Time-lens perspective on fiber chirped pulse amplification systems

Xuanchao Qin; Shaozhen Liu; Yu Chen; Tao Cao; Le Huang; Ziyue Guo; Kailin Hu; Jikun Yan; Jiahui Peng

doi:10.1364/OE.26.019950

1. Introduction

There exists a duality between the equations that describe the paraxial diffraction of light beams in space and the dispersion of optical pulses in media [1–4], and this duality leads to the conclusion that an element introducing a quadratic phase shift in time is an analogy of a thin lens in space, as shown in Fig. 1. On the one hand, a quadratic phase shift is sometimes a pitfall to deal with, especially for ultrashort pulses, because this quadratic phase shift may cause frequency-shifts and distortions of ultrashort pulses. On the other hand, the quadratic phase shift is inherent in the idea of time-lens. Therefore, people may take advantage of this fundamental property of a time-lens to simplify the analysis of processes accompanied with quadratic phase shifts.

Nowadays, chirped pulse amplification (CPA) laser systems are commonly used to amplify ultrashort optical pulses to circumvent limitations caused not only by optical damage issues but also sometimes by the B integral (beam breakup integral), a nonlinear phase shift accumulated over the length of the amplifier [5,6], making it one of the most important technologies for intense lasers. Besides, it is now a common practice for ultrashort pulse fiber lasers as well [7–10]. At the same time, the small core area (∼10² μm²) and the large length of the amplifier (over meters) make the nonlinear phase shift not avoidable even for low power fiber lasers (∼10² mW). Occasionally, this nonlinear phase shift, normally caused by the self-phase modulation (SPM), is used to broaden the pulse spectrum and to further generate shorter or tunable pulses too [11,12]. At the end, this phase shift induced by dispersions and SPM should be compensated in the final compression stage to obtain the shortest pulse achievable, as required in the concept of CPA. Therefore, both nonlinear phase shifts and dispersions of the stretcher and the compressor need to be calculated carefully first while designing a fiber CPA system.

A temporal imaging system is composed of three parts: a prepositive dispersive delay line, an element introducing a quadratic temporal phase, and a postpositive dispersive delay line. The element that imparts the quadratic temporal phase is termed as time-lens [13]. Obviously, a fiber CPA system is nothing but a special case of a temporal imaging system with strong requirements: balances between nonlinearity and dispersions with minimum output pulse distortions. By its nature, time-lens has taken into account of quadratic temporal phase shifts of the system already. Thus, we propose a simple and effective method to design fiber CPA systems based on the temporal imaging theory. Dispersions of the stretcher and the compressor need to satisfy the imaging relation of time-lens caused by nonlinear phase shifts introduced by SPM in the amplification stage. The output pulse duration is ready to be derived simply from the magnification relation of the imaging system, with nonlinearity included.

Fig. 1 The comparison of a spatial lens and a time-lens. (a) The concept of thin lens imaging, where a thin lens imparts a spatial quadratic phase shift to the beam [14]; (b) The concept of time-lens imaging, where a time-lens imparts a temporal quadratic phase shift to the pulse. D₁ and D₂ are GDDs before and after the time-lens, respectively. D_f is the focal length of the time-lens.

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In this paper, we first introduce the basic concept of this method, and then present their rigorous mathematical relations. At last, we compare the numerical results calculated via this simple design method with experimental results obtained from a home-built femtosecond fiber CPA system. Results show that it is a simple, effective and practical design method, especially for ultrashort pulse fiber amplification systems.

2. Imaging theory of time-lens

In a retarded time frame, the frame moving with the pulse, the propagation of an optical pulse in a lossless, linear medium is described by

\frac{\partial a}{\partial z} = - i \frac{β_{2}}{2} \frac{\partial^{2} a}{\partial τ^{2}},

where a refers to a slowly varying envelope of an optical pulse, τ = t− z/β₁ is the retarded time, β₁ is the group velocity, and β₂ is the group velocity dispersion. By comparing with the equation of the paraxial propagation of scalar, monochromatic fields along the z direction

\frac{\partial u}{\partial z} = - i \frac{1}{2 k} (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}),

where u refers to a slowly varying envelope function in space, we can see that the paraxial propagation of one-dimensional monochromatic planwave and pulse propagation are described by the same mathematics [4,15], space-time duality.

