1. Introduction

The control of optical pulse shapes is an enabler in fields such as femtochemistry, coherent control, optical telecommunications, and generation of high-bandwidth electric signals. Optical waveforms with picosecond and femtosecond variations can be generated by properly tailoring the spectral amplitude and/or phase of a coherent optical source. Such control can be achieved by pulse shapers based on zero-dispersion lines [1], acousto-optic longitudinal spectral filters [2], or other combinations of linear and nonlinear elements [3–5]. There is a fixed phase relation between the different spectral components of the electric field of a coherent source. Pulse shaping of coherent sources is a deterministic process, where the electric field of the output waveform is directly calculated from the input electric field and the parameters of the pulse shaper. There is no phase correlation between the optical-frequency components of an incoherent source. Linear pulse shaping of broadband incoherent sources leading to programmable modulation of the cross-correlation of the electric field of a shaped source with an unshaped reference source has also been performed [6,7]. In this case, the shaped source is still a noisy incoherent source, but the field correlation of the shaped source with the mutually coherent reference source is deterministic. The ability to generate temporally shaped waveforms with incoherent sources has been studied [8,9]. Using results of a temporal equivalent of the van Cittert–Zernike theorem [10], a shaped temporal intensity can be obtained by modulating the optical spectrum of an incoherent source, gating the source with a temporal modulator, and propagating the gated source in a dispersive medium. The temporal intensity of the dispersed gated waveform is a scaled representation of the spectral density of the incoherent source, i.e., the time-to-frequency mapping introduced by the dispersive medium makes it possible to shape the average temporal intensity of the output waveform by modulating the spectral density of the incoherent input source. Spectral density modulation and time-to-frequency mapping have already been used to generate shaped waveforms starting from a short optical pulse (i.e., a broadband coherent source for which different spectral components are fully correlated) [11]. Shaped optical and electrical waveforms have recently been generated by incoherent pulse shaping [12,13].

The applicability of pulse-shaping techniques depends heavily on their signal-to-noise ratio (SNR). A previous analysis of incoherent pulse shaping based its evaluation of the SNR on the correlation time increase of the shaped waveform [9]. This does not allow one to estimate the SNR of the shaped waveform using the commonly accepted definition for such a quantity, i.e., a quantification of the relative fluctuations of the intensity around its expected value. This paper presents a consistent statistical analysis of incoherent pulse shaping. A formalism describing incoherent pulse shaping and the temporal van Cittert-Zernike theorem based on the representation of an incoherent source by complex spectral components with random phases is first developed. This formalism is then used to derive the intensity probability density function (pdf) of the shaped waveforms, making possible a full statistical study of the SNR of incoherent pulse shaping. Since the intensity pdf of the shaped waveform is the same as the pdf of an incoherent process, its SNR is restricted to 1. Finally, the evolution of the SNR after polarization multiplexing, averaging over multiple realizations of the incoherent process, and averaging of the intensity over a given time interval is described.

2. Incoherent pulse shaping of a polarized incoherent optical source

2.1. General description of incoherent pulse shaping

The notations for the Fourier transform adopted in this derivation are that the temporal and spectral representations of an electric field are related by

and

where all integrals extend from -∞ to +∞. Figure 1(a) shows the general principle of incoherent pulse shaping. The electric field of an incoherent process with spectral density S _inc(ω) is gated by a temporal gate with complex amplitude g(t). It then propagates in a dispersive medium with second-order dispersion φ ₂. The temporal intensity of the resulting electric field E ₂ is shaped by the modulation of the spectral density of the incoherent process and the time-to-frequency relation induced by chromatic dispersion t = φ ₂ ω.

 

Fig. 1. (a) Principle of incoherent pulse shaping based on temporal gating and spectral dispersion. (b) Example of one realization of the shaped intensity (black line) and scaled spectral density of the incoherent process (red line).

