1. Introduction

Holography in the spatial domain is a well-known two-step imaging process that was first introduced by D. Gabor in 1948 [1]. In the proposed technique, a suitable coherent reference wave interferes with the light diffracted by or scattered from an object. In this way, the entire information (i.e., amplitude and phase) of the diffracted or scattered waves can be recorded and subsequently reconstructed, in spite of the fact that recording media (e.g., a photosensitive film) respond only to the intensity of light [2, 3]. Some years later, computer generated holograms (CGH) were introduced [4, 5], providing the possibility to generate and process images without the need of the real object. In this method, the desired interference pattern, i.e., the hologram, is calculated computationally and transferred to the photosensitive film through a printing or plotting device.

Spatial holography has been widely investigated for a broad range of applications in many different fields, including three dimensional imaging, optical signal processing, data storage, and microscopy [6–9]. Recently, there is an important trend towards the development of dynamic holography, e.g. to achieve 3D images in motion. These systems are based on conventional spatial-holography schemes where the time variable is additionally considered. Dynamic holographic displays have been realized based on spatial light modulators [10, 11] and photorefractive polymers [12]. However, all these developments are focused on displaying and/or processing spatial-domain information. On the other hand, classical holography concepts have been also used to process time-domain information by first transferring this information into the spatial domain, e.g. through a conventional diffraction grating – based configuration [13], or by directly imprinting a temporal interferogram along the spatial domain in an holographic recording material [14].

In this paper, we introduce and demonstrate the exact time-domain counterpart of classical (spatial-domain) holography. This concept essentially involves the recording, generation and/or processing of complex (amplitude and phase) time-domain signals using intensity-only temporal detection and/or modulation optical devices. The resulting procedures greatly simplify present approaches aimed to similar generation and processing tasks. The concept presented here can be interpreted as a new milestone within the context of the so-called space-time duality [15]. The general space-time duality theory builds up over the mathematical equivalence between free-space paraxial diffraction and narrow-band temporal dispersion. The space-time duality has enabled researchers to identify and create temporal equivalents of a large number of signal-processing tools previously developed in spatial optics, including the time-lens concept, temporal imaging and self-imaging systems, real-time Fourier transformation and filtering etc [15–20]. However, to the best of our knowledge, the time-domain equivalent of classical holography is introduced and formalized in our paper for the first time.

The problems of detection, generation and processing of complex time-domain signals have become increasingly important in several fields, particularly in high-speed optical telecommunications and ultrafast information-processing systems [21–24]. In these systems, the desired information is now typically encoded in both amplitude and phase temporal variations. Additionally, the impairments undergone by data signals in an optical communication link or a signal-processing device affect both the amplitude and the phase temporal signal profiles. The time-domain equivalent of the recording process in holography involves photo-detection of the interference between the temporal complex waveform under analysis (information signal) and a reference optical local oscillator, typically a continuous-wave, CW, light at a wavelength properly shifted from the spectral content of the information signal. This procedure is already employed for complex optical signal characterization [25] and it is usually referred to as heterodyne detection. In a conventional heterodyne detection scheme, the amplitude (or intensity) and phase temporal profiles of the optical information signal are numerically recovered from the recorded interferogram using an algorithm based on Fourier transforms. The time-domain holography scheme introduced here goes one important step further by using the detected electrical interferogram (“temporal hologram”) for generation (or additional processing) of an exact replica of the original complex information signal directly in the optical domain. This is achieved in a fairly simple fashion, by intensity-only temporal modulation of CW light with the recorded interferogram, e.g. using a single electro-optic (EO) Mach-Zhender modulator (MZM), combined with a suitable band-pass optical filter. As anticipated, this methodology significantly simplifies present CW-light modulation approaches for optical complex signal generation, which typically require the combination of at least two different, properly synchronized EO modulation processes, e.g. I/Q temporal modulation or amplitude plus phase temporal modulation [21–24]. Alternatively, the modulating interferogram signal can be also computationally designed for generation of a desired, user-defined optical complex waveform using the described single intensity-modulation scheme. This process can be interpreted as the time-domain counterpart of CGH.