Based on this space-time duality, concepts in Fourier optics can be employed in describing characteristics of propagation of optical pulses in dispersive media, including the concept of time-lens. Similar to how the focal length is related to the spatial quadratic phase shift of a conventional thin lens, the focal length of a time-lens is related to the corresponding quadratic phase shift in the same form and can be written as [16]

ϕ_{TimeLens} (τ) = e^{i \frac{τ^{2}}{2 D_{f}}},

where τ, again, is the retarded time and D_f is the focal length of the time-lens. For a time-lens imaging system shown in Fig. 1(b), we define D₁ and D₂ to be the total group delay dispersion (GDD) that the input pulse experiences before and after the time-lens, respectively, and D_f to be the focal length of the time-lens. Then the imaging condition in the time domain can be expressed as

\frac{1}{D_{1}} + \frac{1}{D_{2}} = \frac{1}{D_{f}} .

Once the temporal imaging condition [Eq. (4)] is satisfied, each temporal sample at the input pulse is mapped to a unique temporal sample at the output pulse, i.e. the output pulse holds the same shape of the input pulse but with a pulse duration magnification (or demagnification) [13].

Meanwhile, the output pulse spectrum Ã′(ω) is ready to be obtained from the input pulse spectrum Ã(ω), with terms of phase shifts caused by dispersions and time-lens effects

{\tilde{A}}^{'} (ω) = \frac{1}{2 π} {[\tilde{A} (ω) e^{i \frac{D_{1} ω^{2}}{2}}] * ℱ (e^{i \frac{τ^{2}}{2 D_{f}}})} e^{i \frac{D_{2} ω^{2}}{2}},

where * is the convolution operator. In this way, we can obtain the output pulse a′(τ) in the form of

a^{'} (τ) = \frac{1}{2 π} ℱ^{- 1} {[\tilde{A} (ω) e^{i \frac{D_{1} ω^{2}}{2}}] * ℱ (e^{i \frac{τ^{2}}{2 D_{f}}})} * ℱ^{- 1} (e^{i \frac{D_{2} ω^{2}}{2}}) .

Once the imaging restriction Eq. (4) is imposed, the output pulse a′(τ) is proportional to a scaled version of the input pules a(τ) [13]

a^{'} (τ) = \frac{1}{\sqrt{M}} e^{i \frac{τ^{2}}{2 {MD}_{f}}} a (\frac{τ}{M}),

where the magnification in the time domain (M) is expressed as

M = - \frac{D_{2}}{D_{1}} .

The pulse will be stretched M times in the time domain, where M is a constant determined by GDDs in the input stage and the output stage. When M is negative, the output pulse is an inverse shape of the input pulse in time, corresponding to the condition of an upside-down image in spatial imaging.

3. Time-lens in a fiber CPA system

A fiber CPA system consists of three stages, a stretcher, an amplifier and a compressor. The stretcher and the compressor contribute most of GDDs to optical pulses propagating through, and the amplifier contributes most of SPM induced temporal nonlinear phase shifts to pulses and some GDDs depending on the length of the gain medium [17]. In general, dispersions and nonlinear phase shifts do not necessarily satisfy the temporal imaging condition, because it is a quite strong restriction. However, in most cases what a fiber CPA system requires and produces are nearly transform-limited ultrashort pulses. In other words, if we have a transform-limited Gaussian pulse sent into a fiber CPA system, what we hope to obtain will be a nearly transform-limited pulse with a significant amplification. In this sense, we can view the fiber CPA system as a time-lens system with a stretcher, a compressor and a gain medium corresponding to D₁, D₂ and D_f, respectively, as shown in Fig. 2. One may argue that not all input pulses before the stretchers are transform-limited. However we can always add corresponding dispersions into the imaging system, so this analogy still holds.

