wavelength component traverses as a function of scanning mirror tilt angle is calculated (taking into account the fact that the beam traverses the path y four times) and multiplied by 2= to convert from displacement to phase shift. This phase shift can also be written as a function of angular optical frequency !:

where !0 is the center angular optical frequency. From Eq. (12) and the definition of phase delay, t ¼ ð!0Þ=!0, the phase delay is given by t ¼ 4 x=c. This corresponds to the free-space phase pathlength difference l (referenced from a zero scan angle),

From the definition of group delay, tg ¼ @ð!Þ=@!j!¼!0 , the group delay is given by

This expression for group pathlength difference can alternatively be derived by evaluating the change in pathlength experienced by the center wavelength as a result of an arbitrary mirror tilt angle. It can be seen that the group pathlength difference is equal to the phase pathlength difference plus an additional term that is a function of the properties of the lens, grating, and light source. Both the phase pathlength difference and the group pathlength difference are proportional to the tilt angle of the scanning mirror, so an angular scan of the mirror effects a scan of both phase and group pathlength differences. The second term in the expression for group pathlength difference is dominant in typical configurations; thus the group pathlength difference is much larger than the phase pathlength difference for a given angle . This feature is key to the value of this delay line. The mirror need be moved only a very small amount to generate a large scan of the group pathlength difference. It is also possible to adjust the offset x for a desired center frequency without significantly affecting the scan length of the group pathlength difference. If the center wavelength is located at the mirror pivot (i.e., x ¼ 0), then the phase pathlength difference term l vanishes and there is no phase pathlength scanning. This allows for the application of an external phase-modulating element that is truly independent of the group delay scanning if such a configuration is desired.

Differentiating Eq. (13) with respect to time and using Eq. (6), the center frequency of the interferogram measured using the Fourier domain scanning delay line is given as

Similarly, differentiating Eq. (14) with respect to time and using Eq. (8), the bandwidth of the interferogram measured using the Fourier domain scanning delay line is

The design equations for the high speed OCT system using this delay line are Eqs. (14)–(16). Equation (14) is used to specify the grating pitch, the lens focal length, and the maximum angular excursion of the scanning mirror for a given

120 Rollins and Izatt

source center wavelength and desired group delay scan. Equations (15) and (16) are used to calculate the center frequency and bandwidth, respectively, of the OCT signal for a given mirror offset x and mirror scan rate @ðtÞ=@t. If the mirror is tilted at a fixed angular frequency !, then ðtÞ ¼ !t and @ðtÞ=@t ¼ !. Estimation of the signal bandwidth f from Eq. (16) can be simplified by recognizing that for a typical

configuration, 2x 2lf 0=p.

A resonant scanning mirror tilts as a sinusoidal function of time,

ðtÞ ¼ b sinð2 fmtÞ

where b is the maximum excursion of the tilt angle and fm is the resonance frequency of the scanning mirror. Thus, from Eq. (15) the center frequency becomes

This sinusoidal variation of center frequency and bandwidth with time is the major disadvantage of using a resonant scanner in the delay line. The advantage is that resonant scanners can operate at much higher repetition rates than galvanometermounted scanners. Using a resonant scanner, a reasonable compromise between duty cycle and nonlinearity is to record the interferometric signal only during the middle two-thirds of the forward scan, resulting in an overall duty cycle of 33%. Double-sided scanning increases the duty cycle to 67%. A linear scanner can achieve a duty cycle of nearly 100% (at the expense of scan speed). Unless a tracking bandpass filter is used, as discussed in Section 5.2.3, the use of a resonant scanner with a reduced duty cycle requires the detection bandwidth to be increased compared to an equivalent linear scanner in order to accommodate the varying center frequency. This corresponds to degradation of the SNR. Thus, a resonant scanner is advantageous only in cases where the needed scan rate cannot be achieved with a linear scanner.

As discussed above, if the distance from the grating to the lens is not one focal length, then GVD is introduced. This requires careful alignment to prevent broadening of the cross-correlation due to dispersion mismatch. This property of the RSOD also provides a convenient means to compensate for dispersion mismatch due to dissimilar components in the reference and sample arms. For example, an endoscopic probe in the sample arm may not include an air gap to match the air gap in the RSOD. This dispersion mismatch requires compensation, which can be introduced by the RSOD [28,30,31].

This ODL has few disadvantages. Besides complexity, the major disadvantage is optical power loss. In a double-passed configuration, the light is diffracted into the first order four times. A grating blazed at the appropriate angle can increase the efficiency of diffraction into the first order, but the aggregate loss will be significantly greater than most of the other ODLs reviewed here. In the typical OCT configuration using a Michelson interferometer, the reference ODL must be attenuated to