Integration of liquid crystal optical delay and mechanical stage optical delay for measurement of ultrafast autocorrelations and terahertz pulses

To improve the temporal resolution in an optical delay system that uses a conventional mechanical delay stage, we integrate an in-line liquid crystal (LC) wave retarder. Previous implementations of LC optical delay methods are limited due to the small temporal window provided. Using a conventional mechanical delay stage system in series with the LC wave retarder, the temporal window is lengthened. Additionally, the limitation on temporal resolution resulting from the minimum optical path alteration (resolution of 400 nm) of the conventionally used mechanical delay stage is reduced via the in-line wave retarder (resolution of 50 nm). Interferometric autocorrelation measurements are conducted at multiple laser emission frequencies (349, 357, 375, 394, and 405 THz) using the in-line LC and conventional mechanical delay stage systems. The in-line LC system is compared to the conventional mechanical delay stage system to determine the improvements in temporal resolution relating to maximum resolvable frequency. This work demonstrates that the integration of the in-line LC system can extend the maximum resolvable frequency from 375 to 3000 THz. The in-line LC system is also applied for measurement of terahertz pulses.


Introduction
This paper investigates an optical delay measurement technique, whereby, a method of optical delay using a mechanical delay line, is improved in terms of its temporal resolution through the inclusion of a liquid crystal (LC) wave retarder device.The overall system is characterized and demonstrated Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.through interferometric autocorrelation measurements of an ultrafast laser.Ultimately, the investigated optical delay measurement technique is shown to have drastically improved performance over benchmark traditional ultrafast measurement systems that are based solely on mechanical delay lines.
Interferometry is a fundamental optical technique, typically achieved through mechanical delay lines.Interferometry manifests itself in Fourier transform infrared spectroscopy [1][2][3], terahertz time domain spectroscopy [4][5][6], and autocorrelation measurements of ultrafast lasers [7,8].For many applications, small temporal step size is crucial.In particular, methods based on interferometry with sub-micron wavelengths are susceptible to measurement error due to step size limitations [9].A limiting factor for traditional mechanical-delay-based systems is a coarse minimum incremental motion (minimum step size of the mechanical delay stage, limiting the temporal resolution) [8].
Serna et al approached this problem by implementing an electrically-focused tunable lens to measure fine temporal step sizes in ultrafast laser autocorrelations [10].While this measurement technique is elegant, the actuation that alters the lens focus, includes the compression of a polymer membrane.This can change the focus of the beam as it moves through the tunable lens, creating additional complexity for various optical applications.In contrast, Song et al developed an electrooptic delay method using a lithium niobate substrate [11].This work is instrumental in greatly improving solid state optical delay methods.However, this method requires high operation voltages, and requires precise beam coupling to the optical delay line, making integration into optical systems complicated.Jullien et al approached the challenge of achieving fine temporal step sizes by using a variable optical delay using an LC cell [12].Here, the authors perform theoretical simulations of such an element with an experimentally demonstrated variable optical delay, over short duration time-scales (100 fs).However, typical interferometric experiments require a temporal window that is beyond this small temporal measurement window (i.e. at or beyond 750 fs) [13,14].Of particular interest is the approach of achieving optical delay through optical delay within LC materials, thus reducing the dependence on mechanical delay methods.This is advantageous as it has minimal modification to beam parameters (e.g.beam shaping, fiber optic coupling [10,11,15]).Optical techniques integrating LC devices have improved acquisition parameters [16,17].
This work builds upon our preliminary results [18] and we greatly expand the temporal measurement window through an integration of an LC wave retarder device and mechanical delay line.Our measurement technique places in series an optical delay line for a fine resolution and a mechanical delay line for a coarse resolution of optical delay.The fine resolution introduced by the optical delay line aids in achieving a heightened temporal resolution.The coarse resolution introduced by the mechanical delay line aids in achieving a large temporal measurement window.We provide a detailed characterization of the angle and phase response of the LC wave retarder.Ultimately, we demonstrate this measurement technique over a wide variety of laser emission frequencies, and we provide measurements of a (benchmark) conventional mechanical delay stage system for comparison.The conventional mechanical delay stage system suffers greatly from aliasing due to the low sampling frequency (notated F s in this work) associated with the coarse minimum incremental motion, particularly for frequencies above 375 THz.This is due to the folding frequency for the mechanical delay system (notated F s /2 in this work) being near the laser emission frequency (notated F 0 in this work).In stark contrast, the explored system (optical delay in series with mechanical delay) does not suffer from aliasing effects for the nearinfrared frequencies (349-405 THz) and produces a folding frequency (notated F sLC /2 in this work) well beyond these frequencies.
The remainder of this manuscript is organized as follows: section 2 describes the concept.Section 3 provides a characterization of the angular dependencies of the measurement system components.Section 4 provides a characterization of the optical retardation.Section 5 discusses the performance of the in-line LC system method by contrasting the conventionally used mechanical delay stage method of optical delay.Section 6 applies the in-line LC system to measurement of terahertz pulses.Lastly, this manuscript provides concluding remarks in section 7.

