Modeling for Generating Femtosecond Pulses in an Er-Doped Fiber Using Externally Controlled Spectral Broadening and Compression Mechanisms

Abstract

A model for generating femtosecond laser pulses from a low-power mode-locked laser of moderate temporal half-width was proposed. This was achieved by injecting the pulse into a single-mode inverted-populated Er-doped fiber where self-focusing and absorption were avoided. To initiate spectral broadening, the pulse was phase-modulated by subjecting a part of the fiber to an electric field of suitable intensity and frequency generated into a circular capacitor. To stimulate temporal compression, the phase-modulated pulse was introduced into a combination of two prism sets located symmetrically with respect to the x-axis. After passing the pulse through the first prism set, its spectral components were spatially separated in the y-axis. The spectral phases were manipulated by redirecting the spectral components through a slab cross-section that was subjected to a spatially modulated DC electric field. After passing the slab, the pulse is directed into the second prism set, where the spectral components were spatially overlapped and propagated outside the compressor with the same slope and dimension as before entering the compressor. Constructive super positioning of the phase-manipulated spectral components gave maximum intensity only at a specified location.

1. Introduction

Since the development of mode-locked lasers, shorter pulse durations have been required for many applications in science, medicine, and engineering. Such applications are given in [1,2,3,4,5,6,7,8,9]. To direct a conventional laser pulse into a femtosecond region, it is mathematically necessary to broaden its spectral bandwidth. Because this is achievable by applying nonlinear phenomena to the pulse, different methods for its realization were presented in [10,11,12,13,14]. According to the dispersion relations of the refractive index, the frequency components comprising the spectral bandwidth of the pulse have different velocities. Each frequency will reach an observer at different times, leading either to compressing or expanding the pulse duration. Since most of the materials have a positive-group-delay dispersion (PGDD) that expands the pulse duration, a material of negative-group-delay dispersion (NGDD) is required to reduce its duration. References [15,16,17,18,19,20,21,22] have illustrated different setups for its practical realization. From the above, it is evident that mutual combinations between broadening and compression methods will theoretically yield a temporal compressed pulse. So, for instance, the authors in [23] used nonlinear phenomena to increase the spectral range of a 90 fs laser pulse in a fiber and compressed the broadened pulse by a grating compressor to 30 fs. A pulse compression ratio of 12 (from 5.4 ps to 450 fs) was reported in [24] by using self-phase modulation in a 30 m optical fiber to broaden the spectrum and a grism compressor. The result of a theoretical analysis based on the solution of the nonlinear Schrödinger equation concerning pulse compression ratio and compression pulse quality as a function of fiber length and input pulse intensity was given in [25]. The study was carried out on an optical pulse chirped in a single-mode fiber by self-phase modulation and compressed with a grating compressor. Self-phase modulation in bulk material for spectral broadening the laser pulse and chirped mirror combined with two prisms for its compression was given in [26].

From the previous cited references, it is evident that most of the researchers used:

(1)

Powerful laser pulses for internally driving nonlinear phenomena to initiate the spectral broadening of the laser pulse;

(2)

Compressors with moveable compressor elements for creating compressed pulses at a desired location.

Aim of the Work

To generate a femtosecond laser pulse from a mode-locked laser pulse of low power and moderate halfwidth, it is necessary to externally stimulate nonlinear phenomena. The present article developed a model for their realization. Section 2.1 is devoted for finding a mathematical expression for an electric field that externally phase-modulates the laser pulse during its propagation in the fiber and a setup for its experimental realization. Section 2.2 is devoted to finding a method for external variation of the spectral optical path lengths. Therefore, the spectral phases are controlled such that the pulse compression occurs at a desired location without movement any of the compressor elements. A suggested setup for the experimental realization is also included.

2. Theory

Erbium-doped fibers (EDF) were chosen for generating femtosecond laser pulses for several reasons. They show high gain, polarization-independent amplification, low intrinsic optical noise [27], stable long-term performance, and possibility of being unidirectionally and bidirectionally pumped [28]. To avoid absorption and self-focusing of the laser pulse, the selected fiber was a single-mode and inverted populated. To achieve the cited goal, the pulses had to be externally spectrally broadened during propagation in the EDF. Two prism sets and a slab (for external manipulation of the spectral phases) were responsible for temporally compressing the spectrally broadened laser pulse.

