## Abstract

We report on a Fourier Domain Mode Locked (FDML) wavelength swept laser source with a highly linear time-frequency sweep characteristic and demonstrate OCT imaging without k-space resampling prior to Fourier transformation. A detailed theoretical framework is provided and different strategies how to determine the optimum drive waveform of the piezo-electrically actuated optical bandpass-filter in the FDML laser are discussed. An FDML laser with a relative optical frequency deviation Δν/ν smaller than 8·10^{-5} over a 100 nm spectral bandwidth at 1300 nm is presented, enabling high resolution OCT over long ranging depths. Without numerical time-to-frequency resampling and without spectral apodization a sensitivity roll off of 4 dB over 2 mm, 12.5 dB over 4 mm and 26.5 dB over 1 cm at 3.5 µs sweep duration and 106.6 dB maximum sensitivity at 9.2 mW average power is achieved. The axial resolution in air degrades from 14 to 21 µm over 4 mm imaging depth. The compensation of unbalanced dispersion in the OCT sample arm by an adapted tuning characteristic of the source is demonstrated. Good stability of the system without feedback-control loops is observed over hours.

©2008 Optical Society of America

## 1. Introduction

Optical coherence tomography (OCT) is a depth resolved biomedical imaging technique, providing high-resolution, cross-sectional and three-dimensional images of tissue microstructure [1]. Recently, the introduction of frequency domain (FD) detection techniques in OCT enabled a dramatic increase in imaging speed, while still maintaining high system sensitivity [2–4]. In FD-OCT the echo delay time of backscattered light from the sample is measured via spectrally resolved detection of the interference signal between light from the sample and light from a reference arm. Spectrometer based FD-OCT (spectral OCT) systems are already widely used, especially for ophthalmic applications in the 800 nm wavelength range.

Alternatively, the application of FD-OCT systems based on rapidly swept, narrow band light sources (swept source OCT (ss-OCT)/optical frequency domain imaging (OFDI)) [5–11] offers the additional advantages of dual balanced detection, potentially longer ranging depth and higher imaging speeds, compared to spectral OCT. The advent of Fourier domain mode locked (FDML) lasers [12] as light sources for swept source OCT enabled high imaging speeds of up to 370.000 lines/s [13], combined with good phase stability [14] and long ranging depths [12]. Especially because of their high speed, FDML lasers have been applied to numerous applications like optical coherence microscopy (OCM) [15], phase sensitive profilometry [14], retinal imaging in ophthalmology [16], high speed spectroscopy [17, 18] and art conservation studies [19].

However, a general disadvantage of most FD-OCT systems, including FDML based devices, is the requirement to resample or recalibrate the detected OCT interference-fringe signals prior to Fast Fourier transformation (FFT), in order to provide data evenly sampled in optical frequency. Spectrometer based FD-OCT systems and swept source OCT systems based on light sources with polygon scanners [20], exhibit small non-linearity, but they still require correction for the frequency to wavelength relation ν=c/λ. Swept source OCT systems based on mechanically resonant filters [21] or typical FDML lasers [12, 22] have an even more pronounced non-linear sweep operation, due to the typically sinusoidal excitation of the piezo controlled fiber Fabry Perot tunable filter (FFP-TF) in the laser cavity.

There are five potential problems in ss-OCT caused by the non-linear time-frequency sweep characteristic. (a) First of all, the numerical recalibration step is usually done in software with the computer by resampling the data after analog-to-digital conversion and it might be challenging to perform this step in real time. OCT with state of the art swept sources generates data rates of more than 0.8 Gbyte/s, a data rate that can hardly be handled with standard personal computers today. Field programmable gate array solutions might solve this problem, but are less flexible and more complex to implement. (b) Second, the oversampled parts towards the edges of the sweep (for a sinusoidal sweep) require system memory but, depending on the recalibration algorithm, carry not proportionally more information. Especially extremely large 3-dimensional datasets, acquired in comprehensive OCT applications [23, 24], are memory critical and efficient use of system memory is desired. (c) A third problem occurs if the recalibration step is performed in hardware. In such systems the recalibration is performed by clocking the analog to digital converter (ADC) with an electronic signal generated by a second interferometer [24], so the sampling is performed with an uneven spacing in time, accounting for the non-linear sweep operation. This avoids the problems mentioned before, however, it requires complex electronic hardware and usually isn’t easily adjustable to different operation frequencies of the source. (d) A further problem, common to both, numerical and hardware recalibration, is the noise introduced by the resampling step itself [25] and by phase errors in the recalibration step [10]. The phase errors can lead to a timing jitter, which causes a white noise floor, consequently reducing the dynamic range of the system. (e) Another issue of a non-linear sweep operation is the excess exposure on the tissue. Usually OCT is operated with the maximum permitted power on the sample, in order to achieve the best sensitivity. However, in the oversampled parts of the spectrum, i.e. the parts where the source sweeps slower, the incident light intensity does not increase the sensitivity proportionally, even though it contributes to the total energy per sweep. The power on the sample and with it the sensitivity has to be reduced unnecessarily.

Besides these problems with a non-linear and uncontrolled sweep operation of the source, a highly linear and adaptive time-frequency sweep characteristic may have further advantages. Direct hardware based Fourier transformation of the signal can be envisioned. Such a solution would be faster and at the same time would save a factor of 2 of data transfer bandwidth, because the amplitude values of the FFT only yield half the number of points. As the transfer speed from the analog-to-digital converter to the system memory or data-storage devices currently limits the OCT performance, such an improvement could be beneficial for future OCT devices. Furthermore an adjustably adaptive time-frequency sweep characteristic can be used to partially compensate sample arm dispersion in real time, without adding material into the reference arm or using complex numerical resampling algorithms. Dispersion compensation by an adaptive time-frequency sweep characteristic will be demonstrated in this paper.

#### 1.2 Quality parameter and error tolerance

In order to analyze and compare the quality of the sweep linearity over a defined temporal range, an appropriate error-parameter has to be defined. The analysis and discussion presented in this work is based on the integrated relative frequency error χ defined as a measure for the sweep linearity:

Here Δν (t)=ν (t)-ν_{Lin} (t) is the deviation from the perfect linear time-frequency characteristic ν_{Lin} (t) that can be calculated by a linear fit to ν(t) in the given time interval of interest Δt. Summations are done over all sample points N within Δt.

The main problem with efforts to linearize the time-frequency characteristic of a light source for swept source OCT, or equivalently to linearize the angle dispersion per optical frequency in spectral OCT systems, is the high requirement in linearity. Only very small deviations from a perfectly linear dependence are acceptable, in order to maintain a long ranging depth and a high axial resolution in OCT, if a resampling step should be avoided.

In spectral OCT a concept for linearization of the fringe signals has been demonstrated by correcting the ν=c/λ relation with an additional prism element [26]. For wavelength swept sources a concept to linearize the sweep characteristic of a laser source based on a polygon mirror filter has been demonstrated, but no OCT imaging without recalibration was performed [27].

