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(a) Intensity and phase of the inverse Fourier transform of the spectral interferometric pattern. (b) Reconstructed spectral intensity and phase relative to the frequency center of the input pulse at 192.31 THz. (c) Reconstructed temporal intensity and phase of the input pulse. The reconstructed intensity profile is compared with the estimated Gaussian profile based on optical autocorrelation (solid curve). (d) Reconstructed temporal intensity and phase profiles of the input pulse using a linear frequency-to-time conversion. The reconstructed pulse is significantly broadened.
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In the complete reconstruction of ultrashort optical pulses based on temporal interferometry, the chromatic dispersion and the optical time delay are two key factors, which determine the measurement accuracy. Due to the higher order dispersion, the wavelength-to-time mapping becomes nonlinear, leading to a nonuniformly spaced interference pattern a...
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... over all the successive frequency compo- nents separated by within the whole spectrum. An experi- ential evaluation of the spectral shear is , where is the spectral FWHM of the PUT. On the other hand, the time delay difference should be large enough relative to the pulsewidth to perform the Fourier-transform algorithm, as will be shown in Fig. 8(a). Finally, for a given bandwidth of the receiver, the second-order dispersion can be chosen to op- timize the sensitivity of the ...
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... employ the phase-retrieval algorithm [21] to reconstruct the intensity and phase of the input ultrashort pulse. As the first step, the spectral interferogram in Fig. 7(a) is inverse Fourier transformed. The result is shown in Fig. 8(a). Then, the right sideband in Fig. 8(a) is selected, which represents the interfer- ence term in (9). The selected sideband is shifted by to the origin, then Fourier transformed to retrieve the spectral inten- sity and to calculate the spectral phase by integrating the rel- ative phase at discrete frequencies, as shown in Fig. 8(b). As ...
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... employ the phase-retrieval algorithm [21] to reconstruct the intensity and phase of the input ultrashort pulse. As the first step, the spectral interferogram in Fig. 7(a) is inverse Fourier transformed. The result is shown in Fig. 8(a). Then, the right sideband in Fig. 8(a) is selected, which represents the interfer- ence term in (9). The selected sideband is shifted by to the origin, then Fourier transformed to retrieve the spectral inten- sity and to calculate the spectral phase by integrating the rel- ative phase at discrete frequencies, as shown in Fig. 8(b). As the final step, the temporal intensity ...
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... is shown in Fig. 8(a). Then, the right sideband in Fig. 8(a) is selected, which represents the interfer- ence term in (9). The selected sideband is shifted by to the origin, then Fourier transformed to retrieve the spectral inten- sity and to calculate the spectral phase by integrating the rel- ative phase at discrete frequencies, as shown in Fig. 8(b). As the final step, the temporal intensity and phase of the original pulse is obtained by calculating the inverse Fourier transform of the spectrum in Fig. 8(b), with the result shown in Fig. 8(c). It has been pointed out that the original pulses and the stretched pulses have the same spectrum; thus, if the temporal resolution of an ...
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... to the origin, then Fourier transformed to retrieve the spectral inten- sity and to calculate the spectral phase by integrating the rel- ative phase at discrete frequencies, as shown in Fig. 8(b). As the final step, the temporal intensity and phase of the original pulse is obtained by calculating the inverse Fourier transform of the spectrum in Fig. 8(b), with the result shown in Fig. 8(c). It has been pointed out that the original pulses and the stretched pulses have the same spectrum; thus, if the temporal resolution of an OSC is not high enough to characterize the original pulses, it would not be able to characterize the stretched pulses either [12]. However, in this paper, the ...
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... to retrieve the spectral inten- sity and to calculate the spectral phase by integrating the rel- ative phase at discrete frequencies, as shown in Fig. 8(b). As the final step, the temporal intensity and phase of the original pulse is obtained by calculating the inverse Fourier transform of the spectrum in Fig. 8(b), with the result shown in Fig. 8(c). It has been pointed out that the original pulses and the stretched pulses have the same spectrum; thus, if the temporal resolution of an OSC is not high enough to characterize the original pulses, it would not be able to characterize the stretched pulses either [12]. However, in this paper, the required sampling rate is de- termined ...