Fig. 2 If the nonlinear phase shift produced by the SPM effect can be equivalent to a quadratic phase factor, the gain medium can be regarded as a time-lens, and the whole CPA system can be regarded as a temporal imaging system. D₁ is the GDD of the stretcher and D₂ is the GDD of the compressor.

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For a fiber CPA system, the propagation of an optical pulse in a gain medium can be expressed as [17]

\frac{\partial u}{\partial z} = i γ P_{0} e^{- α z} {| U |}^{2} U,

where U is the normalized amplitude, γ is the nonlinear parameter of the medium, P₀ is the peak power, α is the loss parameter of the medium that can be replaced by −g for gain medium (g is taken as a constant here). With the effect of gain taken into account, the focal length of the time-lens can be described by (detailed derivations can be found in Appendix A)

D_{f} = - \frac{\sqrt{π}}{2 E_{0} τ_{0}^{3} γ L_{eff}} {| D_{1} |}^{3},

where E₀ is the input pulse energy, τ₀ is the input pulse duration (

\frac{1}{e}

half width of the intensity).

L_{eff} = \frac{e^{g L} - 1}{g}

is the effective length, where L is the length of the medium with the SPM effect.

When the gain saturation occurs, the gain coefficient (g = −α) in Eq. (9) is no longer a constant. Therefore, Eq. (10) does not represent the focal length of the equivalent time-lens in a fiber CPA system.

For this location dependent gain coefficient, we can assume the local gain to be a constant along a short propagation distance. Then, the gain medium can be divided into many segments of equal length L, as shown in Fig. 3. The gain coefficient in each segment is [18,19]

g = \frac{g_{0}}{1 + \frac{E}{E_{S}}} \cdot \frac{ν_{0}^{2}}{ν_{0}^{2} + Δ ν_{0}^{2}},

where g is the gain coefficient in this segment, g₀ is the small signal gain coefficient, E is the pulse energy input into this segment, E_S is the saturation energy, and ν₀ and Δν₀ are the center frequency and bandwidth of the laser, respectively. The saturation energy E_S is [15,19]

E_{S} = \frac{h c A_{eff}}{λ_{0} σ f_{Rep} τ_{G}},

where h is the Planck’s constant, c is the speed of light in vacuum, λ₀ is the central wavelength, A_eff is the effective beam cross-section in the gain medium, σ is the emission cross-section of the doped ions in the gain medium, τ_G is the level lifetime of the doped ions, f_Rep is pulse repetition frequency.

Fig. 3 A schematic diagram of the amplification stage considering the gain saturation effect: The gain medium is divided into many segments with equal length L. The gain coefficient within each segment can be considered as a constant. Different segments have different gain coefficients. In this figure, E₀ is the input pulse energy to the CPA system, τ is the retarded time, g_j is the gain parameter in the j_th segment, A⁽ⁿ⁾(0, τ) is the envelope function that input into the n_th segment, and A⁽ⁿ⁾(L, τ) is the envelope function of the output pulse from the n_th segment.

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In this method, we can find the focal length of a time-lens with gain saturation to be (detailed derivations can be found in Appendix B)

D_{f} = - \frac{\sqrt{π}}{2 E_{0} τ_{0}^{3} γ (\sum_{j = 1}^{n} e^{\sum_{k = 1}^{j - 1} g_{k} L} L_{eff}^{(j)})} {| D_{1} |}^{3},

where parameters g_k and

L_{eff}^{(j)}

are the gain coefficient of the k_th segment and the effective length of the j_th segment respectively. By setting a proper quantity of each segment L, we can obtain the focal length of the time-lens with sufficient accuracy through Eq. (13) without lengthy calculations.

Once we view the element introducing nonlinear phase shifts caused by SPM as a time-lens with focal length D_f (Fig. 2), where dispersions introduced by the stretcher D₁ and the compressor D₂ need to satisfy the imaging condition described by Eq. (4). This provides a simple method to design a fiber CPA system.