Concept
This work proposes an interferometric autocorrelation system.The photodetection in the interferometric autocorrelation system is achieved using a gallium phosphide photodiode (Thorlabs DET25-K2) as a photo-detector.This method leverages two photon absorption (gallium phosphide bandgap of 2.3 eV) to produce a non-linear response to optical intensity.The mathematics and analysis of two-photon interferometric autocorrelations in gallium phosphide is thoroughly detailed in the work of Chong et al [8].Chong et al use a gallium phosphide detector to produce an interferometric autocorrelation for near-infrared frequencies (315-400 THz).While a gallium phosphide photodetector is a suitable candidate for producing interferometric autocorrelations at near-infrared frequencies, Chong et al notes that they were unable to produce the interference fringes in some of their interferometric autocorrelations due to the 375 THz sampling frequency limitation in the variable mechanical delay stage.Our work extends on the exploration completed by Chong et al to produce an interferometric autocorrelation using a gallium phosphide photodetector.However, our work aims to eliminate the limitation in mechanical delay step size by introducing an in-line LC wave retarder (Thorlabs LCC1115-B).
The laser used for beam emission is a titanium sapphire laser (Spectra Physics Mai Tai), with a tunable emission frequency (F 0 = 288-434 THz).The beam is attenuated to a power of 50 mW before entry into the autocorrelation system.The variable conventional mechanical delay stage in this experiment is the Newport IMS-600CCHA.This delay stage has a minimum incremental step size quoted as 200 nm.Considering there is an incoming path and a reflected path that must travel on the delay stage, each step will alter the total optical path by 400 nm.The limitation imposed by this constraint is the maximum resolvable frequency for an interferometric autocorrelation.The maximum resolvable frequency is calculated by determining the Nyquist frequency (i.e.half the sampling frequency).The sampling frequency (F s ) follows In this equation c is the speed of light in a vacuum, and ∆x is the optical path alteration.Using (1), the maximum sampling frequency of the conventional delay stage system is 750 THz.Using the Nyquist sampling theorem, the maximum measurable frequency is 375 THz (F 0 = F s /2).The LASER is a general representation of a laser head that provides as the point of emission, for the frequency of emission.The LC represents the in-line Thorlabs LCC1115-B liquid crystal wave retarder, for the fine optical delay.The DS represents the conventional mechanical delay stage, for the coarse optical delay.The PD corresponds to the gallium phosphide photodetector used in this experiment.The BS in this schematic is a 50% reflection, 50% transmission beam splitter.The Trn refers to the transmitted path of the beam after passing through the beam splitter.The Ref refers to the reflected path of the beam after passing through the beam splitter.
In figure 1, our implemented experimental design is shown.Here, the mechanical delay line (Newport IMS-600CCHA) is used for coarse optical delay and the wave retarder is used for fine optical delay.The LC wave retarder is placed in the reflected path of the beam splitter.Either path may contain the LC wave retarder, however, this will affect how the retardance must be altered (positively or negatively).To provide benchmark comparison measurements, the LC wave retarder is removed, thus making the measurement system only have a (coarse temporal resolution) mechanical delay line and representing the standard (benchmark) measurement technique used ubiquitously.
For our system, the LC wave retarder is placed in-line to drastically increase the maximum resolvable frequency of the conventional mechanical delay stage system.The in-line LC system provides an increased temporal resolution dependent on the variation in refractive index in response to an applied electric field.To reduce risk of crystal migration we use a voltage controller (Thorlabs LCC25) to produce the electric field.This device produces a 2 kHz voltage signal with protection against the application of a DC offset voltage.
The LC wave retarder used in this experiment is the Thorlabs LCC1115-B, with damage threshold of approximately 23 mJ cm −2 [19].The average transmission of the LCC1115-B is 97% for the laser emission frequencies used in this experiment [19].While the damage threshold can sustain relatively high pulse energies (over those used in the performed experiments, being in the regime of µJ cm −2 ), it is recommended to implement the method of delay on a lowpower branch in a laser system (e.g. the probe branch of a pump-probe experiment), instead of a high-power branch (e.g. the pump branch of a pump-probe experiment).