Figure 1 shows the linearity requirements. Figure 1 (left) depicts the theoretical optical frequency over time for three different wavelength swept laser sources. We assume three types of frequency swept lasers (sweep rate of 57 kHz each): (i) in the first type, the wavelength filter in the laser is swept perfectly linear in time (black line); (ii) in the second case, an optical bandpass filter is applied with a linear time-wavelength dependence, resulting in a hyperbolic time-frequency characteristic due to the ν=c/λ relation. Such a situation would represent a grating-polygon scanner type of laser as described in [20]; (iii) the third case would represent a laser with a sinusoidally driven filter in wavelength, as found in swept lasers with mechanical resonant filters [21], sources with piezo actuated fiber based Fabry Perot filters [10] or typical FDML lasers [12, 22]. In all three cases, the filters are assumed to be swept over a range of 175 nm from 1231 nm to 1406 nm. The blue hatched area indicates the region with lasing operation, we assume a range of 100 nm from 1260 to 1360 nm.

It should be noticed that the duty cycle (in this example: 175 nm scan range compared to 100 nm lasing) does not influence the linearity of the *wavelenth-linear* laser source in contrast to the source that is *sinusoidal in wavelength*. Only for the latter one, a smaller duty cycle can yield better linearity. The lasing range is centered at 1308 nm, the sweep range of the filter is centered at 1318 nm. The chosen 10 nm offset minimizes the non-linearity in case of the sinusoidal sweep, because the slightly non-linear part of the sine can be used to partially cancel the 1/λ dependence. The sweep duration of the sinusoidal drive waveform equals 3.4 µs. Figure 1 (center) displays the relative frequency error Δν/ν for these three sources relative to a perfectly linear evolution over the time range where the laser source is active (1260 nm to 1360 nm - see hatched area). All three curves from Fig. 1 (left) were fitted linearly in the 100 nm lasing range (blue area) and the relative frequency error was plotted. Because the effective sweep speed of the sinusoidal source is higher, the sweep duration is shorter. Naturally, for the perfectly linear evolution (black line) the relative frequency error is zero. Remarkably, for this specific example, the sinusoidally driven source (green) already exhibits a deviation comparable to the wavelength-linear source, as a direct result of the large drive amplitude of the filter (175 nm) compared to the lasing range. The integrated relative frequency error for the sinusoidal driven source (green line) is χ_{sine}=3.9·10^{-4}, for the wavelength-linear source (red line) χ_{lin_lambda}=4.3·10^{-4} respectively.

These values suggest a high degree of linearity. However, considering the point spread functions (PSF) in OCT application (Fig. 1 (right)) at a ranging depth of 1 mm (Hanning window spectral shape), both, the wavelength-linear as well as the sinusoidally driven source, exhibit significant degradation in peak values and in full width at half maximum (axial OCT resolution). This implies that an even better linearity than 4·10^{-4} is required and an extremely high accuracy and repeatability of the filter is needed.

The approach followed in this paper is to theoretically determine an optimum electronic drive waveform for the piezo actuated fiber Fabry Perot tunable filter, accounting for the mechanical response of the filter, the electronic response of the drive circuit and limitations given by the experimental setup. Constraints and boundary conditions set by the experiment are maximum sweep rate in frequency per second due to electronic bandwidth limitations, maximum permissible power for the filter etc. An experimental setup will be described to measure the frequency dependent mechanical response of the filter. We will discuss and quantify interfering effects, like the mechanical non-linear response of the piezo-electric transducer of the FFP-TF and the repeatability of its mechanical oscillatory motion. The effect of these effects on OCT image quality will be discussed.

## 2. Experimental setup

#### 2.1 Laser and interferometer setup

Figure 2 (left) shows the experimental setup of the FDML laser used for this study. A semiconductor optical amplifier (SOA, Covega Corp.) with a gain maximum centered at 1310 nm is used as a broadband gain medium. Two isolators (ISO) ensure uni-directional lasing. A fiber based tunable Fabry-Perot-filter (FFP-TF, Lambda Quest, LLC.) provides spectral filtering. The filter is driven periodically with a sweep rate of 56.902 kHz. Arbitrary waveforms can be applied to the filter by a digital function generator and a power amplifier as waveform driver. The AC-signal from the programmable function generator is amplified and superimposed to an adjustable DC-voltage to set the sweep’s center wavelength. In order to synchronize the second harmonic of the optical roundtrip time of the light circulating in the laser cavity with the FFP-TF tuning frequency, a 3.6 km long fiber is used in the resonator. The design is based on a sigma-ring configuration. The circulator (CIR) couples light from the ring into the linear part and returns it back into the ring. After forward propagation through the 3.6 km length of fiber, light is backreflected by a Faraday mirror (FRM) and the polarization state is rotated by 90°. Thus, birefringence effects in the 3.6 km fiber are cancelled during back-propagation. Overall 75% of the laser power is coupled out of the laser resonator by two 50/50 couplers. One output is used for analyzing the laser with a Mach-Zehnder interferometer (MZI) in a dual balanced configuration with a detection bandwidth of 350 MHz. The other output is post-amplified with a second SOA and is used for OCT imaging. The two outputs were not used simultaneously. During imaging it was not necessary to monitor the laser with the MZI. Polarization controller paddles (PC) are used to control the polarization state of light entering the SOA (~16dB polarization dependent gain).

#### 2.2 Filter response

One major issue in applying non-sinusoidal drive waveforms at high frequencies (several 10 kHz) to the FFP-TF is the non-flat phase and amplitude response of the filter and the electronic drive circuitry. For low frequencies (<1 kHz) far away from mechanical or electronic resonances, this response function is flat, which means, a certain applied electronic drive waveform will cause the filter to tune in exactly the same time evolution as the applied voltage. However, at higher frequencies near mechanical resonances, the higher harmonics of the drive waveform exhibit a different amplitude and phase response. This means, the mechanical response of the filter will dramatically deviate from the applied drive waveform. The frequency dependent response function of the filter has to be accounted for.

In Fig. 2 (right) a setup for the computer controlled measurement of the response is presented. A sinusoidal drive waveform with an amplitude of approximately 2.4 V (after power amplifier) is applied to the filter. The drive frequency is successively increased from 1 kHz to 199 kHz with a 500 Hz increment. In order to analyze (a) the amplitude response of the filter (including electrical circuitry), the amplified stimulated emission (ASE) of an SOA is used as light source and the spectral width of the transmitted intensity is analyzed with an optical spectrum analyzer (OSA). The result is read out with a personal computer (PC) and the measurement is repeated for the next drive frequency. The frequency dependent amplitude response is plotted in Fig. 3 (left).

