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

where the subscript 0 denotes the center optical frequency or optical wavelength and the subscript D indicates the center (Doppler shift) electronic frequency. Equation (45) approximately gives the Q desired for the bandpass filter in terms of the wavelength bandwidth of the light source.

2.3.2Transimpedance Amplifier

The photodetector is followed by a transimpedance amplifier, shown in Fig. 4, which converts the photocurrent i into an output voltage v. For low frequencies, the output voltage is simply

The capacitance C in parallel with the feedback resistance is necessary for amplifier stability and causes the amplifier to roll off at 20 dB per decade above the dominant pole at

2.3.3Bandpass Filter

A bandpass filter follows the transimpedance amplifier (after a simple single-pole RC circuit that removes the DC photocurrent) to separate the band-limited interferometric signal from noise. The filter is designed with a center frequency f~D and quality factor Q given by Eqs. (36) and (45), respectively. In the discussion below, the tilde indicating electrical frequencies will be dropped. Design strategies for both an active and a passive filter will be given.

Active Sallen and Key Cascade Filter

An active bandpass filter is most useful for barrier frequencies below 100 kHz. For higher frequencies, active filters may be limited by the slew rate of the operational amplifier and passive network filters are required. An active bandpass filter may be created by cascading a low-pass Sallen and Key biquad filter followed by a high-pass one.

Figure 5 depicts the low-pass Sallen and Key filter in a unity gain configuration. The transfer function is given by the standard second-order form

Figure 4 Transimpedance amplifier.

using the simplification R ¼ R1 ¼ R2.

The resonance frequency !n is chosen to be equal to the signal carrier frequency !D. In practice the quality factor Q is chosen to be smaller than the value given by Eq. (45) so that the filter bandwidth is larger than the signal bandwidth. A filter bandwidth that is too narrow will widen the signal in time and arbitrarily limit the axial point spread function. Section 2.5 will show that a filter bandwidth that is too large will let in more noise and reduce the minimum detectable reflectivity or dynamic range of the system.

The unity gain Sallen and Key high-pass filter is exactly the dual or the lowpass filter except that resistor are interchanged with capacitors (Fig. 6). The high-

Figure 6 Sallen and Key high-pass filter.

where

1

!n ¼ p ð52Þ

C R1R2

and

s

Q ¼ 1 ¼ 1 R1 ð53Þ 2 2 R2

using the simplification C ¼ C1 ¼ C2. The resonance frequency and quality factor are chosen to match those for the low-pass filter.

The bandpass filter HbpðsÞ created by the cascade of the low-pass and high-pass filters has two sets of pole pairs at

Thus, the frequency response rolls off at 40 dB per decade in both directions outside of the resonance peak.

Passive Network Butterworth Filter

A passive network bandpass filter may be designed for high frequency operation using the standard techniques for broadband-matching network synthesis. The problem is formulated as follows in relation to Fig. 7: Given a Thevenin equivalent voltage source defined by Vt and Rt, synthesize a lossless two-port network that is matched to the load impedance R2 and has the desired transfer function power gain characteristic jHðj!Þj2 ¼ jV2ðj!Þ=Vtðj!Þj2.

The solution to the general problem has been explored in detail, and only one particular solution using resistive termination will be given here. A passive network bandpass filter can be designed easily by starting with a prototypical low-pass filter and then ‘‘frequency warping’’ this filter into the bandpass domain. The Butterworth low-pass filter works well for this purpose because an LC ladder can always be obtained to match a purely resistive source and load and achieve the nth order

The Butterworth filter with DC gain K and low-pass cutoff !c is maximally flat in the passband and has evenly spaced poles placed in the left half plane (LHP) on a circle centered at the frequency origin. Furthermore, explicit recursive formulas exist for

Figure 7 Network synthesis problem.

Figure 8 LC ladder network of order n.

the element values of the LC ladder, shown in Fig. 8, which simplifies the design process. (An odd-order filter is realized by removing the terminating capacitor.)

Under the condition of equal source and load resistances ðRt ¼ R2Þ, a simple nonrecursive formula is applicable. Using this restriction, in order to realize a Butterworth filter of order n with cutoff frequency !c and a DC gain of unity, the mth component values of the LC ladder shown in Fig. 8 are given by the following definitions:

To realize a bandpass filter with low-pass cutoff !1 and high-pass cutoff !2, one first designs a normalized, prototype low-pass filter with a cutoff frequency of !c ¼ 1 rad/s. The following frequency transformation is then used to warp the filter

We have defined B ¼ !2 !1 as the desired bandpass filter bandwidth and !m ¼ !1!2 as the midband (geometric mean) frequency. It is straightforward to verify that the transformation described by Eq. (58) involves replacing inductors in the prototypical filter (Fig. 8) by an inductor and a capacitor in series and replacing capacitors in the prototypical filter by an inductor and a capacitor in parallel. The necessary component values for this low-pass to bandpass frequency warping are shown in Fig. 9. Resistors remain unchanged. The low-pass to bandpass frequency warping process doubles the number of poles in the system.