Angular measurement characterization
In this section, our measurement procedure is described.The measurement procedure uses a fringe counter to identify the slow axis and fast axis of the LC wave retarder with a linearly polarized transmitted beam.This measurement procedure is used to identify the ideal rotation angle of the LC wave retarder if both the axes of the LC wave retarder and the polarization of the laser beam are unknown.The rotation angle of the LC wave retarder is aligned such that the slow axis and the linearly polarized light are parallel.In this configuration, as voltage is applied, the LC wave retarder provides the corresponding phase retardance to the highest percentage of the laser pulse.Otherwise, there will be components of the laser pulse that are not delayed and will cause elliptical polarization, thus the LC wave retarder will act instead as a variable waveplate.Overall, the measurement procedure is advantageous as no additional components are required for the characterization of the LC wave retarder.
To create the fringe counter, the beam emitted from the laser is directed into a beam splitter to branch the beam into two paths (reflected and transmitted) as seen in figure 1. Mirrors are placed to reflect both transmitted and reflected paths back into the beam splitter to produce an interferometer.When the reflected and transmitted paths have equal path lengths, there will be an interference pattern detectable through the recombined beams.This will provide a frame of reference to compare when the LC wave retarder is placed in either the transmitted or reflected path.Once the LC wave retarder is placed in either path, the opposite path must be lengthened to compensate for the additional time required by the laser pulse to travel inside the wave retarder.The interferometer is aligned such that the interference pattern produced in the recombined path is strong (although sensitive to vibration).This produces clear contrast in results for determining the LC wave retarder axes.To observe the change in interference (intensity) as a function of applied voltage to the LC wave retarder, a photodiode is placed in the recombined path.To identify the ideal alignment (slow axis parallel to linear polarization of the beam) of the LC wave retarder, a fringe count is recorded at 10 • rotation increments of the LC wave retarder.The result of this procedure is displayed in figure 2.
In figure 2(a), fringe counts conducted with the laser beam polarization parallel to the slow and fast axis of the in-line LC wave retarder are displayed.The axes of the in-line LC wave retarder are determined as follows: the data representing each axis are conclusively determined through analyzing the root mean square error (RMSE) of each data set.The RMSE of the data set is calculated to identify the data set that is most constant (i.e.all values in the data set are equal to the mean), which will correspond to the point at which the laser beam polarization is parallel to the fast axis.Each data set is normalized to remove inconsistencies in the DC value (which can affect the RMSE) between data sets.Once the fast axis is determined, the slow axis can be identified as the angle that is 90 • rotated from the fast axis, due to the uniaxial composition of the LC wave retarder.The fast axis fringe count is displayed in black.This data set is relatively constant and does not display any sinusoidal behavior that can be expected if one of the beam paths were altered by the in-line LC wave retarder.In contrast, the slow axis fringe count displayed in red, produces the expected sinusoidal behavior that should be observed if the beam path is altered by the LC wave retarder.
In figure 2(b), the full characterization of the LC wave retarder is displayed.The LC wave retarder is rotated at 10 • increments to provide for a full rotation (360 • ) to verify symmetry about the fast and slow axes.The characterization confirms there are two points where the fast axis is parallel to the polarization to the beam (0 • and 180 • ) and thus alignment can be achieved.In addition, the characterization displays symmetry about the fast axes, revealing a uniaxial material property, indicating that the method to characterize the LC wave retarder is valid.For the interferometric autocorrelation experiments conducted in this work, 90 • is chosen as the rotation on the LC wave retarder for use as the in-line optical delay stage.