To analyze (b) the phase response of the filter, the transmitted light is coupled into a MZI and the interference signal is measured with a differential detector. The signal is digitized with a high speed ADC and a Hilbert transformation enables the localization of the turning points of the filter in time. Thus, the phase shift of the mechanical oscillatory motion of the filter with respect to the applied sine wave can be obtained for each scan frequency automatically (black curve in Fig. 3 (right)). The phase response of the electronic circuits is directly measured by comparing the phase shift of the electronic drive waveform from the function generator with the voltage measured directly at the filter. The phase shift value is directly read out from the oscilloscope. From these two measurements, the response of the filter itself can be extracted. In Fig. 3 (right) the red curve indicates the phase shift due to the phase response of the filter only.

In the amplitude and the phase plot, two distinct resonances can be identified at 55 kHz and at its third harmonic at 169 kHz (resonance peaks in Fig. 3 (left) and phase jumps in Fig. 3 (right)). The pronounced peak around 55 kHz represents the fundamental axial resonance of the filter and an amplitude gain of a factor of 8 can be observed. It is also interesting to note that the third harmonic at about 169 kHz exhibits an amplitude response similar to the response at very low frequencies. To limit the amount of electric power to drive the filter, the fiber length in the FDML resonator was adjusted such that the first order FDML sweep frequency of 56.902 kHz (red crosses in Fig. 3) is close to the main resonance peak. Furthermore the third order harmonic (170.706 kHz, blue crosses in Fig. 3) lies in the vicinity of the second resonance peak. The relatively fast roll off for frequencies above 180 kHz suggests that the harmonics of the applied drive waveform should be limited to <180 kHz for this filter. Thus, driving the filter with the fourth or even higher harmonic is hardly practicable for a fundamental of 55 kHz and this type of filter. The different amplitude response values of 40 nm/V, 4.9 nm/V and 6.8 nm/V as well as the phase shift of 0.93 π, 1.70 π and 0.46 π for first, second and third harmonic have to be accounted for when an optimized drive waveform is applied. Once an optimum drive waveform is determined, the amplitudes for the harmonics are divided by the respective amplitude values of the response function, the phases are subtracted.

## 3. Semi-analytical/non-iterative approach

#### 3.1 General considerations

With the known filter response function, an optimized drive waveform has to be found within the physical constraints.

A first simple approach to realize a linear time-frequency characteristic could be to apply a periodic AC-voltage with a triangle or sawtooth waveform to the filter. Obviously, this would be a poor solution, due to the frequency dependent response function of the filter. If the waveform would be corrected for the filter response, the excitation of the high harmonics would result in excessive thermal load to the filter. Furthermore, even a perfectly linearly driven PZT would not yield a linear time-frequency characteristic because the elongation of the FFP-resonator is proportional to wavelength, so due to the ν=c/λ relation only a sweep, linear in wavelength, would be generated.

In order to find the optimum drive waveform and quantify the error, we will choose several parameters typical for FDML lasers in OCT. First, the wavelength interval where the laser is active has to be determined. The spectral bandwidth should be 100 nm, where the interval of laser operation ranges from 1260 nm to 1360 nm (centered around the gain maximum of the SOA at 1310 nm).

Second, we will choose an appropriate sweep range of the filter. On the one hand, it makes sense to choose a sweep range that exceeds the lasing interval of 100 nm significantly which already improves the linearity in the interval of laser activity. On the other hand, there are several reasons that argue against an excessively increased sweep range. (1) The maximum mechanical stress on the PZT should not be exceeded in order to maintain good long term stability. (2) The non-linear mechanical response will increase at very high drive amplitudes and (3) the duty cycle and the sweep duration become too short, increasing requirements for the ADC. A good compromise for the presented setup is a sweep range of approximately 175 nm. As discussed before, in order to limit the amount of electric energy dissipation in the filter, another necessary restriction is to limit the number of harmonics that should be applied. In our case we focus the analysis to three harmonics.

#### 3.2 Semi-analytical/non-iterative approaches

In this section, a semi-analytical approach to find the optimum waveform is presented. Considering just the first three harmonics, the time-frequency characteristic is given by:

Here, λ_{0} is the wavelength offset, i.e. the center wavelength of the sweep, c is the speed of light in vacuum, A_{i} represent the amplitudes of the respective components in wavelength; φ_{i} are the different phases of the three harmonics respectively and ω=2π·56.902 kHz is the drive frequency of the filter.

#### 3.3 Simple Fourier expansions

The most obvious approach to determine the unknown parameters would be to perform a Fourier expansion of a triangular time-frequency characteristic (series of hyperbolic branches in wavelength) ν_{lin}(t) with a wavelength offset λ_{0} and a sweep range A. All Fourier-components up to third order will be considered, higher orders are neglected, yielding the A_{i}s and φ_{i}s. A_{1} is chosen such that ν(t) exhibits a sweep range of 175 nm.

The wavelength offset λ_{0} is chosen analytically such that the inflection point of the time-frequency characteristic ν^{(1)}(t)=c/λ^{(1)}(t) (case A_{2}=0 and A_{3}=0 in Eq. (2)) is centered in the frequency range of laser activity, reducing non-linearity induced by the ν=c/λ relation (∂^{2}ν^{(1)}(t_{in})/∂t^{2}=0 and c/ν^{(1)}(t_{in})=1308 nm). Figure 4 shows the results of this approach. Figure 4 (left) depicts the triangular linear time-frequency characteristic ν_{lin}(t) for a perfectly linear sweep (black curve) and the resulting frequency representation of the discrete Fourier series of c/ν_{lin}(t) up to third order with a sweep range of 175 nm (red curve). The inflection point of ν^{(1)}(t) has been calculated to be 10 nm below sweep center wavelength. Therefore λ_{0} is chosen to be 10 nm larger than 1308 nm, yielding 1318 nm. The two curves were fitted linearly in the 100 nm lasing range (hatched blue area). The relative frequency errors Δν/ν (relative deviation from fit) are drawn in Fig. 4 (center). The integrated relative frequency error χ as a measure for the non-linearity induced by neglecting the higher order (n>3) of Fourier components, has a value of 9.41·10^{-4}. Remarkably, this value of χ even exceeds the integrated relative frequency error resulting from a sinusoidal drive waveform (sweep range 175 nm) by a factor of more than 2 (χ_{sine}=3.9·10^{-4}, see section 1). The result for OCT applications is depicted in Fig. 4 (right), where the related PSFs are shown for an imaging depth (delay in Michelson interferometer) of 1 mm (Hanning window spectral shape).

The full width at half maximum (axial OCT resolution) as well as the height of PSF exhibit a considerable degradation compared to the PSF of the perfectly frequency-linear sweep. This means that by simply applying the first 3 harmonics of a perfectly linear waveform in frequency, an unacceptably poor linearity with even higher error than a sine waveform is generated.