For a fiber CPA system, given D_f, substituting Eq. (10) into Eq. (4), the GDD in compressor is required to be

D_{2} = - D_{1} + \frac{2 E_{0} τ_{0}^{3} γ L_{eff} D_{1}}{sgn (D_{1}) \sqrt{π} D_{1}^{2} + 2 E_{0} τ_{0}^{3} γ L_{eff}},

where “sgn” is the sign function. The second term on the right of Eq. (14) can be regarded as the GDD caused by the nonlinearity introduced by the amplifier, which is

D_{SPM} = - \frac{2 E_{0} τ_{0}^{3} γ L_{eff} D_{1}}{sgn (D_{1}) \sqrt{π} D_{1}^{2} + 2 E_{0} τ_{0}^{3} γ L_{eff}} .

From the time-lens imaging theory shown in Sec.2, the magnification for the input pulse of such a fiber CPA system will be

| M | = {| 1 + sgn (D_{1}) \frac{2 E_{0} τ_{0}^{3} γ L_{eff}}{\sqrt{π} D_{1}^{2}} |}^{- 1} .

Clearly, it is straightforward to obtain dispersions needed for the stretcher and the compressor of a fiber CPA system, once its amplification parameter is known, which is a parameter set in advance in general. Moreover, the output pulse duration is ready to be predicted when the dispersions are known.

When the gain saturation effect can not be neglected, the gain coefficient g is no longer a constant, and the focal length of time-lens is given by Eq. (13). In the same manner, by substituting Eq. (13) into Eq. (4), the relation between D₁ and D₂ can be found to be

D_{2} = - D_{1} + \frac{D_{1}}{sgn (D_{1}) μ D_{1}^{2} + 1},

where μ is defined as

μ = \frac{\sqrt{π}}{2 E_{0} τ_{0}^{3} γ (\sum_{j = 1}^{n} e^{\sum_{k = 1}^{j - 1} g_{k} L} L_{e f f}^{(j)})} .

The GDD caused by the nonlinearity introduced by the amplifier will be

D_{SPM} = - \frac{D}{sgn (D_{1}) μ D_{1}^{2} + 1} .

The magnification of the pulse duration will be

| M | = {| 1 + \frac{sgn (D_{1})}{μ D_{1}^{2}} |}^{- 1} .

Based on the imaging theory of the time-lens mentioned above, we propose a novel CPA design method which will be specified later. With certain pulse energy and pulse duration to be obtained, this method gives the suggested values of GDDs for the stretcher and the compressor. The detailed process consists of three steps:

Step 1: Set a goal of a CPA system, including the output pulse energy and the output pulse duration magnification;
Step 2: Find out related parameters including: a) gain fiber’s small gain parameter g₀; b) gain fiber’s saturation energy E_S; c) seed pulse duration τ₀; d) seed pulse energy E₀. If the gain saturation effect can be neglected, the gain parameter can be considered as a constant during the amplification process, thus b) is not needed.
Step 3: According to the time-lens imaging theory, choose GDDs for the stretcher and the compressor.

4. Experimental results

To verify the proposed method, we built an Erbium doped fiber CPA system with parameters determined via the design method shown above. Nonlinearity plays a significant role in ultrashort pulse fiber amplification systems, and sometimes it is not something to avoid but to make use of. For example, to generate pulses with durations less than 100 fs for a fiber laser, it is common to introduce strong SPM to broaden the spectrum and compensate the total GDD at the end of the amplifier. For this reason, we set a scenario that a fiber CPA system with strong SPM. A relay of amplifiers instead of a single stage of amplifier is set to simulate general cases.

Fig. 4 (a) A structure diagram of the fiber CPA system. The gain medium of this amplification system is an Erbium doped fiber (EDF). This system consists of two parts: a preamplifier and a main amplifier. There is a dispersion compensation segment behind the main amplifier for the pulse compression. (b) The designed total GDD of all elements before the main amplifier D₁ = −89381 fs², and the designed total GDD of the main amplifier and the compensation segment is D₂ = −16103 fs². The dispersion of the amplifier is the GDD of the EDF in the amplifier.