Optical retardation measurement characterization
To effectively use the in-line LC system, a calibration curve for applied voltage and retardance is developed.This section describes a method to determine the correct calibration curve (relating applied voltage and phase delay) to implement in the voltage controller.The voltage controller calibration curve is found using the fringe counter setup developed to characterize the axes of the LC wave retarder.By sweeping the voltage controller voltage over a range of values (1.66-3.07V in this experiment) a stretching sinusoidal pattern will emerge in the fringe count.Considering the setup is built to determine the fringes of a set wavelength laser, the pattern appears as a sinusoid with a frequency of emission.Five equations for the frequencies of interest (F 0 = 349, 357, 375, 394, and 405 THz) are developed to input into the voltage controller frequencies.The results of this are shown in figure 3.
Figure 3(a) displays the raw data collected from the fringe counter setup.As the voltage is varied one beam path is delayed based on the response of the LC wave retarder.If the response were linear, the graph in figure 3(a) should appear as a sinusoid, instead of a stretched sinusoid.Leveraging this information, a fit (calibration curve) can be applied to the data set to identify the necessary voltage values to input to retrieve the desired optical delay.The calibration curve equation used to generate the data displayed in figure 3(b) follows ( In the data displayed in figure 3 In this equation, V final pk and V initial pk represent the driver voltage values corresponding to the final and initial peaks given by the sinusoidal fit equation derived from figure 3

Temporal autocorrelation
In this subsection, sample temporal autocorrelations are produced from the conventional mechanical delay stage and the in-line LC wave retarder systems.The sample temporal autocorrelations are conducted at a laser emission frequency of 349 THz (corresponding to a wavelength of λ = 860 nm).Displayed in figure 4 are the sample temporal autocorrelations using (a) the conventional mechanical delay stage system and (b) the in-line LC wave retarder system.
In figure 4(a) the temporal autocorrelation measured using the conventional mechanical delay stage system is displayed.
The temporal autocorrelation is observably close to the threshold for an under sampled signal.This is expected because the temporal spacing used for the conventional delay stage system is 1.3 fs (∆x = 400 nm).It is important to consider that the Nyquist sampling theorem (F s ⩾ 2F 0 ) is the guideline set for the analysis of signal spectral features, and not signal time domain features [20].Dossi et al identify multiple applications of time domain analysis that require up to eight times the frequency of interest (F s ⩾ 8F 0 ).Considering the temporal spacing in the data presented in figure 4(a) marginally satisfies the Nyquist sampling theorem (F s = 2.15F 0 ), many data points in the temporal autocorrelation are missing.This nearly insufficient sampling frequency is evidently demonstrated in the inset of figure 4(a), where no semblance of a sinusoid is present.Given this deviation from the strict requirement of Dossi et al the peak to background ratio retrieved is 6.2:1 (less than the ideal 8:1 autocorrelation ratio).
In figure 4(b) the temporal autocorrelation measured using the in-line LC system is displayed.Due to the higher sampling frequency the in-line LC system provides, the peak of the autocorrelation has a higher probability of being sampled.This higher sampling rate reveals a higher peak to background ratio (7:1).This is closer to the ideal 8:1 autocorrelation ratio.This is expected because the temporal spacing used for the in-line LC system is 0.16 fs (∆x = 50 nm).This temporal spacing results in a sampling frequency that significantly surpasses the minimum spectral resolution requirement described by Nyquist sampling theorem (F s = 17.2F 0 ).This abundantly sufficient sampling frequency is evidently demonstrated in the inset of figure 4(b), where a sinusoid with the correct frequency is present.