A simple explanation for the poor linearity is based on the fact that a Fourier series converges in quadratic mean but not pointwise. This means, the Fourier expansion is an optimization for the waveform over the whole cycle period. However, since high linearity is not required over the entire period, but only during a relative small temporal interval, adding higher orders might even degrade linearity in the desired part of the sweep, though improving the deviation over the whole period in total. Particularly considering the fact that high linearity is only needed for either forward or backward sweep, but not for both, performing a Fourier expansion and skipping higher orders is not a good approach to find the optimum waveform.

## 4. Numerical methods A: constant sweep duration

As detailed above, the method of a simple Fourier expansion doesn’t yield an acceptable solution for a linear time-frequency characteristic. Thus, we will focus on a numerical, iterative optimization process. For the non-linear fit process, an optimization parameter (error parameter) has to be defined that describes the quality of linearity in the predetermined frequency interval (1260 nm to 1360 nm) of laser activity. We will use the integrated relative frequency error introduced in section 1 as error parameter. To avoid divergence of the optimization algorithm, certain constraints and boundary conditions have to be set, usually given by the experiment.

A very typical situation for OCT applications is a certain desired sweep duration. This could be the case for buffered FDML setups [13], where a defined integer fraction of the roundtrip time is desired. Another example where optimization with a constant sweep duration can make sense, is a condition where the maximum sampling rate of the ADC and the minimum ranging depth set a lower limit for the sweep duration, in order to enable a certain number of sampling points per sweep (A-scan in OCT).

With the method of constant sweep duration, only solutions are considered which provide lasing over a certain time range within each drive cycle. The main advantage implementing an optimization algorithm with the method of constant sweep duration is that the lasing duration during a drive cycle is fixed and known. So the numerical treatment simply integrates the error over a certain time range for a drive cycle.

Including only three harmonics, there are a total of seven parameters that have to be optimized: the amplitudes of the first three orders A_{1}, A_{2}, A_{3}, the phases of the first three orders φ_{1}, φ_{2}, φ_{3} and the wavelength offset λ_{0}. Since sweep duration and frequency interval are given, two boundary points in the time-frequency domain can be defined. We can assume a linear sweep from high to low frequency. Hence, the integrated relative frequency error χ can be calculated as described in section 1. χ has to be minimized finding the ideal parameter combination by a non-linear numeric fit procedure.

In order to gain more insight into the optimization problem and the dependence of the error on the parameters, first, second, third order harmonic amplitude and the wavelength offset were kept invariant as fixed parameters during optimization. A nonlinear fit algorithm is applied to optimize the remaining three variables (first, second and third order harmonic phase) for each given parameter combination. This process is repeated for different values of first, second and third harmonic amplitude and the offset in form of a raster scan. With this approach, the minimized χ can be plotted dependent on different parameters in form of error maps.

Figure 5 shows such an error map from the results of an exemplary numerical optimization with a constant sweep duration of 4.385 µs. The chosen sweep duration equals a quarter of the drive cycle duration for f_{drive}≈57 kHz, which would yield a 228 kHz repetition rate for a 4× buffering scheme [13]. In order to ensure that the sweep range is close to the experimentally reasonable value of 175 nm, the amplitude of the first order harmonic is set to 87.5 nm in this example. The amplitude of the second order harmonic and the third order harmonic were stepped with 0.6 % increments (relative to the amplitude of the first order harmonic) ranging from 0 % to 30 % and 0 % to 12 %, respectively. The wavelength offset is changed in steps of 5 nm ranging from 1265 nm to 1355 nm. Figure 5 shows the minima of the integrated relative frequency errors acquired by optimization with different wavelength offsets for each combination of second and third order harmonic amplitudes (Matlab V7.4.0. function “fmincon”).

The first interesting feature is that for the given parameter set an absolute minimum of χ can be found for A_{2}=13.2%·A_{1} and A_{3}=0.6%·A_{1} (Fig. 5, black cross) with the corresponding phases φ_{1}=0.024 π, φ_{2}=0.980 π and φ_{3}=1.324 π. The minimum of χ corresponds to a wavelength offset λ_{0}=1315 nm, indicating that the nonlinearity induced by the ν=c/λ relation is reduced by choosing an offset slightly above the center wavelength of laser activity of 1308 nm (see section 3).

The value of χ in this error map ranges from 3.232·10^{-3} to the minimum of 3.346·10^{-5}, which is approximately one order of magnitude improvement over wavelength linear sources (see section 1). It should be noted that the result obtained for the chosen parameter set leads to a minimum of χ located in an area of rather high amplitudes of second harmonic, resulting in relatively high drive voltages due to the weak response of the filter. Solutions with smaller amplitudes of the higher harmonics would be preferred.

Simulations with differing values of first order amplitude A_{1} (from 82.5 nm to 102.5 nm) result in similar error plots. However, higher values of A_{1} yield smaller minimum values for χ, so a compromise between electro-mechanical load and required minimum error has to be made.

## 5. Numerical methods B: arbitrary sweep duration

In cases where the sweep duration is not pre-determined and in order to get a general understanding of the linearization problem, a more flexible approach is applied without the constraint to a constant sweep duration. This method provides full access to sweep duration, wavelength offset and phases for locally optimized solutions.

Again, the integrated relative frequency error χ is used as optimization parameter that has to be minimized adjusting the amplitude and phase parameters of the three harmonics, including wavelength offset. In contrast to the previously presented approach, a predetermined evaluation interval in the time-frequency domain cannot be set.

Therefore, in order to calculate χ, a search algorithm has to evaluate the temporal interval reaching from 1260 nm to 1360 nm (lasing range) and a linear fit to the curve serves as an ideally linear function to determine the error χ. Thus, this optimization approach is computationally more expensive compared to the approach of fixed sweep duration and requires more computing time.

A simultaneous optimization of all parameters at the same time is not performed for the reason described in the previous chapter. Furthermore, here χ is getting arbitrarily small for increasing A_{1} as well as for increasing A_{2} and A_{3}, the problem diverges towards higher amplitudes. So A_{1}, A_{2}, A_{3} and the wavelength offset λ_{0} are kept invariant during optimization and are changed in a reasonable raster to plot an error map. φ_{2} and φ_{3} are optimized with a nonlinear optimization algorithm (Matlab V7.4.0 function “fminsearch”) for each parameter configuration. Since the time interval of laser activity is found by the search algorithm, the absolute phase is not essential and one can arbitrarily chose the phase of the first harmonic φ_{1} to 0.

In Fig. 6 the results of the “arbitrary sweep duration” simulation are shown. Just like in the simulation of section 4, A_{1} is set to 87.5 nm, consistent with the considerations in section 1. The raster is chosen such that A_{3} ranges from 0 % to 12.5 % of A_{1} in 0.5 % steps and A_{2} is adjusted from 0 % to 24 % of A_{1} in 0.8 % steps. The wavelength offset λ_{0} is changed from 1275 nm to 1345 nm in 5 nm steps. In Fig. 6(A) the minimum of the integrated relative frequency errors χ acquired by optimization of φ_{2} and φ_{3} for the different wavelength offsets λ_{0} is depicted in a surface plot versus the second and third order harmonic amplitudes. Figure 6(B) shows the related plot illustrating the optimum λ_{0} for the same simulation. The corresponding distributions of sweep duration, optimum phase of second harmonic and optimum phase of third harmonic are presented in Fig. 6(C), Fig. 6(D) and Fig. 6(E) respectively.