2.1. Broadening of the Spectral Bandwidth of the Laser Pulse Using an Externally Triggered Phase Modulation

A laser pulse can be coherently spectrally broadened when it is subjected to nonlinear effects either internally through self-phase modulation or externally by amplitude, frequency, or phase modulation. For a small intensity, where self-phase modulation would be ineffective, external induction of wide-range spectral broadening would be required. This was achieved by an external high-frequency electric field that modulated the refractive index of the fiber according to the following equation [29]:

$$\mathrm{n}\left(\mathsf{\Omega }\right)={ \mathrm{n}}_0\left(\mathsf{\Omega }\right)+{ \mathrm{n}}_2{\mathrm{E}}^2\left(\mathrm{t}\right),$$

(1)

where $\mathsf{\Omega }$ is the angular frequency of the electric field of the laser pulse,${ \mathrm{n}}_0\left(\mathsf{\Omega }\right)$ and ${\mathrm{n}}_2$ are the linear and nonlinear refractive indices of the fiber, respectively, and E(t) is the time-dependent external applied electric field. To create a femtosecond laser pulse, the induced spectral bandwidth (Δν), according to the Heisenberg uncertainty principle, must be in the order of 1/Δt, where Δt is the desired laser-pulse duration. To fulfill this criterion, the nonlinear part of Equation (1), ${\mathrm{n}}_2{\mathrm{E}}^2\left(\mathrm{t}\right)$, has to be in the order of the linear part ${\mathrm{n}}_0\left(\mathsf{\Omega }\right)$. This can be achieved by subjecting the fiber to a high frequency and high electric-field strength. This electric field was created by solving the Maxwell’s equations within a circular-shaped capacitor in which a part of the fiber was wrapped. The required value of the electric-field strength was obtained for certain conditions concerning the radius of the capacitor and the frequency of the generator. Due to the considered symmetry, the Maxwell’s equations have been written in cylindrical coordinates as: (detailed solution see Appendix A)

$$\frac{\partial \mathsf{{\rm E}}\left(\mathrm{r},\mathrm{t}\right)}{\partial \mathrm{r}}= \mathsf{\mu }\frac{\partial \mathrm{H}\left(\mathrm{r},\mathrm{t}\right)}{\partial \mathrm{t}},$$

(2)

$$\frac{\partial \mathrm{H}\left(\mathrm{r},\mathrm{t}\right)}{\partial \mathrm{r}}+\frac{1}{\mathrm{r}}\mathrm{H}\left(\mathrm{r},\mathrm{t}\right)= \mathsf{\epsilon }\frac{\partial \mathsf{{\rm E}}\left(\mathrm{r},\mathrm{t}\right)}{\partial \mathrm{t}}.$$

(3)

With

$$\mathrm{H}\left(\mathrm{r},\mathrm{t}\right)= \mathrm{Re}\left[\mathrm{H}\left(\mathrm{r}\right)\exp \left(\mathrm{i}\mathsf{\omega }\mathrm{t}\right)\right],$$

(4)

$$\mathsf{{\rm E}}\left(\mathrm{r},\mathrm{t}\right)= \mathrm{Re}\left[\mathrm{E}\left(\mathrm{r}\right)\exp \left(\mathrm{i}\mathsf{\omega }\mathrm{t}\right)\right],$$

(5)

it follows after setting Equations (4) and (5) into Equations (2) and (3) and applying some mathematical manipulations,

$$\frac{{\partial }^2\mathrm{H}\left(\mathrm{r}\right)}{\partial {\mathrm{r}}^2}+\frac{1}{\mathrm{r}}\frac{\partial \mathrm{H}\left(\mathrm{r}\right)}{\partial \mathrm{r}}-\frac{1}{{\mathrm{r}}^2}\mathrm{H}\left(\mathrm{r}\right)+ \mathsf{\epsilon }\mathsf{\mu }{\mathsf{\omega }}^2\mathrm{H}\left(\mathrm{r}\right)=0,$$