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The experimental setup of the proof-of-principle system is shown Fig. 4(a). The input spectrum is shown in the left lower part of Fig. 4(a), which shows a standard soliton mode-locked laser. For a central wavelength of 1550 nm, E₀ = 46.5 pJ, and τ_FWHM = 388.6 fs input pulse, we expect pulses to be amplified to nJ level with pulse durations compressed to less than 100 fs via SPM spectrum broadening. τ_FWHM and τ₀ ( $\frac{1}{e}$ half width of the intensity) are related in the form of $τ_{FWHM} = 2 \sqrt{\ln 2} \cdot τ_{0}$ [20], so τ₀ = 233.4 fs. Thus, the lengths of gain fibers are set to be 0.55 m for the preamplifier and 2.95 m for the main amplifier, respectively. The measured output pulse energy is found to be 2.0 nJ. Due to the SPM effect, the output pulse spectrum can be significantly broadened to 60 nm, shown in the middle of the lower part of Fig. 4(a). After the main amplifier, there is a segment of fiber for the purpose of dispersion compensation. The autocorrelation curve of the pulse after the dispersion compensation segment is shown in the right of the lower part of Fig. 4(a), and the measured pulse duration is about 82 fs with a Gaussian pulse shape assumption.

Because the fiber lengths are impossible to be obtained as the exact designed values when splicing fibers together. The actual dispersions are a little bit different from the designed ones. The designed values of D₁, D₂ are −89381 fs² and −16103 fs² respectively, while the actual value of D′₁ and D′₂ are −89160 fs² and −16365 fs² (D_f is −13645 fs²). The GDD of each part of this fiber CPA system is shown in Fig. 4(b), where the average gain coefficient g = 1.07 m⁻¹ and the nonlinear parameter of the gain fiber γ = 0.002 W⁻¹m⁻¹.

Fig. 5 The blue dashed line represents the relation between D₁ and D₂ given by Eq. (14). The red solid line represents the relation between D₁ and the magnification(|M|) given by Eq. (16). The vertical dashed line is D₁ = −89381 fs², which is the designed value of the GDD of the stretcher when |M| = 0.1802. The vertical coordinate of the intersection point between the vertical line and the blue dash line is D₂ = −16103 fs², which is the designed value of the GDD of the compressor.

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From the temporal imaging condition described by Eq. (4), we can easily obtain the dependency of the pulse duration magnification |M| on the GDD in the stretcher, shown in Fig. 5 with the red solid line. According to the relation, the GDD of the stretcher needs to be about −89381 fs² to generate the demanded output pulse duration. The relation between D₂ and D₁ can be found too, shown in Fig. 5 with the blue dashed line, where we take the focal length of the time-lens as D_f = −13645 fs². With the GDD of the stretcher (D₁) and the focal length of the time-lens (D_f) being determined, the GDD in compressor (D₂) can be found to be −16103 fs².

In order to verify whether the output pulse is nearly transform-limited, we adjust the fiber length in the compensation stage to obtain the shortest pulse duration. Under this circumstance, the dispersion introduced by the dispersion compensation segment that can minimize the output pulse duration is measured to be −64400 fs² (corresponding to D′₂ = −16365 fs²), with an output pulse duration (FWHM) of 82 fs, while the designed GDD of the compressor is D₂ = −16103 fs² and the designed pulse duration is 70.0 fs. For an all-fiber ultrashort pulse amplification system, this is in good agreement, especially when the exact fiber lengths are hard to achieve very good accuracy. Therefore, this agreement does verify the practicability of our design method.

At the same time, the measured pulse duration magnification is |M′| = 82 fs/388.6 fs = 0.21, while the ratio of the GDD in the compressor to the GDD in the stretcher is |D′₂/|D′₁| = 0.18 not far from the measured value. This equivalence confirms that a CPA system can be considered as a time-lens imaging system. Thus, the amplification stage in a fiber CPA system can be viewed as a time-lens.