Frequency domain autocorrelation analyses
In this subsection, the increased temporal resolution provided by the in-line LC system is evaluated using the power spectral density of the temporal autocorrelation measurements of five laser emission frequencies: F 0 = 349, 357, 375, 394, and 405 THz. Figure 5 contrasts the conventional mechanical delay stage performance with the in-line LC system, using the power spectral density of their corresponding temporal autocorrelations.The subfigures correspond to different laser emission frequencies being below (349 THz), marginally below (357 THz), equal to (375 THz), marginally above (394 THz), and above (405 THz) the folding frequency (F s /2 = 375 THz) of the conventional mechanical delay stage system.For figures 5(a)-(e) all the laser emission frequencies are (well below) the folding frequency of the in-line LC system (F sLC /2 = 3000 THz).
Three metrics being, error in emission frequency, fullwidth-half-maximum (FWHM) error, and average noise floor, are provided to quantify system performance.This is shown in table 1. Error in emission frequency quantifies the difference between center frequency provided by the fast Fourier transform of the interferometric autocorrelation data and the actual emission frequency.FWHM error quantifies the difference between the calculated FWHM using the experimental data and the FWHM calibration data for the Spectra Physics  Mai Tai laser used in the experiment.Averaged noise floor quantifies the noise in the experimental data, calculated using the average of a sample region in the power spectral data that should have a value of zero in all data sets (33-320 THz).
The average noise floor is defined, in this work, in three regimes being high (average noise floor ⩾ 0.15), moderate (0.10 < average noise floor < 0.15) and low (average noise floor ⩽ 0.10).Scenarios are highlighted in which the in-line LC system provides a benefit to the system beyond what can be quantified in the conventional delay stage, as it is unable to resolve the emission frequency.This is noted with not applicable (N/A) in table 1.In general, the in-line LC system performs better than or comparable to the mechanical system.Figure 5(a) displays power spectral density results from (top) the conventional mechanical delay stage autocorrelation and (bottom) the in-line LC temporal autocorrelation for a laser emission frequency of 349 THz (below the folding frequency).In the conventional mechanical delay stage power spectral density, there is a small peak (due to the high average noise floor of 0.23) corresponding to the laser emission frequency observable at a frequency of 349 THz.This is expected because this laser emission frequency is below the folding frequency of 375 THz (related to the mechanical limitations on temporal resolution).In the in-line LC autocorrelation the power spectral density displays a significantly more distinct peak (due to the low average noise floor of 0.10) at 349 THz, in addition to a second peak corresponding to the first harmonic.The conventional mechanical delay stage temporal autocorrelation is able to resolve the laser emission frequency, however, we can see that higher frequencies may not be resolvable (seen later in this work).Additionally, both systems were highly accurate in the calculated frequency of emission with the conventional mechanical delay and the in-line LC systems achieving a near-zero emission frequency error (0.23% and 0.16%, respectively).However, the in-line LC system demonstrates a reduction of the FWHM error from the 63% provided by the mechanical delay system, to 26%.
Figure 5(b) displays power spectral density results from (top) the conventional mechanical delay stage autocorrelation and (bottom) the in-line LC temporal autocorrelation for a laser emission frequency of 357 THz (marginally below the folding frequency).In the conventional mechanical delay stage power spectral density, there is a distinct peak (due to the moderate average noise floor of 0.13) corresponding to the laser emission frequency observable at a frequency of 357 THz.This is expected because this laser emission frequency is marginally below the folding frequency of 375 THz.In the in-line LC autocorrelation the power spectral density displays a significantly more distinct peak (due to the low average noise floor of 0.067) at 357 THz, in addition to a second peak corresponding to the first harmonic.The conventional calculated frequency of emission with the conventional mechanical delay stage temporal autocorrelation is still able to resolve the laser emission frequency as it approaches the folding frequency.Both systems were highly accurate in the mechanical delay and the in-line LC systems achieving a near-zero emission frequency error (0.26% and 0.12%, respectively).The in-line LC system continues to demonstrate a reduction of the FWHM error from the 63% provided by the mechanical delay system, to 26%.
Figure 5(c) displays power spectral density results from (top) the conventional mechanical delay stage autocorrelation and (bottom) the in-line LC temporal autocorrelation for a laser emission frequency of 375 THz (at the folding frequency).In the conventional mechanical delay stage power spectral density, there no distinct peak corresponding to the laser emission frequency observable at a frequency of 375 THz.This is expected because this laser emission frequency is equal to the folding frequency of 375 THz.For the in-line LC wave retarder autocorrelation, the power spectral density displays a significantly distinct peak (due to the low average noise floor of 0.077) at 375 THz, in addition to a second peak corresponding to the first harmonic.The conventional mechanical delay stage temporal autocorrelation is no longer able to resolve the laser emission frequency as it approaches the folding frequency.The in-line LC system achieves a near-zero emission frequency error (0.065%).The FWHM is increased to 45% for this emission frequency but remains lower than the previous FWHM errors calculated for the conventional mechanical delay system.However, considering the conventional mechanical delay system can no longer resolve the frequency of emission, the emission frequency error, and corresponding FWHM error, is represented as a N/A value.
Figure 5(d) displays power spectral density results from (top) the conventional mechanical delay stage autocorrelation and (bottom) the in-line LC wave retarder temporal autocorrelation for a laser emission frequency of 394 THz (marginally The power spectral density data of both systems is displayed for the laser emission frequency of 375 THz.The Nyquist sampling frequency of the conventional mechanical delay stage system is provided the in figure and represented as Fs/2.The frequencies corresponding to the first and second harmonic of the emission frequency of the laser are displayed above their corresponding data point which are represented as 2F 0 and 3F 0 , respectively.above the folding frequency).In the conventional mechanical delay stage power spectral density, there is a distinct peak corresponding to the aliasing effect produced by the mixing of the sampling frequency and emission frequency of the laser at 356 THz, being F s − F 0 .This is expected because this laser emission frequency is marginally above the folding frequency of 375 THz.In the in-line LC autocorrelation the power spectral density displays a significantly distinct peak (due to the low average noise floor of 0.058) at 394 THz.The conventional mechanical delay stage temporal autocorrelation is no longer able to resolve the laser emission frequency.The inline LC system achieves a near-zero emission frequency error (0.062%).The FWHM error is decreased back to 26% for this emission frequency.
Figure 5(e) displays power spectral density results from (top) the conventional mechanical delay stage autocorrelation and (bottom) the in-line LC temporal autocorrelation for a laser emission frequency of 405 THz (above the folding frequency).In the conventional mechanical delay stage power spectral density, there is a distinct peak corresponding the aliasing effect at 345 THz, F s − F 0 .This is expected because this laser emission frequency is above the folding frequency of 375 THz.In the in-line LC autocorrelation the power spectral density displays a significantly distinct peak (due to the low average noise floor of 0.058) at 405 THz.The conventional mechanical delay stage autocorrelation is no longer able to resolve the laser emission frequency.The in-line LC system has a near-zero emission frequency error (0.23%).The FWHM error is decreased remains at 26% for this emission frequency.
From the above results, it is apparent that the conventional mechanical delay stage autocorrelation has limitations for near-infrared wavelengths, particularly frequencies above 375 THz.However, the in-line LC system did not experience limitations over the near-infrared laser emission frequencies tested being (349-405 THz).
To illustrate the extremely high folding frequency of the in-line LC delay system, figure 6 shows a measurement for a laser emission frequency displayed up to the folding frequency of 3000 THz.The harmonics of the signal are observable over many octaves.This shows tremendous improvement over the mechanical delay stage system.