Figure 6(A) shows that the smallest values for χ are found in a U-shaped “valley of minima”, where the two branches (label “i” and “ii”) are spanning a square of relatively small χ (blue area in the surface plot). Outside this square χ is considerably larger. In contrast to the approach with fixed sweep duration, here χ is getting smaller for higher amplitudes. The absolute minimum of χ within these parameter-limits is the local minimum at the end of the branch i) with χ=2.373·10^{-6}. Nevertheless, simulations with an extended range of A_{2} and A_{3} indicate that even smaller values of χ can be achieved in the branch ii). Branch ii) is unlimited towards higher amplitudes and is getting broader for higher A_{2} and A_{3}. Interestingly, like in the previous section, regions of minimal values of χ correspond to a wavelength offset λ_{0} of ~1315 nm, indicating that the nonlinearity induced by the ν=c/λ relation can be reduced by choosing an offset slightly above the center wavelength of laser activity of 1308 nm (1318 nm for A_{1}=87.5 nm). Thus, the two branches can also be identified in Fig. 6(B). The plot of the related sweep durations in Fig. 6(C) indicates that the optimum sweep duration in vicinity of the upper branch equals about 4.5 µs. This fact is consistent with the simulation of constant sweep duration (4.385 µs), yielding a minimal χ in the same area of the error map. On the other hand, the sweep duration along the lower branch ranges from about 3.6 µs to 2.9 µs in the error map and is further decreasing for higher amplitudes (improved linearity, but shorter sweep duration yielding a very asymmetric time-frequency characteristic). By analyzing Fig. 6(C) and Fig. 6(A), one can determine which areas in the error map yield low values of χ for a specific sweep duration. Therefore the ansatz of arbitrary sweep duration offers a more general result than the approach presented in chapter 4. Simulations with differing values of A_{1} are resulting in qualitative similar error maps. Nevertheless, the values of χ can be made arbitrarily small by increasing A_{1}, the problem diverges towards higher amplitudes.

Using these error maps, the ideal operation point for the filter can be chosen within the experimental constraints. To study the feasibility of OCT with k-space linear FDML, here a specific sweep duration is not mandatory. We will focus on regions on the map with small values of A_{2} and A_{3} in order to avoid very high driving voltages for the higher harmonics, preventing excessive electro-mechanical stress on the piezo of the FFP-TF. Hence, the operation point will be set in the lower branch in the region near zero 2nd order amplitude (A_{2}=2.4 % and A_{3}=5 %; indicated as point 4 in Fig. 6(A)). This solution is a good compromise between having small amplitudes and sufficient linearity. The expected value of χ is 2.307·10^{-5}, this is 20× smaller than the equivalent χ, achieved with a sinusoidal or linear time-wavelength sweep characteristic (see section 1).

In order to understand the dependence of the integrated frequency error on the phases of second and third harmonic, Fig. 7 shows three surface plots where the calculated χ is plotted versus φ_{2} and φ_{3} (raster of 2°) for the parameter combinations representing three characteristic points in the error map (Fig. 6(A), points number 4,8,9). The absolute minimum of each plot is indicated with crosses. Figure 7(A) depicts the chosen operation point (point 4) with relative small values for A_{2}=2.4 %, A_{3}=5 % and an ideal λ_{0}=1315 nm. Figure 7(B) is illustrating a case with high A_{2}=14.4 %, small A_{3}=0.5 % and ideal λ_{0}=1310 nm (point 8). Finally Fig. 7(C) demonstrates a case with high A_{2}=14.4 %, high A_{3}=9 % and ideal λ_{0}=1320 nm (point 9). In all three cases the domain of high linearity in the plots is an S-shaped blue curve but the orientations differ. Considering point 4, the range of φ_{3} yielding small χ is much smaller than the corresponding range of φ_{2} whereas the situation is inverse at point 8.

There are two conclusions that can be drawn from Fig. 7: On the one hand, it becomes clear that χ is critically dependent on the phases, underlining the benefit of numerical simulation to predetermine the best point of operation. On the other hand, it can be seen that for a given point in the error map, there is more than one good solution yielding a sufficiently small χ, because χ is only slightly varying over the s-shaped curve. For the waveform with the smallest error, the phases are most critical.

## 6. Experimental results and comparison to theory

#### 6.1 Measurements confirming numerical simulation

In the following paragraph it will be demonstrated that the results obtained with the numerical approach presented before can be used to successfully realize a k-linear FDML laser, i.e. an FDML laser with highly linear time-frequency sweep operation. The experimental data will support the fact that the filter can be driven with sufficient repeatability for OCT without a resampling or recalibration step. Effects like non-linear coupling, parasitic mechanical oscillations and drift are sufficiently small at the demonstrated operation point. A comparison between the experimentally achieved linearity and that predicted by theory will be given.

In order to gain information about the time-frequency characteristic of the FDML-laser, the interference signal obtained from a MZI (dual balanced detection (see Fig. 2 (left)) is analyzed by Hilbert transformation. To adjust the experimental parameters according to the simulation, the wavelength range of laser activity is monitored with an OSA and adjusted by varying the SOA current. In our case, an SOA current of ~350 mA is applied, resulting in the desired lasing range from 1260 nm to 1360 nm (10 dB drop in intensity on OSA). The AC-voltage amplitude of the first order harmonic is adjusted such that A_{1} equals 87.5 nm. The wavelength offset λ_{0} is set to the appropriate value by tuning the applied DC-voltage of the piezo of the FFP-TF. Both, first order harmonic amplitude A_{1} and wavelength offset λ_{0}, can directly be monitored with the OSA. In order to derive the relative frequency error Δν/ν and calculate the integrated frequency error χ, a 100 times averaged interference signal is recorded with the oscilloscope and a Hilbert transformation is performed at the part of the signal where the envelope drop is smaller than 10 dB (≈ 1260 nm to 1360 nm). The accumulative phase evolution of the interference fringes can be extracted and fitted linearly. So Δν/ν and χ result from the phase deviation Δφ (phase of the electronic beat signal from the MZI), assuming a wavelength range of the analyzed data from 1260 nm to 1360 nm and setting all ν_{i}s in Eq. (1) to a center frequency ν_{M}=c/1310 nm.