Strictly speaking, the time-lens approximation requires $D_{1} ≫ τ_{0}^{2}$ , the “far field condition” of imaging. Actually in our case, this condition is not satisfied, because $D_{1}^{'} / τ_{0}^{2} = 1.64 < 10$ . Given the calculation does not deviate too much from the experimental results, the time-lens approximation above still holds. In the fiber CPA above, the stretching factor of the stretcher is less than 2, so in general the time-lens approximation would work with any stretch factors larger than that. This approximation won’t work when pulses are distorted, e.g. the interference between dispersive waves and solitons, fissions of solitons, or breaking up of pulse. These effects will limit the application of time-lens in a fiber CPA [21–23], however these conditions are not what people want in general for lasers and to some extent people can avoid these effects via modifying input pulse spectra.

In addition, for a soliton mode-locked laser, the generated pulse is very close to a nearly transform limited pulse. Ideally the image of a transform limited pulse would be a transform limited one. The calculated relative chirp of the output pulse is fairly small (< 0.02). Therefore, output pulses of a fiber CPA system that satisfy the temporal imaging condition can be considered as nearly transform-limited. The detailed derivations can be found in Appendix C.

Because the GDD of fiber amplification systems is mainly determined by the lengths of fibers, and the length is not a continuously adjustable parameter for cutting and splicing processes. From the comparison between the experimental results and designed parameters, people will find our method helpful to determine lengths of fibers for each segment in a close range. For a pre-chirp managed amplification laser [24], it is quite close to the relay amplifiers above in concept. So this time-lens perspective may be found useful in those applications too.

5. Conclusion

Based on the temporal imaging theory, we present a novel design method for a fiber CPA system. Given the initial pulse duration, the pulse energy, the gain and the nonlinearity of the gain medium, the relation between GDDs of the stretcher and the compressor can be easily found under the imaging condition, with the pulse duration magnification or demagnification of the input pulse duration ready to be obtained. The nature of the time-lens takes nonlinear phase shifts into account already, which makes it automatically suitable for those CPAs with pulse duration compression needed. A fiber CPA system with two amplifier stages and pulse duration demagnificant of one fifth was made to verify the proposed designing method. Experimental results and calculated parameters were in good agreements. It shows that this simple, effective and practical method can be used to design multistage amplification systems with a good performance, and also provides a new perspective for designing a CPA system.

A. The focal length of a time-lens produced by SPM without considering gain saturation effects

The relation between a normalized amplitude U and a slowly varying pulse envelope A is given by

U (z, τ) = \frac{A (z, τ)}{\sqrt{P_{0}}} e^{\frac{α z}{2}},

where α is the loss parameter of the medium that can be replaced by −g for gain medium (g is taken as a constant here).

The solution of Eq. (9) is [17]

U (L, τ) = U (0, τ) e^{i Φ_{NL} (L, τ)},

where L is the total length of the gain medium, in which the SPM effect occurs. The nonlinear phase shift can be written as

Φ_{NL} (L, τ) = {| A (0, τ) |}^{2} \frac{γ}{g} (e^{g L} - 1),

where γ is the nonlinear parameter of the medium.

The envelope function of a non-chirped Gaussian pulse in a retarded time frame is

A_{0} (τ) = \sqrt{\frac{E_{0}}{τ_{0} \sqrt{π}}} e^{- \frac{τ^{2}}{2 τ_{0}^{2}}},

where τ₀ is the pulse duration (

\frac{1}{e}

half width of the intensity) and E₀ is the pulse energy. The GDD of the stretcher is D₁. At this time, the envelope function of this pulse can be expressed as

A_{0} (τ) = \sqrt{\frac{E_{0}}{τ_{0} \sqrt{π}}} \frac{τ_{0}}{\sqrt{τ_{0}^{2} - i D_{1}}} e^{- \frac{τ^{2}}{2 (τ_{0}^{2} - i D_{1})}} .