Terahertz measurement
This section demonstrates an additional ultrafast photonic application to provide further experimental data where implementation of the in-line LC system provides additional time domain sampling (i.e. higher sampling frequency), beyond the capabilities of the conventional mechanical delay stage.This additional ultrafast photonic application is measurement of a terahertz time domain pulse and is performed using both the conventional mechanical delay and in-line LC systems.In figure 7 the terahertz time domain acquisition using both systems are shown.
The terahertz measurement data in figure 7(a) corresponds to the scan acquired using the conventional mechanical delay system for optical delay.The inset displays the time data between 0.937 and 0.941 ps, with the maximum time domain resolution consisting of only three data points.Figure 7(b) displays the terahertz measurement data acquired using the inline LC system for optical delay.The inset in figure 7(b) displays the same time domain interval as the inset in figure 7(a) between 0.937 and 0.941 ps, only here it contains 27 data points, far exceeding the three data points from the inset of figure 7(a).The large contrast in number of data points, being 27 in comparison to three, demonstrates the fine time domain resolution of the in-line LC system.The increase in data points from the conventional mechanical delay system to the in-line LC system corresponds to an 800% increase in time domain sampling.This demonstrates the quality of the in-line LC system.
Beyond terahertz measurement, the in-line LC system measurement system could also have further impact.In particular, the field of sub-cycle control of ultrafast laser pulses could benefit.Such systems take measurement of attosecond duration ultrafast pulses [21][22][23].Therefore, extremely fine temporal resolutions are required to accurately measure these pulses [13].