In order to verify the numerical simulation in vicinity of the chosen operation point in the error map of Fig. 6(A), an analysis of the interference signal for an arm-length mismatch of the MZI of 2 mm corresponding to the points 1 to 7 was performed. Therefore, the corresponding voltage amplitudes resulting in an A_{2} of 2.4 %·A_{1} and the respective values of A_{3} have to be applied, accounting for the amplitude response of the filter (see paragraph 2).

The wavelength offsets are adjusted to the expected values of the theory respectively (1315 nm for point 1 to 4, 1310 nm for point 5, 1295 nm for point 6 and 1285 nm for point 7). In each case, prior to signal acquisition, the phase of second harmonic and third harmonic AC-voltage (with respect to first harmonic) is optimized online by minimizing the width and simultaneously maximizing the height of the Fourier transform of the fringe signal on the oscilloscope.

In Fig. 8 (left) the measured integrated frequency error χ of the corresponding interference signals is plotted versus third order harmonic amplitude A_{3} (blue). Here, the dominant error is caused by the inaccuracy in the determination of the wavelength range. The red points indicate the values of χ expected from simulation. Figure 8 (right) presents the expected optimum phases of the second and third order harmonic AC-voltage according to the simulation (red points: third order harmonic phases; pink points: second order harmonic phases). The phases are corrected for the phase shift due to the filter response. The optimum phases for second and third order harmonic found in experiment are plotted as blue line (third order harmonic phase) and as dark blue line (second order harmonic points). The error bars indicate the error for the second order harmonic phase, the third order harmonic error is too small to be drawn.

The qualitative and quantitative good agreement between theory and experiment in Fig. 8 (left) confirm that the filter motion can be predicted theoretically very well. Parasitic mechanical oscillations, drift etc. are small enough to not significantly affect the measured coherence properties of the source. The integrated relative frequency errors χ, obtained from the measurement, are comparable to those predicted by the simulation, the local minimum in the range from 5 % to 6 % of third harmonic amplitude can clearly be identified.

#### 6.2 Discrepancy between theory and experiment – non-linear coupling

However, besides the good agreement in *total errors*, discrepancies in theoretically and experimentally optimized *phase* can be identified in Fig. 8 (right), especially for the last four amplitude points. To understand this observation, one has to recall the phase maps of Fig. 7, demonstrating that for the parameter set of a given point on the error map, there are several phase combinations (S-shaped curve) yielding almost identical error values χ.

In the vicinity of the chosen operation point, the range of second order phase of this S-shaped curve is much larger than the third order phase range. So equivalent χ can be achieved, reducing third order phase as well as second order phase down to the predicted values. Thus, due to small inaccuracies in experimentally chosen parameters, like errors in the phase response measurements of the filter, the true minimum might vary along this S-shaped curve.

In this measurement the wavelength offset was strictly set to the values given by the simulation. However, if λ_{0} is slightly increased, the linearity can be improved, yielding smaller values of χ. This particularly applies for the last three points. Here, the predicted decrease of optimum offset from 1310 nm to 1285 nm could only be observed to a minor degree. Furthermore, the two polynomial fits (third order) in Fig. 8 (left) indicate that the minimum of the experimental error curve is shifted to higher amplitudes (absolute deviation of approximately 1 %). This result is also underlined by the following comparison.

In Fig. 9 (left, blue curve) the measured evolution of the relative frequency error Δν/ν is plotted for the optimum parameter combination found in the experiment along the points number one to seven from Fig. 6(A). So only the amplitude of the second order harmonic was kept constant, the other parameters were optimized in the experiment. Additionally, the relative frequency error of the optimum parameters expected from the simulation (point 4, red curve) is drawn. In agreement with the findings before, the best solution can be found for a larger amplitude of third order harmonic corresponding to an A_{3}=6 % instead A_{3}=5 %. The optimum offset is about 1320 nm instead of 1310 nm. The optimum phase of the second order harmonic is 1.138 π, the corresponding phase of the third order harmonic is 1.351 π respectively. As can be seen in Fig. 9 (left) the two solutions are in very good agreement, yielding nearly the same characteristic and sweep duration (3.5 µs instead of 3.6 µs expected from simulation). The resulting integrated frequency error of the experimental data is 2.11·10^{-5} and therefore in very good agreement with the optimal χ of 2.31·10^{-5} predicted by simulation.

In Fig. 9 (center) the corresponding PSFs are plotted for a Michelson interferometer with 1 mm inbalance in arm lengths (2 mm for a MZI) and a spectral envelope shaped to a Hanning function from 1260 nm to 1360 nm.

Both, the measured PSF (blue) and that calculated from the simulation result (red) are in very good agreement with the PSF of a perfectly linear time-frequency characteristic (black). Comparing the PSFs of Fig. 9 (center) to the PSFs of Fig. 1 (right) for a sinusoidal sweep and a linear time-wavelength characteristic, a dramatic improvement can be seen. Figure 9 (right) shows the corresponding measured fringe signal (no apodization, original envelope) for the presented solution, where the asymmetric envelope of the fringe signal can be seen.

In summary, besides the very good agreement between the absolute error values of theory and experiment, a slight deviation in phase and offset can be observed. The observed discrepancies are probably due to the nonlinearity in the mechanical response of the filter, meaning that the first order harmonic amplitude also leads to contributions in second or third order harmonic amplitude.

#### 6.3 Required accuracy and procedure to set k-space linear FDML

Figure 10 shows the critical dependence of the quality of linearity on the adjusted phase. The interference signal is recorded with a MZI at a 2 mm arm length mismatch (corresponding to 1 mm imaging depth) and the parameter-set of the optimum operation point described above.

The optimum PSF is plotted as black curve and the measured PSFs with slightly detuned phases of 5° (1.7° of first harmonic phase) and 10° (3.3° of first phase) are plotted as red and blue lines respectively. The detuning introduces timing errors of 80 ns and 166 ns respectively. It can be seen in Fig. 10 that width and height of the PSFs significantly degrade, even for such small phase deviations. These findings are again in good agreement with the simulations in Fig. 7(A).

In conclusion, the measurements and comparison to theory in paragraph 6 suggest that a viable approach for a successful linearization of the sweep operation can consist of the following four steps: (1) The complex (amplitude and phase) filter response function of the FFP-TF filter is measured with a setup comparable to the one presented in section 2. (2) Error maps plotting the integrated error dependent on the amplitudes A_{i} are calculated with a numerical fit procedure. (3) A good operation point within the experimental constraints is chosen on the error map, with an error as small as necessary and drive amplitudes as small as possible. (4) The system is set to amplitude and phase values predicted by the simulation and a fine adjustment of the offset center wavelength and the phases is used for final optimization, using online feedback from an oscilloscope. Possible small discrepancies between theory and experiment due to effects like for example non-linear coupling can be corrected. Usually the width of the PSF generated by the built in FFT-function of an oscilloscope is sufficient. With this method, an integrated frequency error of χ=2.11·10^{-5} for a sweep duration of 3.5 µs was achieved which is about a factor of 20x improvement compared to a wavelength-linear or sinusoidal sweep characteristic (chapter 1). For the online analysis on the oscilloscope a simple, inexpensive fixed MZI, built of two fusion spliced 50/50 couplers, could be applied.