The intensity function of the pulse is

{| A_{0} (τ) |}^{2} = \frac{E_{0} τ_{0}}{\sqrt{π}} \frac{1}{\sqrt{τ_{0}^{4} + D_{1}^{2}}} e^{- \frac{τ^{2}}{τ_{0}^{2} + \frac{D_{1}^{2}}{τ_{0}^{2}}}} .

When the dispersion is large enough (

D_{1} ≫ τ_{0}^{2}

), Eq. (26) can be expanded in Taylor series. Then we take the first order approximation of the Taylor expansion, the intensity function can be expressed as

{| A_{0} (τ) |}^{2} \approx \frac{E_{0} τ_{0}}{| D_{1} | \sqrt{π}} (1 - \frac{τ_{0}^{2}}{D_{1}^{2}} τ^{2}) .

After being stretched, the pulse enters the gain medium where the SPM effect begins to occur. Substituting Eq. (27) into Eq. (23), we can get the time varying part of the nonlinear phase shift expressed as

Φ_{TimeLens} (τ) = - \frac{E_{0} τ_{0}^{3}}{{| D_{1} |}^{3} \sqrt{π}} \frac{γ}{g} (e^{g L} - 1) τ^{2} .

This is a time-varying quadratic phase factor, for which the amplification stage can be seen as a time-lens. By comparing Eq. (3) with Eq. (28), the focal length of the time-lens produced by the SPM effect in the amplification process can be obtained as

D_{f} = - \frac{\sqrt{π}}{2 E_{0} τ_{0}^{3} γ L_{eff}} {| D_{1} |}^{3},

where the effective length L_eff is

L_{eff} = \frac{e^{g L} - 1}{g} .

B. The focal length of a time-lens produced by SPM considering gain saturation effects

As shown in Fig. 3, the length of each gain medium segment is L. The pulse energy launched into the first segment is E₀. So the gain coefficient g₁ of this segment is [for simplicity, here we omit the coefficient described by Eq. (11)]

g_{1} = \frac{g_{0}}{1 + \frac{E_{0}}{E_{S}}} .

If the peak power of the pulse is written as P₀ at the beginning of amplification, the equation that govern the pulse propagation in the first segment of the gain medium is

\frac{\partial U}{\partial z} = i γ P_{0}^{(1)} e^{g_{1} z} {| U |}^{2} U .

After the first segment, the pulse turns into the following form:

{\begin{cases} U^{(1)} (0, τ) = \frac{A^{(1)} (0, τ)}{\sqrt{P_{0}^{(1)}}}, \\ Φ_{NL}^{(1)} = {| U^{(1)} (0, τ) |}^{2} γ L_{eff}^{(1)} P_{0}^{(1)}, \\ U^{(1)} (L, τ) = U^{(1)} (0, τ) e^{i Φ_{NL}^{(1)}}, \\ A^{(1)} (L, τ) = A^{(1)} (0, τ) e^{i Φ_{NL}^{(1)}} e^{\frac{g_{1} L}{2}}, \\ {| A^{(1)} (L, τ) |}^{2} = {| A^{(1)} (0, τ) |}^{2} e^{g_{1} L}, \end{cases}

where the effective length of the first segment

L_{eff}^{(1)}

is

L_{eff}^{(1)} = \frac{e^{g_{1} L} - 1}{g_{1}} .

In general, after the first segment of the gain medium, the shape of the pulse remains the same. The peak power and the pulse energy change to be e^g₁L times those of the input pulse, and the nonlinear phase shift produced by the SPM is

Φ_{NL}^{(1)}

.

The evolution of the pulse through the second segment of the medium should have the same form as Eq. (33), i.e.