Conclusion
This manuscript contributed a methodology to incorporate an LC wave retarder for fine spatial resolution within coarse spatial resolution systems for ultrafast interferometry.This included a method to identify the optimal LC wave retarder orientation for optimal optical delay.The work also proposed a method to identify the correct voltage calibration curve to a voltage controller which provided correct and calibrated increments in optical delay.The work provided a thorough comparison of the conventional mechanical delay system implementation and the presented in-line LC system.This comparison was completed through the integration of the in-line LC system into ultrafast experimentation via an interferometric autocorrelation.The in-line LC system demonstrated the ability to accurately increase the maximum resolvable frequency of the conventional mechanical delay stage system with folding frequency increased from 375 THz to 3000 THz.This work is an important step in replacing mechanical delay lines with optical delay lines for drastically improved performance.The in-line LC system was also applied for measurement of terahertz pulses, showing fine temporal spacing.The in-line LC system may benefit applications such as sub-cycle control of ultrafast laser pulses.

Figure 1 .
Figure 1.A visual schematic of the in-line LC system is displayed.The LASER is a general representation of a laser head that provides as the point of emission, for the frequency of emission.The LC represents the in-line Thorlabs LCC1115-B liquid crystal wave retarder, for the fine optical delay.The DS represents the conventional mechanical delay stage, for the coarse optical delay.The PD corresponds to the gallium phosphide photodetector used in this experiment.The BS in this schematic is a 50% reflection, 50% transmission beam splitter.The Trn refers to the transmitted path of the beam after passing through the beam splitter.The Ref refers to the reflected path of the beam after passing through the beam splitter.