#### 6.4 OCT performance

In order to examine the performance of k-space linear FDML for OCT, we analyze the PSF roll off over imaging depth in OCT application. The optimum waveform presented above is applied to the filter and the interference signal is acquired with the MZI for several delays. In Fig. 11 (left) the resulting PSFs are plotted versus corresponding image depth (image depth=0.5× difference in MZI arm lengths). The average output power of the laser is 9.2 mW. It should be noted that here no recalibration or resampling step and also no spectral reshaping or apodization was performed prior to FFT.

Figure 11 (right) shows the logarithmic representation, adjusted such that the peak values of the PSFs represent the measured sensitivity.

The sensitivity values were calibrated with an additional measurement using a neutral density filter with 46.4 dB attenuation in one interferometer arm. The values are adjusted for coupling losses in the interferometer. The red circles in Fig. 11 (right) indicate the drop of fringe contrast of the signal extracted from the mean fringe intensity of the acquired intensity envelopes (by Hilbert transformation, see [28]).

The best axial resolution in air (3 dB width of Gaussian fit) was 14.0 µm, in theory the spectrum would support about 13.7 µm in air. For some larger delays, the resolution is labeled in Fig. 11 (left), exhibiting only a 7 µm increase over 4 mm. The maximum sensitivity at short delays is 106.6 dB, about 5 dB below the theoretical shot noise limit of 111.5 dB for 9.2 mW optical power incident on the sample at a detection bandwidth of 350 MHz [4]. The sensitivity is indicated in Fig. 11 (right) for several delays. About 4 dB roll off for 2 mm imaging depth, 12.5 dB for 4 mm and 25.5 dB for 10 mm is measured. Remarkably, the discrepancy between the peaks of the PSFs and the plotted red circles, indicating the drop of fringe contrast, is very small over a wide imaging range. This is an indication that most of the roll off is caused by loss in fringe visibility, rather than added phase error. This means, for the given imaging range, the quality of linearity of the applied waveform is sufficient and the roll off is entirely determined by the coherence properties of the source. This fact supports the assumed model that a possible difference between instantaneous lasing wavelength and the instantaneous center wavelength of the FFP’s passband can be neglected here.

Besides the good sensitivity values, the system also exhibits a dynamic range of about 69 dB which can be considered very high for a swept source OCT system [13, 29, 30], especially since the ADC resolution was only 8 bit (digital oscilloscope), again underlining the advantage in dynamic range of FD-OCT [13].

#### 6.5 Imaging with k-space linear FDML without resampling and apodizing

In the following, OCT-imaging with the described k-space linear FDML laser is presented without recalibration step and spectral apodization prior to FFT and a comparison to a conventional sinusoidally driven filter with and without recalibration is given. For that purpose, as described in paragraph 2, the light is coupled out of the laser cavity after the FFP filter and is post amplified to 22 mW by a booster SOA [10] and used in a standard Michelson setup for OCT imaging (dual balanced detection). A pair of galvo-scanner mirrors (Cambridge Technologies, 6215H) is used for transverse scanning on the sample. The calculated transverse spot size on the sample is 22.5 µm. The average incident power on the sample was 10 mW.

The data acquisition was performed using a 400 MSamples/s ADC with a resolution of 12 bit. The real, actual imaging sensitivity was measured with an OD 1.7 neutral density filter and a mirror as sample. The values are not adjusted for losses in the microscope, mirror, coupler and circulator, explaining the reduced numbers compared to the measurement described above. The measured sensitivity is 103 dB for the k-space linear sweep whereas for a sinusoidal sweep, the sensitivity is with 100 dB slightly smaller, what might be due to the recalibration step.

In Fig. 12 three sets, each consisting of two 2D images of a human finger *in vivo*, are presented for each of three different situations: In Fig. 12(A) and Fig. 12(B) the filter was driven sinusoidally (175 nm scan range) with no recalibration step prior to FFT. For the images in Fig. 12(C) and Fig. 12(D) the same filter drive waveform was used but here a recalibration step was performed. Therefore, an interference signal (single depth) was recorded with the interferometer once and the phase characteristic is extracted and used for recalibration of all lines of the image. Finally, Fig. 12(E) and Fig. 12(F) are demonstrating the case where the k-space linear FDML described in the previous chapters is applied and no recalibration step was used. The presented data sets consist of 4000 lines×1664 samples (1536 samples for Figs. 12(A), 12(C), and 12(E)), each image was acquired in 0.070s. The transversal scan range is 4.3 mm the axial scan range is 6 mm. The animation in Fig. 12(G) is a rendered 3D representation of the human finger, recorded with the k-space linear FDML and no recalibration step. The movie was created from a 3D data set of 512 frames×512 lines×1536 samples, acquired in 4.6 s. The scan range equals 4.3 mm in length, 3.0 mm in width and 6 mm in depth. The images are cropped (3.3 mm×1.7 mm×1.9 mm) for better visibility. The three images Fig. 12(A), Fig. 12(C) and Fig. 12(E) as well as Fig. 12(B), Fig. 12(D) and Fig. 12(E) are acquired at the same sample position.

Obviously, the 2D-images acquired with the sinusoidal drive waveform are of considerably poor quality in the case where no recalibration step is performed, as can be seen comparing Fig. 12 (A) with Fig. 12(C) and Fig. 12(B) with Fig. 12(D). This indicates that for FD-OCT with a typical 100 nm bandwidth at 1300 nm wavelength at ~1-2 mm imaging range, an integrated frequency error of χ=4·10^{-4} is not sufficient to perform OCT without resampling.

Figures 12(E)–12(F) show 2D images acquired with the k-space linear FDML laser and no recalibration. Compared to the images acquired with the standard method (sinusoidal sweep+recalibration) in Figs. 12 (C)–12(D)), no significant difference in image quality can be noticed. The Figs. 12(E)–12(F) might even exhibit slightly higher contrast. However, in the k-space linear case, a more pronounced line artifact can be seen. This artifact originates from an electronic interference signal in the system with a fixed frequency. In the case of recalibration, the signal is washed out over a frequency range, whereas in the k-space linear case (or the sinusoidal case without recalibration), a peak at a certain frequency remains as line artifact. This artifact can be removed by increasing the electronic gain in the system. The 3D image and movie animation shown in Fig. 12(G), again acquired with the k-space linear FDML and no recalibration and apodization, demonstrate the capability of the presented system for high speed 3D imaging. Considering the sweep duration of 3.5 µs, similar image quality can be expected if the buffering technique is applied in order to multiply the sweep rate [13]. The image quality should be equivalent to a buffered FDML configuration with more than 250 kHz line rate, yielding a 3D acquisition time for the same size of data set of about 1 second.