The ideas introduced in this paper are experimentally demonstrated through the generation and subsequent retrieval of user-defined, complex optical temporal waveforms using the predicted, simpler schemes based on time-domain equivalents of holographic concepts. In particular, a sequence of arbitrarily chirped Gaussian-like pulses and a 16-state Quadrature Amplitude Modulation (16-QAM) data stream are successfully generated using intensity-only electro-optic (EO) modulation of a continuous wave (CW) laser.

2. Principle

Figure 1 shows the space-time analogy between classical spatial holography and the proposed time-domain concept. In the spatial-domain holography system (Fig. 1(a)), complex information of a specified monochromatic wave e_s(x,y) (where x and y refer to the spatial variables that are transversal to the chosen light propagation direction) can be stored as an intensity pattern i(x,y) in a photosensitive film after interfering with a reference wave, which is selected to be a mutually coherent uniform plane wave e_LO propagating with a certain angle (${\theta }_y$) with respect to the information wave-front. The intensity pattern i(x,y) can also be designed computationally, e.g. by emulating the mentioned interference process, and this procedure is typically referred to as CGH [5]. The complex information (i.e., amplitude and phase) of the original wave can be subsequently recovered by illumination of the recording film with the same optical reference wave e_LO. The field transmitted through the film is diffracted into different components, which are angularly separated. The propagation angle ${\theta }_y$ is related to the spatial-domain frequency ${\nu }_y={\theta }_y/\lambda$, where $\lambda$ is the wavelength of the information and reference signals. Therefore, considering a sufficiently large value for${\theta }_y$, the original wave-front e_S(x,y) can be successfully recovered [3]. In what follows we establish an analogy between the described processes of recording and retrieval of complex information encoded in spatial-domain waveforms and equivalent processes for temporal waveforms.

 

Fig. 1 Space-time duality between classical spatial holography and time-domain holography: (a) Implementation steps of classical spatial holography; (b) Implementation of the concept of time-domain holography as the temporal counterpart of spatial-domain holography. For the sake of simplicity, the optical temporal signals are represented by their amplitude envelope. OC, optical coupler; PD, photodetector; R, resistor; AWG, arbitrary waveform generator; MZM, Mach-Zehnder modulator.

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2.1 Complex optical signal recording

The complex optical information signal e_S(t) is a purely time-domain variation, defined as$e_S(t)=e_{s0}(t)\exp \left[j({\omega }_{S}t+{\varphi }_S)\right]$, where $e_{s0}(t)=\left|e_{s0}(t)\right|\exp \left[j\angle \left\{e_{s0}(t)\right\}\right]$ is the complex amplitude envelope, denotes the optical carrier frequency, and ${\varphi }_S$ is an arbitrary constant phase. For recording purposes, the signal is made to interfere with a reference signal, which is selected to be a CW $e_{LO}(t)=\sqrt{i_{LO}}\exp \left[j({\omega }_{LO}t+{\varphi }_{LO})\right]$, where i_LO is the constant optical intensity of the CW reference, is the reference optical carrier frequency and ${\varphi }_{LO}$ is an arbitrary constant phase. The interference between the information and reference signals is observed at the output of an optical coupler (OC), and the resulting time-domain intensity pattern i(t) is recorded, e.g. using a high-speed photo-detector (PD) (Fig. 1(b)). The intensity pattern i(t) is given by

This method for complex optical signal detection is extensively employed nowadays, known as heterodyne detection. Typically, two PDs are commonly employed, following a configuration known as balanced photodetection [25]. The signals $e_s(t)$and $e_{LO}(t)$ are added in phase and counter phase in each photodiode, and the resulting electrical waveforms are conveniently subtracted. Only the third term in the RHS of Eq. (1) remains, removing the autocorrelation term. In this case, the bandwidth requirements of the detection process are reduced by two fold, i.e. the photo-detection bandwidth should be at least twice that of the information signal, or, in other words, the intermediate frequency should be fixed to satisfy ${\omega }_i\ge B/2$.