Figure 2 .
Figure 2. Results are shown for the calibration procedure which identifies the ideal alignment of the rotation angle for the LC wave retarder.Displayed in (a) are the data sets from fringe counts conducted with the beam polarization aligned parallel to the fast (black) and slow (red) axis of the LC wave retarder.Shown in (b) is the thorough characterization of the LC wave retarder wave retardance using the RMSE of a fringe count scan produced at each corresponding rotation angle.
(b), the values of a, b and c, in equation (2), are 12.8, −2.20, and −1.09 respectively, for F 0 = 405 THz.The interpreted delay can be validated by using the following equation for calibration ratio, R calib , being (a), displayed in figure3(b).The λ represents the wavelength emitted by the laser (λ = 740 nm, corresponding to 405 THz in this dataset).The k variable corresponds to the number of wavelengths between the initial and final peaks, including the final peak.The calibration ratio calculated in equation (3) should equal approximately one.The final dataset, after the calibration curve is applied, can be viewed in figure3(b).The calibration curve of each frequency chosen for a temporal autocorrelation is displayed in figure3(c).The spacing of the data presented in figure3(b) is nonuniform due to the stretching of the data in figure3(a).The calibration curve, figure3(c), provides data collection with the correct voltage values to provide a uniform optical retardation.

Figure 3 .
Figure 3. Results of the procedure to create each calibration curve for the LC wave retarder voltage controller.In (a), the fringe count data using an emission frequency of 405 THz at LC wave retarder controller voltage values.In (b), the interpreted delay calculated using a fringe count fitting equation, derived through fitting the data in (a) to a sinusoid with a frequency of 405 THz.In (c), each calibration curved is shown for the LC wave retarder voltage controller derived from using the procedure.

Figure 4 .
Figure 4. Results are presented that are produced from sample interferometric autocorrelations conducted using two evaluated methods of optical delay, at a laser emission frequency of 349 THz.The optical delay in each temporal autocorrelation is achieved using (a) the conventional mechanical delay stage system and (b) the in-line LC wave system.Insets are provided in both (a) and (b) between the temporal delay of −0.0368 and −0.0323 ps to provide a visual contrast in sampling.

Figure 5 .
Figure 5.The power spectral density of interferometric autocorrelations performed with the conventional mechanical delay stage (top), and the in-line LC systems (bottom) displayed up to the sampling frequency of the conventional mechanical delay stage system (Fs = 750 THz).The power spectral density data of both systems is displayed for the laser emission frequencies (F 0 ) of (a) 349 THz, (b) 357 THz, (c) 375 THz, (d) 394 THz, and (e) 405 THz.The Nyquist sampling frequency of the conventional mechanical delay stage system is provided on each sub figure and represented as Fs/2.For visual purposes, first harmonic (2F 0 ) is displayed in sub figures (a)-(c).

Figure 6 .
Figure 6.A sample power spectral density of an interferometric autocorrelation performed with the conventional mechanical delay stage (top), and the in-line LC systems (bottom) displayed up to the Nyquist sampling frequency of the in-line LC system (F sLC /2 = 3000 THz).The power spectral density data of both systems is displayed for the laser emission frequency of 375 THz.The Nyquist sampling frequency of the conventional mechanical delay stage system is provided the in figure and represented as Fs/2.The frequencies corresponding to the first and second harmonic of the emission frequency of the laser are displayed above their corresponding data point which are represented as 2F 0 and 3F 0 , respectively.

Figure 7 .
Figure 7. Displayed are the measurements of a terahertz time domain pulse conducted using the two evaluated methods of optical delay.The optical delay in each terahertz measurement is achieved using the conventional mechanical delay stage (a), and the in-line LC wave (b) systems for optical delay.An inset is provided in both (a) and (b) between the temporal delay of 0.937 and 0.941 ps to provide a visual contrast in system sampling frequencies.

Table 1 .
Summary of the temporal autocorrelation measurements performed using the in-line LC and conventional (mechanical) methods of optical delay.