(6)

$$\frac{1}{\mathsf{\mu }\mathrm{i}\mathsf{\omega }}\frac{{\partial }^2\mathrm{E}\left(\mathrm{r}\right)}{\partial {\mathrm{r}}^2}+\frac{1}{\mathrm{r}}\frac{1}{\mathsf{\mu }\mathrm{i}\mathsf{\omega }}\frac{\partial \mathrm{E}\left(\mathrm{r}\right)}{\partial \mathrm{r}}- \mathsf{\epsilon }\mathrm{i}\mathsf{\omega }\mathrm{E}\left(\mathrm{r}\right)=0,$$

(7)

where $\mathsf{{\rm E}}\left(\mathrm{r},\mathrm{t}\right)$ and $\mathrm{H}\left(\mathrm{r},\mathrm{t}\right)$ are the spatial and temporal electric and magnetic field strengths, respectively, E(r) and H(r) are the spatial components of the corresponding fields, respectively, and $\mathsf{\epsilon }$ and $\mathsf{\mu }$ are the electric permittivity and magnetic permeability, respectively.

With $\mathsf{\epsilon }\mathsf{\mu }{\mathsf{\omega }}^2={ \mathrm{k}}^2$, the solution of Equations (6) and (7) are given according to [30], by

$$\mathrm{E}\left(\mathrm{r}\right)={ \mathrm{AJ}}_0\left(\mathrm{kr}\right)+{{\mathrm{A}}^{\prime }\mathrm{Y}}_0\left(\mathrm{kr}\right),$$

(8)

$$\mathrm{H}\left(\mathrm{r}\right)={ \mathrm{BJ}}_1\left(\mathrm{kr}\right)+{{\mathrm{B}}^{\prime }\mathrm{Y}}_1\left(\mathrm{kr}\right),$$

(9)

where ${\mathrm{J}}_0$ and ${\mathrm{J}}_1$ are the first kind Bessel functions of zero and first order, respectively,${\mathrm{Y}}_0{ \mathrm{and} \mathrm{Y}}_1$ are the second kind Bessel functions of zero and first order, respectively, and A and ${\mathrm{A}}^{\prime } \mathrm{as} \mathrm{well} \mathrm{as} \mathrm{B}, \mathrm{and} {\mathrm{B}}^{\prime }$ are integration constants.

To avoid the singularity of ${\mathrm{Y}}_0{ \mathrm{and} \mathrm{Y}}_1$ at $\mathrm{r} =0$, ${\mathrm{A}}^{\prime } \mathrm{and} {\mathrm{B}}^{\prime }$ were set as equal to zero. To acquire B, Ampere’s law was applied:

$$\oint \mathrm{H}\left(\mathrm{r},\mathrm{t}\right)\mathrm{rd}\mathsf{\phi } = \mathrm{Bcos}\left(\mathsf{\omega }\mathrm{t}\right){\mathrm{J}}_1\left(\mathrm{ka}\right)\left(2\mathsf{\pi }\mathrm{a}\right)=\int \frac{\partial }{\partial \mathrm{t}}\mathrm{E}\left(\mathrm{r},\mathrm{t}\right)\left(2\mathsf{\pi }\mathrm{r}\right)\mathrm{dr} = \mathrm{I}\left(\mathrm{t}\right),$$

(10)

where $\mathrm{rd}\mathsf{\phi }$ is an element of the circular path along $\mathrm{H}\left(\mathrm{r}\right)$ with radius r, a is the radius of the capacitor, and $\mathrm{I}\left(\mathrm{t}\right)$ is the feeding current of the capacitor with amplitude ${\mathrm{I}}_{\mathrm{o}}$.