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... is not high enough to characterize the original pulses, it would not be able to characterize the stretched pulses either [12]. However, in this paper, the required sampling rate is de- termined by the frequency of the temporal interferogram. In the inverse Fourier transform as the last step, by adding zero values at both sides of the spectrum in Fig. 8(b), we can improve the sampling rate in the time domain. The reconstructed intensity profile in the time domain is compared with the Gaussian in- tensity profile measured using an optical autocorrelator, an ex- cellent agreement is reached. The product of the temporal and spectral widths (both measured as FWHM) is 0.433, which in- dicates ...
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... is 0.433, which in- dicates that the original pulse is nearly transform limited. To show the impact of the higher order dispersion on the proposed technique, we reconstruct the phase and the intensity profiles of the ultrashort pulse using a linear fitting in the calibration pro- cedure, i.e., only the GVD is considered. The result is shown in Fig. 8(d). As can be seen, the pulse is broadened. The broad- ening is mainly due to the TOD, which has been predicted by (12). Some jumps in the phase profile corresponding to the zero intensity points are also observed. Obviously, without consid- ering the effect of the higher order dispersion, the reconstructed intensity profile of the PUT is ...
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... This method significantly reduces the Doppler effect of high-speed moving objects to the femtosecond level [18]. It can also achieve high measurement accuracy by recording the time-domain and frequencydomain interference fringes separately using an oscilloscope and a spectrometer [19,20]. It holds great promise for application in axial clearance measurements, although no related articles have been reported yet. ...
Rotor-stator axial clearance is a crucial design parameter affecting rotating machines’ efficiency and safety. To accurately measure the dynamic axial clearance in high-speed machinery, a precise method based on time division multiplexing with frequency domain interferometry has been proposed. This method has proven robust and accurate through simulations and experiments. The inclusion of an optical switch enables the utilization of dispersive interferometry(DPI) and time division multiplexing for multiple channels of the light source. It achieves a static accuracy of 1.5 µm for a 15 mm range and a dynamic accuracy of 9 µm at 3000 rpm.
... In the final step, the optical phase can be accumulated by phase delay retrieval algorithm [42], [43]. According to (3), the phase of the spectral interferogram has linear relationship relative to the frequency. ...
... The accumulated phase and linear fitting lines are shown in Fig.5 (b). The accumulated phase of the spectral interferogram is obtained by the phase retrieval algorithm [42], [43]. Note that, the origin of the fitting line is shifted to zero to emphasize the relation between the fitting slope and displacements. ...
A femtosecond laser metrology with nanometer precision and dynamic range of centimeter incorporating time-stretch interferometry and phase delay retrieval method is proposed and experimentally demonstrated. Displacement encoded phase-sensitive temporal interferogram is generated when phase stabilized femtosecond laser pulses transmitting through a time-stretch interferometer. To avoid the chirp of temporal interferogram due to high-order dispersion, time-to-frequency mapping is established to transform the temporal interferogram to the spectral interferogram for high-speed detection. The phase delay of spectral interferogram corresponding to specific displacements is retrieved and accumulated for linear fitting. The fitted slope is the time delay between two arms in the interferometer therefore displacement can be calculated. For precision verification, the preset displacements are measured. Mean error and standard deviation are presented after sliding average of measured results in 4 s.
... The chromatic dispersion up to group delay dispersion performs an optical Fourier transform to the pulse. 35 The reference pulse transmits through an optical delayer and an additional DCF. So considering the dispersion up to group delay dispersion, the complex pulse 35 can be expressed as ...
... 35 The reference pulse transmits through an optical delayer and an additional DCF. So considering the dispersion up to group delay dispersion, the complex pulse 35 can be expressed as ...
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... The reference pulse transmits through an optical delayer and an additional DCF. So considering the dispersion up to group delay dispersion, the complex pulse30can be expressed as 2 2 p p 2 0 p / ( ) exp / 2 t L a t h jt L a (3) 2 I 2 r r 2 0 I r / ( ) exp / 2 t L L a t h j t L L a (4)where 0.5 p 0 2 0 0 0 2 exp h H j L j L and 0.5 r 0 2 0 I 0 0 I 2 exp ( ) h H j L L j L L are the complex amplitude of the two complex pulse. 0 H is a constant of the fiber transmission rate and 0 is the zero-order mode-propagation constant.9For ...