Download Full Size | PDF

Figure 1(b) shows an example of a shaped optical waveform. The spectral density of the incoherent process is chosen as a 20th-order super-Gaussian centered at 1550 nm with full width at half maximum of 50 nm (i.e., approximately 40 ps^-1). The gating is performed by a temporal modulator with a 10-ps-duration Gaussian intensity transmission. The second-order dispersion is 100 ps². The intensity of the shaped waveform is compared to the function S _inc(t/φ ₂); ie, the spectral density of the incoherent process with the frequency axis scaled by the second-order dispersion φ ₂. As can be observed, significant noise is present on the shaped waveform. Noisy electrical waveforms have also been obtained after photodetection [12,13]. The purpose of this paper is to elucidate the signal-to-noise-ratio properties of incoherent pulse shaping.

2.2. Derivation of the electric field of the shaped waveform

Instead of the correlation-function formalism used in Refs. 9 and 10, a description of the incoherent process by uncorrelated frequency components with random phases is used. Since the system is linear, the electric field E ₂(t), corresponding to one realization of the incoherent process after gating and chromatic dispersion, is calculated as the sum of the fields E _{2, ω},(t) obtained for each optical frequency ω in the incoherent source. The initial temporal electric field for one realization of the incoherent process at the frequency ω is

φ _inc is a random variable uniformly distributed between 0 and 2π, and different realizations of the incoherent process are described by different realizations of the variable φ _inc. After gating, the electric field is

which can be written in the spectral domain as

Propagation in the dispersive medium with second-order dispersion φ ₂ leads to the electric field

After some rearrangement, this field is written

Since φ _inc is a frequency-uncorrelated random variable uniformly distributed in the interval [0,2π], φ _inc(ω) + φ ₂ ω ²/2 can be replaced by another equiprobable realization of the same random variable φ _inc(ω) (this is equivalent to considering another realization of the incoherent process). For a given ω, g̃ (ω′ - ω) exp [iφ ₂(ω′-ω)²/2] exp [iφ ₂(ω′-ω)ω] represents the transfer function of the amplitude of the gate after dispersion φ ₂ and time delay φ ₂ ω calculated at the optical frequency ω′-ω. The transfer function of the modulator after dispersion is noted g_chirped, and the propagated electric field is

The electric field of the shaped waveform obtained for one realization of the random process of spectral density S _inc is obtained by summation of the electric fields of Eq. (8),

Equation (9) describes the electric field of one realization of the shaped waveform for a power spectral density S_inc, gate g, and second-order dispersion φ ₂.

2.3. Derivation of the average intensity of the shaped waveform

The average value of the intensity is calculated by averaging over all the possible realizations of the incoherent process as

Since φ _inc is uncorrelated between different optical frequencies, one has 〈exp[iφ _inc(ω)- iφ _inc(ω′)]〉= δ(ω-ω′). This leads to

where G_chirped = ∣g_chirped∣² is formally the intensity transmission of the chirped gate (in practice, this is the temporal intensity of the optical pulse obtained by gating a monochromatic source and measuring the resulting optical intensity after the second-order dispersion φ ₂). The temporal intensity at a given time t is, on average, given by the convolution of the spectral density of the source under test with the temporal transfer function of the chirped gate expressed at time t−φ ₂ ω. Equation (11) shows that shaping the optical spectrum of the incoherent process allows one to shape the average temporal intensity of the waveform after gating and dispersion, and that the temporal resolution is ultimately limited by G_chirped. In the particular case when G_chirped is short compared to the variations of the spectral density, one obtains

where the energy transmission of the gate, i.e., the integrated transmission of the gate, has been normalized to one. Equation (12) links the average temporal intensity of the waveform after gating and chromatic dispersion to the spectral density of the incoherent process via frequency-to-time mapping. It is the basis of incoherent pulse shaping, where a given temporal intensity is obtained by shaping the spectral density of an incoherent process, gating the incoherent process, and propagating it in a dispersive system. Equation (12) is only true on average, i.e., after averaging over the infinite number of realizations of the incoherent process.