In view of future clinical imaging applications, the system shows good stability and reproducibility using the presented method of linearization. Once the optimal waveform is found, the linearity performance does not degrade over a period of several hours and parameters do not have to be readjusted during this time. Aging effects of the piezo or drift do not seem to be a dominant effect on this time scale. On a day to day basis it is necessary to slightly change phases of second and third order harmonic and/or the wavelength offset to obtain the required linearity. This might be due to varying external conditions like temperature. So for a potential commercial system an active feedback loop might be required.

#### 6.6 K-space adaptive FDML – Dispersion compensation

Besides the possibility to linearize the optical frequency sweep over time, the control of the filter motion by an arbitrary waveform offers the unique possibility to adaptively adjust the sweep characteristic. So the possibility to actively control and adapt the time-frequency characteristic of the filter waveform by manipulating amplitude and phase of the three harmonics offers more than implementation of a k-space linear swept laser source. One application could be to dynamically compensate unbalanced dispersion in sample and reference arm of an OCT system. For example in ophthalmic applications, perfect dispersion balancing would require to adjust the amount of dispersive material in the reference arm for different eye lengths of each patient. Usually this problem is solved by numerically compensating dispersion [31] in a post-processing step.

However, in the k-space adaptive FDML approach, dispersion compensation is achieved in hardware by manipulating the waveform applied to the filter and real-time compensation with no additional computation time needed becomes possible. The approach is analogous to the “approximate dispersion compensation” described by Wojtkowski, *et al.* [31], where a simple resampling technique is used. This technique only works perfectly at a single depth and compensation over the entire range cannot be achieved. However there are two situations where such a technique can be particularly useful: (1) In cases where only a very small amount of remaining dispersion imbalance has to be compensated, this technique works very effectively. (2) For swept source optical coherence microscopy (OCM) applications [15, 32], where imaging is only performed in one depth, this technique is also perfectly suitable. Especially, since with k-space adaptive FDML the amount of compensation can be dynamically adjusted, it would be feasible to adjust the amount of dispersion compensation synchronous to the image depth. Because OCM is very sensitive to dispersion due to the high resolution, this approach might be very helpful for future swept source OCM systems.

In order to demonstrate that almost perfect dispersion compensation for one imaging depth is possible with k-space adaptive FDML, first of all the amplitude and phase parameters are adjusted to values yielding a k-space linear FDML laser as demonstrated before. The measured PSF from the fringe signal of a Michelson interferometer for an arm unbalance of 1 mm is shown in Fig. 13 (left) (black curve). In this case, the dispersion in both arms was balanced. In order to simulate sample dispersion, three 0.7 mm thick silicon plates are inserted into the sample arm and the PSF is measured again for the same optical delay. The resulting PSF is depicted in the same graph (red curve). The interference signal was numerically corrected for the losses induced by surface reflection of the plates. For the third PSF, the parameters (phases of the harmonics of the drive waveform) are adjusted by optimizing the width and peak value of the PSF online on an oscilloscope in FFT mode. The PSF resulting from the optimized fringe signal is plotted in Fig. 13 (left) (blue curve). It can be seen that the additional dispersion from the silicon plates leads to a significant degradation of the PSF. By increasing only the third order harmonic phase by approximately 10°, the effect of dispersion is almost perfectly compensated for this particular depth.

To demonstrate partial dispersion compensation over a larger range of imaging depths, a 0.5 mm thick quartz glas plate is inserted into the sample arm of the Michelson interferometer instead of the mirror. Thus, two isolated reflections are detected, one at an imaging depth of 1 mm and one at ~1.8 mm. The result is shown in Fig. 13 (right). The black curve represents the PSFs for the k-space linear laser, again with no silicon. The red curve represents the PSF for the same waveform but now with the silicon plates inserted in one arm. Here again, a clear degradation of the PSF can be observed. The blue line shows the PSFs for the re-optimized filter drive waveform. It can be seen that only a partial compensation is possible at two different depths simultaneously. However, since the amount of added dispersion in this experiment significantly exceeds the typical dispersion imbalance in OCT applications, it can be expected that smaller amounts in real imaging situations can be compensated sufficiently.

## 7. Conclusion and outlook

In conclusion, a detailed analysis of the problem how to linearize the time-frequency dependence of a high speed, wavelength swept FDML laser source is presented. The required experimental setup to analyze the complex response function of the electrically driven, optical bandpass filter in the laser is described, different methods how to optimize the ideal drive waveform are discussed and an experimental guideline how to generally linearize FDML sources is given. A detailed comparison of the theoretical results and the experimental data is used to discuss in how far the response of the filter can be predicted theoretically. In principle, the presented approach could also be used for linearization of non-FDML laser sources.

Based on this analysis, a wavelength swept FDML laser source is demonstrated with 100 nm sweep range at a center wavelength of 1300 nm, a sweep duration of 3.5 µs and a non-linearity of 2·10^{-5}. The high degree of the linearity enables swept source OCT imaging without the step of resampling or of spectral reshaping prior to FFT. A comparison of image quality shows no difference of k-space linear FDML without recalibration or resampling and standard FDML. Good stability and reproducibility of the system is observed over hours. With the resampling step being obsolete, hardware FFT solutions become feasible, paving the way for future ultrafast real time OCT systems for clinical environments. Furthermore, additional noise generated by the resampling step is avoided. The tuning rate restrictions imposed by the requirement to drive with three harmonics can be solved. In order to improve the speed performance and multiply the effective sweep rate, the technique of buffered FDML [13] can be applied and the image quality can be expected to be similar for identical power levels.

The possibility to dynamically adjust the sweep characteristic to deliberately generate non-linear sweeps in order to compensate for imbalanced dispersion in the interferometer is demonstrated. The technique is analogous to the widely used software based dispersion compensation by resampling; however, again the demonstrated “hardware solution” requires no extra processing power. The technique works well for small amounts of dispersion over the whole imaging range or at a single depth even for high dispersion and may become important for future FDML based OCM systems.

This work demonstrates that the steps of software resampling, numerical apodization and numerical dispersion compensation may become obsolete in future high speed OCT systems. Advantages of k-space linear FDML lasers like no additional noise caused by resampling, reduced speed requirements for the ADC, a factor of 2 reduction in data rate, lower storage capacity requirements for large comprehensive data sets and higher sensitivity due to the avoidance of overexposure of the sample may pave the way for future ultrafast real time OCT systems for clinical environments.

## Acknowledgment

We would like to acknowledge support from Prof. Wolfgang Zinth at the LMU Munich, helpful advice from Desmond C. Adler, Vivek J. Srinivasan and James G. Fujimoto from the Massachusetts Institute of Technology and Maciej Wojtkowski from the Nicholas Copernicus University in Torun. This research was sponsored by the Emmy Noether program of the German Research Foundation (DFG) HU 1006/2-1.

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