The fact that the autocorrelation term imposes the need for a detection bandwidth at least four times higher than the bandwidth of the information signal is a well-known issue in traditional holography [3]. Inspired by holography concepts, it is worthy to note that in Eq. (1), the first term in the RHS may be neglected if the reference intensity is sufficiently strong, i.e. whenever $\sqrt{i_{LO}}\gg \left|e_S(t)\right|$, leading to a significant decrease of the bandwidth specifications (at least by two fold). This idea can be readily exploited to relax by half the bandwidth requirements in the typical optical heterodyne phase detection scheme based on a single PD. If the above condition is satisfied, the photo-detection bandwidth should be only twice broader than that of the information signal. As mentioned above, this requirement is similar to that of balanced dual photo-detection schemes.

In an optical heterodyne scheme, the temporal phase profile of the information signal is numerically recovered from the measured time-domain interferogram, typically using a Fourier-transform based algorithm [26]. In exact analogy with classical holography, the work reported here goes one step further by proposing the direct use of the recorded temporal interferogram to re-create optically the original complex information signal using time modulation of the reference CW light with this recorded interferogram (i.e. the “temporal hologram”). Moreover, the temporal hologram may be also created numerically through a process that can be interpreted as the time-domain counterpart of CGH [5]. In this case, we computationally model the recording step of holography by numerically emulating the photo-detected signal in Eq. (1), taking into consideration the desired complex optical information signal, e_s(t) , to be subsequently generated. In this time-domain CGH process, the DC components can be directly omitted, which translates into the above-described two-fold photo-detection/modulation bandwidth decrease. The numerically computed temporal hologram can be practically generated by means of an electrical arbitrary waveform generator (AWG), as shown in Fig. 1(b).

2.2 Optical complex signal reconstruction/generation

The optical CW reference signal, $e_{LO}(t)$, is temporally modulated in intensity by a waveform proportional to the temporal hologram i(t) in Eq. (1). This can be practically implemented using a single Mach-Zehnder modulator (MZM) driven by a voltage v(t) proportional to i(t). A graphic example of the spectrum of the optical signal at the MZM output is represented in Fig. 1(b). In exact analogy with its spatial-domain counterpart, Fig. 1(a), the generated optical signal has a frequency spectrum composed by the four terms in Eq. (2), but spectrally shifted by the reference optical frequency,${\omega }_{LO}$.

In line with our discussions above, the first term in the RHS of Eq. (3) may be neglected whenever $\sqrt{i_{LO}}\gg \left|e_S(t)\right|$. The bias point of the MZM must be chosen so that the bias voltage plus the DC value of the electrical signal (whose modulated spectral response is described by the second term in the RHS of Eq. (3)) correspond to the MZM’s minimum transmission point, eliminating the strong discrete tone at ${\omega }_{LO}$ and ensuring the required changes of phase in the sinusoidal temporal component (third and forth terms of Eq. (3)). In any case, the complex-field (amplitude and phase) information optical signal can be recovered from the modulated waveform by simply filtering in the corresponding spectral component, using a suitable optical band-pass filter centered at ${\omega }_S$. The filter should ideally exhibit a flat-top spectral amplitude response and a linear spectral phase profile over the bandwidth of the information signal. It is important to note that this band-pass filtering procedure is the time-domain equivalent of the image selection process by angular diffraction in spatial holography. Indeed, in the space-time duality framework, the base-band temporal frequency variable plays the analog role of the angular frequency variable.

The frequency of the CW source used to generate the complex waveform ultimately determines the central optical frequency of the information term to be filtered in. This procedure easily enables to locate the generated optical waveform around the wavelength of interest, effectively implementing a wavelength conversion process. Notice also that the modulating signal is composed by the target signal and its temporal conjugate. Thus, if the band-pass filtering procedure is implemented to select the spectral component corresponding to the conjugated signal, one could directly achieve temporal phase conjugation (TPC) of the original optical waveform. TPC has proved useful for linear impairment (e.g. dispersion) compensation in fiber-optics telecommunication links [27].