Accordingly, $\mathrm{B}$ and $\mathrm{H}\left(\mathrm{r}\right)$ are determined, respectively, as:

$$\mathrm{B} ={ \mathrm{I}}_{\mathrm{o}}/\left[{\mathrm{J}}_1\left(\mathrm{ka}\right)2\mathsf{\pi }\mathrm{a}\right],$$

(11)

$$\mathrm{H}\left(\mathrm{r}\right)=\left[{\mathrm{I}}_{\mathrm{o}}/\left(2\mathsf{\pi }\mathrm{a}\right)\right]\left[{\mathrm{J}}_1\left(\mathrm{kr}\right)/{\mathrm{J}}_1\left(\mathrm{ka}\right)\right].$$

(12)

From Equation (3) and considering Equation (12), $\mathrm{E}\left(\mathrm{r}\right)$ is calculated as follows:

$$\mathrm{E}\left(\mathrm{r}\right) ={ \mathrm{I}}_{\mathrm{o}}{\mathrm{kJ}}_0\left(\mathrm{kr}\right)/\left[\left(\mathrm{i}\mathsf{\omega }\mathsf{\epsilon }\right) \left(2\mathsf{\pi }\mathrm{a}\right){\mathrm{J}}_1\left(\mathrm{ka}\right)\right],$$

(13)

With A =${ \mathrm{I}}_{\mathrm{o}}\mathrm{k}/\left[\left(\mathrm{i}\mathsf{\omega }\mathsf{\epsilon }\right)\left(2\mathsf{\pi }\mathrm{a}\right){\mathrm{J}}_1\left(\mathrm{ka}\right)\right]$, the relation between A and B is as follows:

$$\mathrm{B} = \mathrm{iA}\sqrt{\mathsf{\epsilon }/\mathsf{\mu }},$$

(14)

and the real parts of $\mathrm{E}\left(\mathrm{r},\mathrm{t}\right)$ and $\mathrm{H}\left(\mathrm{r},\mathrm{t}\right)$ can be written, respectively as

$$\mathrm{Re}\left[\mathrm{E}\left(\mathrm{r},\mathrm{t}\right)\right]= \mathrm{Re}\left[{\mathrm{AJ}}_0\left(\mathrm{kr}\right)\exp \left(\mathrm{i}\mathsf{\omega }\mathrm{t}\right)\right]={ \mathrm{I}}_{\mathrm{o}}{\mathrm{kJ}}_0\left(\mathrm{kr}\right)\sin \left(\mathsf{\omega }\mathrm{t}\right)/\left[\left(\mathsf{\omega }\mathsf{\epsilon } \left(2\mathsf{\pi }\mathrm{a}\right){\mathrm{J}}_1\left(\mathrm{ka}\right)\right)\right],$$

(15)

$$\mathrm{Re}\left[\mathrm{H}\left(\mathrm{r},\mathrm{t}\right)\right]= \mathrm{Re}[\mathrm{iA}\sqrt{\mathsf{\epsilon }/\mathsf{\mu }}{\mathrm{J}}_1\left(\mathrm{kr}\right)\exp \left(\mathrm{i}\mathsf{\omega }\mathrm{t}\right)]={ \mathrm{I}}_{\mathrm{o}}\mathrm{k}\sqrt{\mathsf{\epsilon }/\mathsf{\mu }}{\mathrm{J}}_1\left(\mathrm{kr}\right)\cos \mathsf{\omega }\mathrm{t}/\left[\mathsf{\omega }\mathsf{\epsilon } \left(2\mathsf{\pi }\mathrm{a}\right){\mathrm{J}}_1\left(\mathrm{ka}\right)\right].$$

(16)

From Equations (15) and (16), the time average of the Poynting vector within the capacitor was equal to zero. This result was in accordance with the assumption of the neglected polarization, absorption losses within the fiber, and the radiation into the surrounding. The latter can be achieved by setting the distance between the capacitor plates to be in the order of 1/100 of the wavelength of the modulating electric field. From Equation (15), it was evident that the electric field could attain large values as ${\mathrm{J}}_1\left(\mathrm{ka}\right)$ approached small values. Since this was possible for a discrete infinite number of $\left(\mathrm{ka}\right)$ values, we had to choose between those provided easily available frequencies and suitable capacitor dimensions. From the obtained results, the electric-field strength of the laser pulse was phase-modulated according to the following relation:

$${\mathrm{E}}_{\mathrm{laser}}\left(\mathrm{z},\mathrm{t}\right)={ \mathrm{c}}_1\exp \left[-{\mathrm{c}}_2{\left(\left(\mathrm{t}-{\mathrm{t}}_{\mathrm{s}}\right)/\Delta {\mathrm{t}}_{\mathrm{s}}\right)}^2\right]\times \exp ({\mathrm{i}\mathsf{\Omega }}_0\mathrm{t})\exp \left[(\frac{2\mathsf{\pi }\mathrm{i}}{{\mathsf{\lambda }}_0}) \left({\mathrm{n}}_0\left(\mathsf{\Omega }\right)\mathrm{Z}+{\mathrm{n}}_2{\mathrm{E}}^2\left(\mathrm{t}\right){\mathrm{Z}}_{\mathrm{e}}\right)\right],$$

(17)

where $\exp \left[-{\mathrm{c}}_2{\left(\left(\mathrm{t}-{\mathrm{t}}_{\mathrm{s}}\right)/\Delta {\mathrm{t}}_{\mathrm{s}}\right)}^2\right]$ is the time-dependent amplitude of the electric-field strength of the laser pulse to be phase-modulated; t_s and $\Delta {\mathrm{t}}_{\mathrm{s}}$ are the time of maximum amplitude and temporal width of the electric-field strength of the laser pulse, respectively, ${\mathsf{\Omega }}_0$ and ${\mathsf{\lambda }}_0$ are the central vacuum angular frequency and the wavelength of the laser pulse, respectively, ${\mathrm{E}}^2\left(\mathrm{t}\right)$ is the square of the phase-modulating electric field obtained in Equation (15), which is multiplied by ${\mathrm{n}}_2$ to be in the order of ${\mathrm{n}}_0\left(\mathsf{\Omega }\right)$, ${\mathrm{n}}_0$ and ${\mathrm{n}}_2$ are the linear and nonlinear refractive indices, respectively, $\mathrm{Z}$ is the total length of the fiber; and ${\mathrm{Z}}_{\mathrm{e}}$ is the length of the fiber subjected to the alternating electric field in the capacitor. To avoid sparking within the capacitor, it had to be placed in a grounded container that had been highly evacuated. This container consists of getter material to adsorb gases that were adsorbed on the surface of the material and might be free during the operation. To maintain a constant fiber temperature, the housing has to be subjected to an air-cooled Peltier cooler.

2.1.1. Computations

By keeping the denominator of Equation (15) constant, the fast Fourier transform (FFT) algorithm was applied on Equation (17) at ${\mathrm{Z}}_{\mathrm{e}}=$ 2 m for an amplified laser pulse of a temporal maximum amplitude at 1.8 W, duration $\Delta {\mathrm{t}}_{\mathrm{s}}=1\times {10}^{-7}$ s, ${\mathrm{t}}_{\mathrm{s}}=4\times {10}^{-7}$ s, ${\mathsf{\Omega }}_0=1.2\times {10}^{15}$ 1/s,${\mathsf{\lambda }}_0=1.531 \mathsf{\mu }\mathrm{m}$, phase-modulated with electric-field strength $6.8\times {10}^7$ V/m, frequencies of 100 MHz, 1.2 GHz, ${\mathrm{n}}_2= 3.2\times {10}^{-16} {\mathrm{m}}^2/{\mathrm{V}}^2$ [31], and ${\mathrm{n}}_0\left(\mathsf{\lambda }\right)$ given by Sellmeier’s equation [32].

${\mathrm{n}}_0{}^2\left(\mathsf{\lambda }\right)=\left[1+{\mathrm{B}}_1{\mathsf{\lambda }}^2/\left({\mathsf{\lambda }}^2-{\mathrm{C}}_1\right)+{\mathrm{B}}_2{\mathsf{\lambda }}^2/\left({\mathsf{\lambda }}^2-{\mathrm{C}}_2\right)+{\mathrm{B}}_3{\mathsf{\lambda }}^2/\left({\mathsf{\lambda }}^2-{\mathrm{C}}_3\right)\right]$ with ${\mathrm{B}}_1=0.6968$, ${\mathrm{B}}_2=0.40817$, ${\mathrm{B}}_3=0.8949$, ${\mathrm{C}}_1=6.853\times {10}^{-2}{ \mathsf{\mu }\mathrm{m}}^2$,${ \mathrm{C}}_2=0.1161\times {10}^{-2}{ \mathsf{\mu }\mathrm{m}}^2$ and ${\mathrm{C}}_3=9.914{ \mathsf{\mu }\mathrm{m}}^2.$

2.1.2. Results and Discussion

For computing the Fourier spectrum, the limited computer capacity made it impossible to account for more than $8\times {10}^6$ samples within the frequency range of $7\times {10}^{14 }$–$1.7\times {10}^{15}$1/s for each of the modulating frequencies.