2.4. Derivation of the intensity probability density function of the shaped waveform

Reconsidering Eq. (9), which describes the temporal electric field obtained for one realization of the incoherent process at a given time t, it is apparent that the electric field at this particular time t is analogous to the electric field of an incoherent process. This is because the temporal electric field is obtained as the Fourier transform of a given spectral function $\left[\sqrt{S_{\mathrm{inc}}\left(\omega \right)}{g}_{\mathrm{chirped}}\left(t-{\phi }_2\omega \right)\right]$ with an associated random uncorrelated spectral phase φ _inc(ω) uniformly distributed over [0,2π]. From the pdf of the intensity of thermal polarized light, or similarly considering that the obtained intensity is the modulus square of sum of complex numbers with random phases [14,15], the probability density function of the intensity of the field at a given time t is

where 〈I〉 is given by Eq. (11). This is a fundamental description of the SNR of incoherent pulse shaping, which is used in Sec. 3 to calculate the effect of various operations on the intensity of the shaped waveform.

The signal-to-noise ratio associated with the probability density function of Eq. (13) is equal to 1, indicating that the shaped waveform is extremely noisy, as can be judged by Fig. 1(b). The pdf of the shaped intensity at a given time and the associated SNR do not depend on the gate duration or the dispersion. In Ref. 9, the SNR is defined using temporal correlation functions. With such definition, the SNR is impacted by the duration of the gate and the second-order dispersion. These choices impact the coherence time of the shaped waveform (as seen below) and not the SNR, which is commonly defined as the fluctuations of a quantity around its mean value and not by the correlation of its fluctuations at different points. Figure 2 displays the pdf calculated considering the intensity between −1.5 ns and 1.5 ns and 100 realizations of the incoherent process. The agreement with the theoretical prediction of Eq. (13) is excellent on both linear and logarithmic scales.

 

Fig. 2. Probability density function of the intensity of the waveforms of Fig. 1 (black line), with the theoretical prediction of Eq. (13) (red markers).

Download Full Size | PDF

2.5. Derivation of the coherence time of the shaped waveform

The intensity correlation of the shaped waveform can be evaluated around a given time t ₀. Considering Eq. (9) and assuming that the power spectral density of the incoherent source is such that S _inc(ω) is constant over the interval where g _chirped(t-φ ₂ ω) is non-zero, one has

The two-time correlation function of the electric field around t ₀ can be written as

Because of the absence of correlation between two distinct optical frequencies of the incoherent process, one has 〈exp[iφ _inc(ω)−iφ _inc(ω′)]〈 = δ(ω−ω′), and Eq. (15) is simplified into

Using the frequency representation of g _chirped leads to the expression

The integral in the right-hand side of Eq. (17) can be calculated as

where G is the intensity transmission of the gate, i.e., G(t) = |g(t)|². The amplitude of the degree of coherence

is

Equation (19) has been previously derived with a different approach as Eq. 17 of Ref. 10 (which has a typo since no factor of 2 should be present with the adopted definition of the second-order dispersion) and Eq. 8 of Ref. 9. It is the temporal equivalent of the van Cittert–Zernike theorem, which shows that the correlation of the fluctuations of the electric field of the gated incoherent after dispersion are linked to the Fourier transform of the temporal intensity transmission of the gate. The analogy of Eq. (19) with the expression of the degree of coherence of an incoherent process as a function of the optical spectrum of the process shows that the coherence time of the shaped electric field is approximately T ₂ = φ ₂ /T_G, where T_G is the duration of the transmission of the temporal gate.