Figure 10 shows an example of a second-order prototypical Butterworth lowpass filter that is transformed into a four-pole bandpass filter. For Rt ¼ R2 ¼ 5 k and design parameters of a center frequency of 400 kHz and a quality factor of 10,

Figure 9 Low-pass to bandpass frequency warping.

Figure 10 Second-order Butterworth bandpass filter.

the following component values are obtained: L1 ¼ 28:1 mH, C1 ¼ 5:64 pF, L2 ¼ 141 H, and C2 ¼ 1:13 nF. The frequency response of the resulting bandpass filter is shown in Fig. 11. The filter has two zeros located at the origin and two complex pole pairs located at

p1;2 ¼ 0:092 j2:60 and p3;4 ¼ 0:086 j2:42 rad=s 106 ð59Þ

Thus, the frequency response rolls off at 40 db per decade in both directions off the main peak.

2.3.4Demodulation

Amplitude demodulation can occur by either mixing or envelope detection. Both models will be briefly introduced here so that noise propagation through the electronics can be understood for both methods.

Mixing

Demodulation by mixing entails multiplication with a reference sinusoidal signal of the correct phase followed by low-pass filtering. Mixing is easily implemented by

Figure 11 Frequency response of Butterworth bandpass filter.

of the minimum detectable reflectivity of the OCT system. The sensitivity, or signal-

to-noise ratio (SNR), for any system is defined simply as the signal power Psignal divided by the noise process variance var fnðtÞg. If the noise process is zero-mean

WSS, then

which shows that the SNR is equal to the signal power divided by the mean noise power, where the noise power is the integral of its power spectral density Snð!Þ over all frequencies.

The signal-to-noise ratio will be derived below by propagating the various sources of noise described in Section 2.4 through the detection electronics described in Section 2.3. In the shot noise limit, the noise power is dominated by the shot noise of the reference beam, also termed the local oscillator. The signal power is given by the correlation of the power reflected from the sample with the power in the local oscillator. Weak reflections from within the sample are therefore multiplied by the local oscillator field to create a relatively strong interference signal at the detector. In this manner, a relatively high SNR can be achieved. Figure 12 displays the variables used for the noise process, power spectral densities, and noise equivalent bandwidths at various points in the system and will be helpful in the subsequent sections.

2.5.1Photodetection Noise

From Eq. (3), the photodetector current is the sum of the DC power PR ¼ jARj2=2 0 returning from the reference path, the DC power PS ¼ jASj2=2 0 returning from the sample, and an interference term real fESERg from the correlation of the signal and the local oscillator:

We have also added a term idark that represents the dark current of the photodiode. The signal power is completely contained within the term real fESERg, which is

separated from the DC components PR and PS by bandpass filtering and demodula-

Figure 12 Noise variable definitions.

tion. The filtering and demodulation steps, however, do not eliminate the noise associated with these components, which may exist at all frequencies.

Both shot noise [Eq. (66)] and RIN [Eq. (67)] contribute to the photocurrent noise inðtÞ, which can propagate through the detection electronics. Therefore, the power spectral density Sin ð!Þ of this noise can be expressed as the sum of these two uncorrelated components:

where hii is the mean photocurrent and contains contributions from the local oscillator beam, the sample beam, the signal, and the dark current.

2.5.2Transimpedance Amplifier Noise

At the input to the transimpedance amplifier (Fig. 4), thermal noise [Eq. (65)] from the feedback gain resistor R is added to the photocurrent noise. Because all the noise sources are uncorrelated, the power spectral density of the noise current becomes

At the output of the transimpedance amplifier, the noise current niðtÞ is transformed into a voltage nvðtÞ with a gain of R. Because the variance scales as R2, the spectral density Svn ð!Þ of the output noise voltage is

Equation (72) can also be written in terms of the mean output voltage hvi ¼ hiiR as

where a new variable q has been defined as representing the amplitude of the noise spectral density, which is constant across all frequencies. Equation (73) is useful because the voltage output of the transimpedance amplifier is more conveniently measured than the photocurrent.

2.5.3 Bandpass Filter Noise Equivalent Bandwidth

Let the bandpass filter have an impulse response hbpðtÞ and a transfer function HbpðsÞ given by one of the design procedures detailed in Section 2.3.3. The noise input to this filter is given by Eq. (73). Since the input noise is white with a constant spectral density, the output noise will be colored, or shaped according to the characteristics of the filter. Using the standard results for a WSS stochastic process through a linear time-invariant filter, we obtain the power spectral density Sbpð!Þ of the noise process nbpðtÞ at the output of the filter:

The noise variance at the bandpass filter output is