Figure 1 illustrates the modulus of the amplitude spectrum of the electric field of a laser pulse with a maximum power of 1.8 W, duration of $1\times {10}^{-7}$ s, and subjected along two meters to a modulating frequency of 1.2 GHz. The computation was performed at ${\mathrm{Z}}_{\mathrm{e}}$= 2 m using (FFT) algorithm.

Figure 2 shows that according to Equation (17), the normalized temporal distribution of the amplitude of the electric field of the phase-modulated laser pulse had a maximum power of 1.8 W, a duration of $1\times {10}^{-7}$ s, and a modulating frequency of 1.2 GHz, when computed at $\mathrm{Z}$ = 10 m.

2.2. Pulse Compression

To compress the spectral broadened laser pulse, it was directed into a combination of four prisms divided into two identical parallel sets (P1-P2 and P1′-P2′), as shown in Figure 3. The corresponding prisms of the two sets were located at the same x-positions and displaced in the y-axis. The first set acted as a Fourier transformer and was responsible for separating the frequency components comprising the pulse into parallel beams located at different y-positions. The second prism set acted as an inverse Fourier transformer and was responsible for the spatial overlap of the spectral components throughout the environment outside the compressor.

The frequency components comprising the pulse that was emitted from the first prism set were directed via 45° mirror M1 onto the cross section of a slab. The slab, which had a low absorption coefficient as well as a well-defined frequency-dependent Kerr coefficient, was sandwiched between two electrodes. To change the spectral optical paths, each location across the width of the slab was subjected to a DC electric field with a constant value and a variable length in y-axis. This was realized, as shown in Figure 4, by connecting one of the two electrodes to the ground and the other to a DC electric source. The latter electrode was made from a photo-conducting material and was homogeneously illuminated with an array of LEDs. To change the phases of the spectral components, a stretchable computer-controlled opaque screen was used to control the illuminated area across the width of the slab. The emitted radiation from the slab was redirected by the mirror M2 into the lower set of prisms. The location of M2 was arranged such that the spectral components intersected the third prism P2′ at the same locations with respect to the base from which they were emitted from the second prism P2 of the upper set. Due to the symmetry of the configuration, the spectral components were overlapped at the output edge of the fourth prism and had the same slope. The slab’s length was chosen so that the compressor would be approximately quadratic-shaped and the electric field across its thickness would have a moderate value. For all spectral components to be constructively superimposed at a desired location, the spectral phases had to be equal or shifted by multiple of $2\mathsf{\pi }$ at that location. As shown in Figure 3, to fulfill this condition, the spectral optical path lengths across the setup and outside it had to be determined without subjecting the slab to an electric field. By dividing each of the obtained spectral optical path lengths by its corresponding vacuum wavelength $\left(\mathsf{\lambda }\right)$ and ignoring the integer number, a dimensionless fraction of the spectral wavelength was obtained. The corresponding phase of this fraction was obtained as follows:

$$\mathsf{\phi }\left(\mathsf{\Omega }\right)=2\mathsf{\pi }\times \mathrm{Fraction}\left(\mathsf{\lambda }\right),$$

(18)

and the spectral angle $\Delta \mathsf{\phi }\left(\mathsf{\Omega }\right)$ for completing $\mathsf{\phi }\left(\mathsf{\Omega }\right)$ to $2\mathsf{\pi }$ was:

$$\Delta \mathsf{\phi }\left(\mathsf{\Omega }\right)=2\mathsf{\pi }-\mathsf{\phi }\left(\mathsf{\Omega }\right).$$

(19)