Using the expression of the temporal correlation of the intensity of an incoherent process as a function of the amplitude of the degree of coherence of the source [14], the two-time correlation function of the shaped intensity around time t ₀ is

Equation (20) can be normalized as

where τ is the difference variable t′-t and

Figure 3 displays the calculated normalized two-time intensity correlation function compared to the theoretical predictions of Eq. (21) for durations of the gate of 10 ps, 20 ps, and 40 ps. An excellent agreement is obtained in all cases. The correlation time of the intensity is inversely proportional to the duration of the gate, as expected from Eqs. (19) and (21). A close-up on one realization of the intensity of the shaped waveform shows the scale of variations of the intensity in these three cases (note that the shaped waveforms are calculated for different realizations of the incoherent process). Although Eq. (21) implies that the correlation time of the shaped waveform depends on the duration of the gate and the second-order dispersion [9], these parameters do not control the statistical shot-to-shot variations of the shaped intensity.

 

Fig. 3. Intensity correlation function γ for a Gaussian temporal gate of duration 10 ps, 20 ps, and 40 ps (upper row, from left to right). The correlation functions calculated with simulations of the shaped waveforms are plotted with a black line, and the predictions of Eq. (21) are plotted with red markers. The lower row represents close-ups of realizations of the shaped waveforms, where the horizontal dashed line indicates the value of the temporal intensity expected after averaging over an infinite number of realizations of the incoherent process.

Download Full Size | PDF

3. Polarization multiplexing, averaging, and finite temporal resolution in incoherent pulse shaping

3.1 Polarization multiplexing

Polarization multiplexing corresponds to the addition of two uncorrelated, orthogonally polarized incoherent sources after identical gating and dispersive propagation. Such a waveform can be obtained with an unpolarized incoherent optical source followed by a polarization-independent modulator. It can also be obtained by splitting a polarized shaped incoherent source and recombining it along two orthogonal states of polarization with a relative delay longer than the intensity correlation time (this relative delay ensures that two uncorrelated intensities are summed). The temporal intensity of the shaped waveform is the sum of the temporal intensity along each polarization. Since the pdf of the intensity of each shaped waveform is given by Eq. (13), this is formally identical to the probability density function of the intensity of an unpolarized incoherent optical source [14],

The pdf of the corresponding intensity at I = 0 is zero, and the corresponding signal-to-noise ratio is √2, as expected from the sum of two uncorrelated random variables. An illustrative example of one realization of a shaped unpolarized waveform is plotted in Fig. 4(a). The corresponding probability density function is plotted in Fig. 4(b). Polarization multiplexing of two mutually incoherent shaped waveforms increases the signal-to-noise ratio, but significant fluctuation of the intensity around its mean value is still present.

 

Fig. 4. (a) One realization of the unpolarized shaped intensity (black line) and spectrum of the incoherent process plotted versus the temporal variable φ ₂ ω (red line). (b) Intensity probability density function of the unpolarized shaped waveform.

Download Full Size | PDF

3.2 Shaped intensity averaging

The intensity I _1-N = (I ₁ + I ₂ +…I_N)/N obtained as the average of N intensities I ₁, I ₂,…I_N corresponding to realizations of the random incoherent process is considered. This quantity is obtained by detecting and averaging the N independent realizations of the intensity of the shaped waveform. The calculation of the pdf of I _1-N is formally equivalent to the calculation of the intensity resulting from the sum of N uncorrelated speckle patterns [16]. The pdf of I _1-N, the average of N uncorrelated intensities with pdf given by Eq. (13), is

It can be checked that the average intensity of this process is 〈I〉 and its standard deviation is 〈I〉√N. As a consequence of the central limit theorem, p_N asymptotically behaves like a Gaussian probability density function centered at I = 〈I〉 with standard deviation 〈I〉/√N (this property can be used for large values of N for which Eq. (23) cannot be estimated directly). The shaped waveforms obtained as an average over 10, 100, and 1000 realizations of the incoherent process are plotted in Fig. 5 along with calculations of p_N for N = 10, N = 100, and N = 1000. Averaging the uncorrelated shaped waveforms leads to an expected increase in the SNR and a decrease in the width of the pdf around the average value of the intensity.