3. Experiments

In this work, we apply the introduced concepts to the generation of arbitrary, user-defined complex temporal signals by intensity-only modulation of CW light. The modulating electrical signals are numerically computed following the above described steps, i.e. through a procedure that can be interpreted as the time-domain analog of CGH. In particular, two distinct optical complex signals are targeted. Firstly, a stream of 16 arbitrarily chirped optical Gaussian pulses is generated to prove that the proposed technique has the capability to generate optical complex waveforms with purely arbitrary, user-defined temporal phase profiles. Secondly, as a practically relevant example, a telecom 1024-symbol 3-Gbps optical data stream under a complex modulation format (16-QAM) is experimentally generated. The amplitude and phase temporal profiles of the two generated signals are characterized using the aforementioned enhanced heterodyne detection method with a single photo-detector, thus entirely emulating the described time-domain equivalents of spatial holography processes.

The experimental setup is shown in Fig. 2. A CW laser generates a reference signal centered at$f_{LO}={\omega }_{LO}/2\pi =193.381THz,$which is split by a 10/90 OC for further use in the generation and the subsequent detection processes. The CW light acts as a carrier for a 10-GHz dual-drive MZM (5.1-V biased, corresponding to its transmission minimum). The modulator is driven by an electrical waveform $v(t)$(temporal hologram), which is numerically designed using the CGH-concepts described in section 2.1. The time-domain hologram $v(t)$ is then practically generated using an electronic AWG, namely the AWG-7122C from Tektronix with 3dB-bandwidth of 9.6 GHz. The MZM optical output is proportional to $v(t)$and centered at f_LO. Hence, the lobe corresponding to the spectrum of e_S(t), which is shifted by $f_i={\omega }_i/2\pi =4.5GHz,$with respect to f_LO, is centered at $f_S={\omega }_S/2\pi =193.385THz.$ Obviously, the optical central frequency of the generated signal can be easily tuned by correspondingly tuning the oscillator frequency f_LO. After an amplification stage, the resulting modulation signal is band-pass filtered using a tunable optical filter (Santec OTF-350) centered at f_S, and the complex desired data stream is finally generated. To validate this claim, the intensity and phase temporal profiles of the resulting optical signal are measured by the above-described improved recording holographic step, using a single 10-GHz PD attached to a real-time oscilloscope, namely, the DSO90254A Infiniium from Agilent. The power split in the first OC together with the losses in the upper arm of Fig. 2 intrinsically introduce a sufficiently high power difference to satisfy the above stated condition, $\sqrt{i_{LO}}\gg \left|e_S(t)\right|$, leading to the anticipated two-fold increased BW efficiency. In all tested cases, the photo-detected temporal interferogram is nearly identical to the numerically designed modulation signal $v(t)$ (temporal hologram), validating the above-described numerical design procedure and overall time-domain holography theory. Finally, the intensity and phase temporal profiles of the signal under test are numerically recovered from the recorded temporal interferogram using a conventional Fourier transform – based algorithm [26] for further validation through a direct comparison with the target data. Note that, in Fig. 2, the frequency bandwidth of the modulating time-domain waveform (temporal hologram), which in turn determines the bandwidth of the generated complex optical waveform, is limited by either the operation bandwidth of the MZM or the processing speed of the employed AWG.

 

Fig. 2 Setup used to demonstrate the time-domain equivalent of CGH for optical complex signal generation usign a single MZM. The figure also shows the signal in time (black) and frequency (blue) along the setup. The optical path is represented by black lines, whereas the electrical path is represented by red lines. AWG, electrical arbitrary waveform generator; CW, continous wave; MZM, Mach-Zenhder modulator; EDFA, Erbium-doped fiber amplifier; BP, band pass; PD, photodetector; OSC, sampling oscilloscope.