Dual-comb spectroscopy is a promising method for precise optical spectrum analysis with fast data acquisition speed. However, its implementation and applications are often hindered by the complexity of optical comb systems. Here, as a compact and robust system, femtosecond imbalanced time-stretch spectroscopy (FITSS) with simple optical layout is proposed and demonstrated. The time-stretch interferometry from one femtosecond laser builds a mapping from the optical frequency domain to the radio frequency regime. In experiment, the absorption line of a hydrogen cyanide cell is encoded in the probing arm of a Mach-Zehnder interferometer (MZI). The down-converted radio frequency comb is transformed from a periodically chirped waveform, which is the interferogram of the MZI with different dispersion values on two arms. By turning the optical filter, the spectrum over a wide range is analyzed.
... At the MZM output, the encoded optical waveforms are propagated down an L km length of the SSMF to a remote location, where the PD is located. Generally, the complex baseband frequency domain transfer function of a SSMF is expressed, in terms of its Taylor series expansion, as [7]: ...
... Therefore a unique linear one-to-one mapping between the optical frequency and time is created thanks to the linear group delay response with respect to the frequency. If higherorder dispersion is taken into account [54][55][56], the frequency-to-time mapping process is still valid so as long as the temporal Fraunhofer approximation is satisfied. In this scenario, however, the mapping is no longer linear due to the higher-order frequency dependency of the group delay. ...
... This technique relies on spectral interferometry measurement and its update rate is limited by the refresh rate of the optical spectrometer (typically up to ~ 10 kHz), hence not capable of real-time single-shot characterization of ultrashort optical pulses. A promising solution is real-time spectral interferometry [54,91], which combines SPIDER and DFT techniques, hence enabling pulseby-pulse characterization of the complex field (amplitude and phase) with a significantly higher update rate of up to several GHz. ...
... This makes the system not stable due to the inherent high sensitivity of the interferometer to environmental perturbations. To remove this difficulty, in [54,91], an alternative real-time spectral interferometry method without using optical interferometers has been reported [65]. Figure 4 illustrates the principle of the method. ...
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... If the dispersion up to the third order is considered, then a new wavelength-to-time mapping function would be used, which is given by [28] ω = tΦ − Φ ::: ...
In this paper, techniques to generate microwave arbitrary waveforms based on all-fiber solutions are reviewed, with an emphasis on the system architectures based on direct space-to-time pulse shaping, spectral-shaping and wavelength-to-time mapping, temporal pulse shaping, and photonic microwave delay-line filtering. The generation of phase-coded and frequency-chirped microwave waveforms is discussed. The challenges in the implementation of the systems for practical applications are also discussed.Highlights► Techniques to generate arbitrary microwave waveforms are reviewed. ► Microwave arbitrary waveform generation (AWG) based on direct space-to-time is discussed. ► Microwave AWG based on spectral-shaping and wavelength-to-time mapping is discussed. ► Microwave AWG based on temporal pulse shaping is discussed. ► Microwave AWG based on photonic microwave delay-line filtering is discussed. ► Examples for generation of frequency-chirped and phase-coded microwave waveforms are provided.
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... using the Fourier transform [14], [15] or Hilbert transform [17], [18] algorithms. In the self-referencing time-domain interferometric techniques [12], [14], [15], [17], [18], since the interferogram can be recorded in the time domain by using a high-speed real-time oscilloscope, a complete characterization of an ultrashort optical pulse with a high update rate of up to a gigahertz can be achieved. However, since an optical interferometer implemented by using discrete components, such as a glass plate [14], a fiber-optic Sagnac interferometer [15], or a free-space Michelson interferometer [12], [17], is usually used to generate two time-delayed interfering optical pulses, the performance of pulse characterization methods is greatly affected by the poor stability of the interferometer due to its inherent high sensitivity to environmental perturbations, leading to considerable errors in the phase measurement process. ...