 

Fig. 5. Examples of shaped intensity averaged over 10, 100, and 1000 realizations of the incoherent process (upper row, from left to right) and corresponding pdf p ₁₀,p100, and p1000 (lower row, from left to right).

Download Full Size | PDF

3.3. Temporal multiplexing

Temporal multiplexing corresponds to a situation where N shaped waveforms are summed. One approach is splitting a single-shaped waveform into N identical-shaped waveforms and adding the corresponding intensities with relative delays longer than the intensity correlation time, for example, by adding the photocurrents obtained by photodetection. This is done at the expense of temporal resolution and can be seen as a discrete version of the convolution process studied in the next subsection. Another approach is to use a dispersive element that induces identical second-order dispersion φ ₂ and group delay T to different bands of optical frequencies centered at ω ₀, ω ₀ + Ω,…ω ₀ + (N-1)Ω. With such an element, if one uses identical incoherent spectral density in each interval [ω ₀ + (j-1/2)Ω, ω ₀ + (j + 1/2)Ω], identical waveforms are on average generated after gating and chromatic dispersion. The resulting intensity after photodetection with a bandwidth smaller than the separation Ω is only sensitive to the sum of the intensities of the optical waveforms. In this case, noise reduction occurs at the expense of the temporal extent of the shaped waveform (which is of the order of φ ₂Ω) since the available bandwidth of each incoherent process is decreased to Ω. With temporal multiplexing, the obtained intensity is the sum of N uncorrelated intensities with pdf given by Eq. (13). The obtained pdf is therefore identical to Eq. (23).

3.4. Detection with finite temporal resolution

Detection of the shaped waveform with a finite temporal resolution corresponds, for example, to the generation of high-bandwidth electrical signals by optical-to-electrical conversion or to the characterization of the shaped optical waveform by nonlinear cross-correlation with a pulse of finite duration. It is mathematically expressed as a convolution of the intensity of the physical waveform with the response function of the detection system R(t), i.e., the measured signal is

A detector with a response time T_R is considered. The probability density function of the shaped intensity I is that of a polarized incoherent process with a coherence time T ₂ = φ ₂/T_G. Since photodetection is an intensity-only process, calculating the pdf of I _convolved is formally equivalent to calculating the pdf of the integrated intensity of an incoherent source with the same coherence time. The latter calculation is rather involved in the general case, and an estimate of the pdf is obtained using various simplifications [15].

The function R is taken as a rectangular function (i.e., the detector integrates the received intensity between -T_R/2 and T_R/2). The interval of extent T_R is assumed to contain a large number of coherence times of the shaped intensity, in which case the convolved intensity is to be expressed as a sum of M uncorrelated intensities, where M is the ratio T_R/T ₂. Finally, M is rounded up to an integer, so that the gamma function is simply Γ(M) = (M−1)!. With these assumptions, the pdf of the convolved intensity is

This pdf is similar to that of Eq. (23) since the corresponding random variables are in both cases averages of a number of uncorrelated intensities with pdf given by Eq. (13). According to Eq. (25), the finite response time of a photodetector converting a shaped optical waveform to a shaped electrical waveform increases the SNR. However, the SNR increases only by the square root of the number of uncorrelated intensities being summed up, i.e., the square root of T_RT_G/φ ₂. This comes at the expense of a decrease in temporal resolution of the pulse-shaping system.

4. Conclusions

A statistical formalism has been developed for incoherent pulse shaping. Using a representation of the incoherent process as a sum of complex contributions with random phases, the probability density function of the shaped intensity has been shown to be identical to the intensity pdf of an incoherent source. A direct consequence of this pdf is that the signal-to-noise ratio of the shaped waveform at a given time is fundamentally equal to 1. Using the derived pdf, the statistical properties of the intensity after various common operations has been calculated. These derivations are useful in estimating the signal-to-noise ratio achievable with incoherent pulse shaping in various experimental conditions.