To account for $\Delta \mathsf{\phi }\left(\mathsf{\Omega }\right)$, as found in Equation (19), the nonlinear part of the refractive index of the slab’s material, given in Equation (1), was activated. This was realized by subjecting a particular length $\mathrm{l}\left(\mathsf{\Omega }\right)$ along the slab at the location corresponding to the frequency across the slab’s width to a DC electric field, according to:

$$\Delta \mathsf{\phi }\left(\mathsf{\Omega }\right)=\frac{2\mathsf{\pi }}{\mathsf{\lambda }}{\mathrm{n}}_2{\mathrm{E}}^2\mathrm{l}\left(\mathsf{\Omega }\right)$$

(20)

Solving Equation (20) for $\mathrm{l}\left(\mathsf{\Omega }\right)$, one obtains

$$\mathrm{l}\left(\mathsf{\Omega }\right)=\frac{\Delta \mathsf{\phi }\left(\mathsf{\Omega }\right)\mathsf{\lambda }}{2{\mathsf{\pi } \mathrm{n}}_2{\mathrm{E}}^2}=\frac{\Delta \mathsf{\phi }\left(\mathsf{\Omega }\right) \mathrm{c}}{{\mathsf{\Omega }\mathrm{n}}_2{\mathrm{E}}^2},$$

(21)

where ${\mathrm{n}}_2$ is the nonlinear Kerr coefficient of the slab’s material that was chosen to be the same as that of the fiber’s material,$\mathrm{l}\left(\mathsf{\Omega }\right)$ is the length along the slab corresponding to the frequency $\mathsf{\Omega }$ that was subjected to the DC electric field, and c is the vacuum velocity of light.

To simulate the temporal distribution resulting from the compressor at a desired location, an inverse fast Fourier transform (IFFT) algorithm was applied on the manipulated spectral phases.

2.2.1. Computations

The computation for determining $\mathrm{l}\left(\mathsf{\Omega }\right)$ given in (21) was performed for equilateral prisms of side length ${\mathrm{l}}_1$ = 0.22 m, height as h = 0.19 m, length between the end of the base of the first prism and the apex of the second one as ${\mathrm{l}}_2$ = 0.05 m, slab length as 0.2 m, slab thickness as 0.003 m, and electric field influencing the slab as 100 V/mm.

2.2.2. Results and Discussion

Figure 5 shows the fitted curve of the spectral length $\mathrm{l}\left(\mathsf{\Omega }\right)$ subjected to the DC electric field that influenced the slab to stimulate the frequency components to be in-phase at the desired location. The computation was carried out for a bandwidth of ${10}^{15}$ 1/s induced by modulating the pulsed laser phase by a frequency 1.2 GHz.

Figure 6 represents the compressed laser pulses computed for the modulating frequencies of 100 MHz and 1.2 GHz. For each of the modulating frequencies, the computation was performed for a fixed number of frequency samples $8\times {10}^6$. The results showed that the compression for the modulation frequency at 100 MHz was better than that found at 1.2 GHz. This behavior was due to the larger ratio of the selected frequencies relative to the actual frequencies at 100 MHz compared to that at 1.2 GHz. This may be attributed to the fact that 1.2 GHz were more under-sampled or strongly aliased. In practical application, all frequencies would be in-phase if the slab is subjected to the desired distribution of the electric field given by $\mathrm{l}\left(\mathsf{\Omega }\right)$. Therefore, in this case, it was suggested that much better compression would be achieved.

3. Conclusions

Although this proposal allowed an ordinary femtosecond laser to be generated, it differed from previous studies in the following respects. The pulses to be compressed were obtained from a mode-locked laser of moderate halfwidth and low power, such that self-phase-modulation could not be initiated. The compression was achieved using a one-stage compressor. The compressed pulses were generated at any desired location without the movement of the compressor elements. Moreover, from the above, it is evident that the spectral broadening of the laser pulse as well as the spectral phase manipulation depend only on ${\mathrm{n}}_2{\mathrm{E}}^2$ that must be in the order of ${\mathrm{n}}_0$. This condition can be easily achieved by the external variation of the electric field. Consequently, this model can be used for any frequency that is not markedly attenuated during propagation in the proposed setup.