We propose and demonstrate a simple method for the full characterization of an ultrashort optical pulse based on temporal interferometry, using an unbalanced temporal pulse shaping (UB-TPS) system. The UB-TPS system consists of a Mach-Zehnder modulator and two dispersive elements (DEs) having opposite dispersion, but nonidentical in magnitude. The entire system can be considered as a typical balanced TPS system for a real-time Fourier transformation to generate two time-delayed replicas of the input optical pulse, followed by a residual DE to perform a second real-time Fourier transformation to convert the two time-delayed pulse replicas to two frequency-sheared optical spectra. The spectral interferometry is performed in the time domain. The spectral magnitude and phase information of the input optical pulse is accurately and unambiguously reconstructed from the recorded temporal interference pattern based on a Fourier transform algorithm. Compared with a conventional pulse characterization system based on linear interferometric measurement using an optical interferometer implemented by using discrete components, the proposed system features better stability, higher adaptability, and single-shot measurement. The use of the proposed system for the characterization of a femtosecond pulse before and after passing through a 60-m-long single-mode fiber is experimentally demonstrated.
... If the dispersive device has only the SOD and the disper- sion is large enough to satisfy , then a linear wavelength-to-time mapping, that is, , would result, where is the time relative to the time delay due to . If the dispersion up to the third order is considered, then a new wavelength-to-time mapping function would be used, which is given by [22] (3) Then, in the proposed system, the signal at the output of the MZM is given by ...
An unbalanced temporal pulse-shaping (TPS) system for chirped microwave waveform generation is proposed and demonstrated. The proposed system consists of an ultrashort pulsed source, a Mach-Zehnder modulator and two dispersive elements. The dispersions of the two dispersive elements are opposite in sign, but not identical in magnitude. The entire system is equivalent to a conventional balanced TPS system with two complementary dispersive elements for real-time Fourier transformation and a third dispersive element to achieve a second real-time Fourier transformation. The key contribution of this work is that the third-order dispersion of the dispersive elements is considered, which leads to the generation of a frequency-chirped microwave waveform. A theoretical analysis is performed in which a mathematical model that relates the second- and third-order dispersion of the dispersive elements and the chirp rate of the generated microwave waveform is developed. The theoretical model is then verified by numerical simulations and an experiment. A chirped microwave waveform with different chirp rates of -0.0535 and 0.715 GHz/ns by tuning the third-order dispersion using a tunable chirped fiber Bragg grating is experimentally demonstrated.
... Linear interferometric measurement can also be done in the time domain based on temporal interferometry where two time-delayed replicas of the input pulse are generated using an interferometer and then temporally stretched in a dispersive medium. The magnitude and phase of the input pulse are reconstructed from the temporal interference of the two time-delayed and stretched pulses using Fourier-transform [5], [6] or Hilbert-transform [7] algorithms. Since a fiber-optic Sagnac interferometer [6] or a free-space Michelson interferometer [7] is usually used to generate two delayed replicas of the input pulse, the performance of pulse characterization methods is greatly affected by the instability of the interferometer due to its sensitivity to environmental perturbations, leading to considerable errors in the phase measurement process. ...
... The magnitude and phase of the input pulse are reconstructed from the temporal interference of the two time-delayed and stretched pulses using Fourier-transform [5], [6] or Hilbert-transform [7] algorithms. Since a fiber-optic Sagnac interferometer [6] or a free-space Michelson interferometer [7] is usually used to generate two delayed replicas of the input pulse, the performance of pulse characterization methods is greatly affected by the instability of the interferometer due to its sensitivity to environmental perturbations, leading to considerable errors in the phase measurement process. A feedback control loop can be introduced in the interferometer to minimize the measurement errors [8]. ...
... The magnitude and phase information of the input pulse to be measured are reconstructed from the recorded temporal interferogram based on a Fourier transform algorithm [5]. Unlike the previous systems in5678, no interferometer is involved in the proposed method. The system stability is significantly improved, enabling complete characterization of a sub-picosecond optical pulse with high accuracy. ...
In this paper, we demonstrate a simple method for the full characterization of an ultrashort optical pulse based on temporal interferometry using an unbalanced temporal pulse shaping (UB-TPS) system. The UB-TPS system is functioning to generate and stretch two time-delayed replicas of the input pulse. The magnitude and phase information of the input pulse is reconstructed from the recorded temporal interference of the two time-delayed and dispersed pulses based on a Fourier transform algorithm.