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3.1 Arbitrarily chirped optical Gaussian pulses

The complex envelope of the target optical stream of 16 chirped Gaussian pulses is given by

_{$e_{S0}(t)=A\cdot {\displaystyle \sum _{m=1}^{16}\exp \left\{-\frac{(1-j{C}_m)}{2}\frac{{(t-m{T}_S)}^2}{T_0^2}\right\}},$} (4)where A is the pulses’ constant peak amplitude, T₀ ( = 260ps) defines the time width of each Gaussian waveform, C_m is the chirp parameter and T_S ( = 1.3ns) is the sequence period. The stream is designed such that each pulse in the data stream exhibits a different chirp value, and in particular, C_m ranges from C₁=2 to C₁₆=-2. The computationally designed temporal hologram $v(t)$has a 3dB bandwidth of ~6 GHz and is plotted in Fig. 3(a). The spectra of the optical modulated signals after the MZM and after the optical BP filter are represented in Fig. 3(b) and 3(c), respectively. The latest one corresponds to the spectrum of the target optical complex pulse stream$e_{s0}(t)$, centered at the desired optical frequency $f_S={\omega }_S/2\pi =193.385THz.$ Figs. 3(d) and 3(e) show the measured intensity and phase temporal profiles of the generated optical signal, together with the target data. The figure shows an excellent agreement between the measured (blue curve) and the target complex waveform (dashed red curve), in both amplitude and phase profiles.

 

Fig. 3 16-symbol sequence of Gaussian pulses arbitrary chirped. (a) Electrical signal v(t) generated by the AWG; (b) Power spectral density (PSD) of the optical signal after the intensity modulation; (c) PSD of the optical signal after the band pass filtering; (d) Intensity of the generated complex waveform (blue) and target intensity (red); (e) Phase of the generated complex waveform (blue), obtained by applying off-line digital signal processing to the electrical interferogram measured in the detection process, and target phase profile (red).

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3.2 16-QAM modulated data stream

Our second example proves the generation of a telecom 1024-symbol 3-Gbps 16-QAM modulated data stream. Its complex envelope is defined as

 

Fig. 4 1024-symbol 16-QAM optical data stream. (a) A portion of the numerically designed temporal hologram, shown over 16 consecutive symbols, as generated by the electrical AWG; (b) Intensity and (c) phase profiles of the generated complex optical signal (blue line) and target data stream (red line) over the signal portion (16 symbols) shown in (a); (d) Constellation of the generated data stream (blue points) and ideal constellation of a16-QAM signal.

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The results presented above convincingly prove the concepts introduced here. Further optimization of the employed experimental setups would enable to minimize some of the observed deviations in the generated optical complex waveforms. A main source of deviations in our specific experimental setup concerns the non-ideality of the used band-pass filter, preventing a complete and accurate filtering out of the unwanted spectral content from the modulated optical signal. On the other hand, the plotted results of the output phase also present some deviations due to the noise in the photo-detection procedure.

4. Conclusions

We have introduced the formal time-domain counterpart of spatial-domain holography. This concept allows the reconstruction of the amplitude and phase temporal profiles of a target complex optical signal in a simultaneous fashion, by use of intensity-only detection and modulation devices. This approach has been employed to experimentally demonstrate the generation of optical waveforms with arbitrary, user-defined complex (amplitude and phase) modulation patterns, e.g. a sequence of arbitrarily chirped Gaussian pulses or a 3-Gbps 16-QAM modulated data pattern, by using a extremely simple setup involving intensity-only modulation of a CW light source and band-pass filtering.

Considering the broad range of applications of classical holography, one could foresee a similarly vast number of interesting uses for its time-domain counterpart. For instance, as briefly mentioned above, the described time recording and generation holographic method could be easily modified to implement phase conjugation of the optical temporal waveform under analysis, potentially enabling new, simplified schemes for impairment compensation in optical telecommunication links.