{\begin{cases} U^{(2)} (0, τ) = \frac{A^{(2)} (0, τ)}{\sqrt{P_{0}^{(2)}}}, \\ Φ_{NL}^{(2)} = {| U^{(2)} (0, τ) |}^{2} γ L_{eff}^{(2)} P_{0}^{(2)}, \\ U^{(2)} (L, τ) = U^{(2)} (0, τ) e^{i Φ_{NL}^{(2)}}, \\ A^{(2)} (L, τ) = A^{(2)} (0, τ) e^{i Φ_{NL}^{(2)}} e^{\frac{g_{2} L}{2}}, \\ {| A^{(2)} (L, τ) |}^{2} = {| A^{(2)} (0, τ) |}^{2} e^{g_{2} L} . \end{cases}

In Eq. (35),

{\begin{cases} A^{(2)} (0, τ) = A^{(1)} (L, τ), \\ P_{0}^{(2)} = P_{0}^{(1)} e^{g_{1} L}, \\ g_{2} = \frac{g_{0}}{1 + \frac{E_{0}}{E_{S}} e^{g_{1} L}}, \\ L_{eff}^{(2)} = \frac{e^{g_{2} L} - 1}{g_{2}} . \end{cases}

Therefore, the envelope function of the pulse output from the second segment of the gain medium is

\begin{array}{l} A^{(2)} (L, τ) & = A^{(2)} (0, τ) e^{i Φ_{NL}^{(2)}} e^{\frac{g_{2} L}{2}} \\ = A^{(1)} (0, τ) e^{i [Φ_{NL}^{(1)} + Φ_{NL}^{(2)}]} e^{\frac{(g_{1} + g_{2}) L}{2}} \\ = A^{(1)} (0, τ) e^{\frac{(g_{1} + g_{2}) L}{2}} \cdot e^{i γ {| A^{(1)} (0, τ) |}^{2} [L_{e ff}^{(1)} + e^{g_{1} L} L_{eff}^{(2)}]} . \end{array}

And the nonlinear phase shift is

Φ_{NL}^{(1)} + Φ_{NL}^{(2)} = γ {| A^{(1)} (0, τ) |}^{2} [L_{eff}^{(1)} + e^{g_{1} L} L_{eff}^{(2)}] .

It can be seen from Eq. (37) that the shape of the pulse remains the same after two segments of the gain medium, while only the amplitude is amplified by

e^{\frac{(g_{1} + g_{2}) L}{2}}

times.

Assume the total length of the gain medium is L₀. Then the number of segments is $n = \frac{L_{0}}{L}$ , and the envelope function A⁽ⁿ⁾(L, τ) of the pulse output from the n_th segment is

A^{(n)} (L, τ) = A^{(1)} (0, τ) e^{\sum_{j = 1}^{n} \frac{g_{j} L}{2}} \cdot e^{i γ {| A^{(1)} (0, τ) |}^{2} (\sum_{j = 1}^{n} e^{\sum_{k = 1}^{j - 1} g_{k} L} L_{eff}^{(j)})} .

In Eq. (39),

{\begin{cases} g_{k} = \frac{g_{0}}{1 + \frac{E_{0}}{E_{S}} e^{\sum_{m = 1}^{k - 1} g_{m} L}}, \\ L_{eff}^{(j)} = \frac{e^{g_{j} L} - 1}{g_{j}} . \end{cases}

Here we have the following convention,

\sum_{k = 1}^{0} g_{k} L = \sum_{m = 1}^{0} g_{m} L = 0 .

Eq. (39) can also be expressed as B-integral. In this case, Eq. (39) can be written as

{\begin{cases} A^{(n)} (L, τ) = A^{(1)} (0, τ) e^{\sum_{j = 1}^{n} \frac{g_{j} L}{2}} \cdot e^{i {| U^{(1)} (0, τ) |}^{2} (\sum_{j = 1}^{n} B^{(j)})}, \\ B^{(j)} = γ L_{eff}^{(j)} P_{0} e^{\sum_{k = 1}^{j - 1} g_{k} L} . \end{cases}

where U⁽¹⁾(0, τ) is the normalized amplitude of A⁽¹⁾(0, τ), B^(j) is the B-integral of the j_th segment.

Thus the total nonlinear phase shift is

Φ_{NL}^{Total} = γ {| A^{(1)} (0, τ) |}^{2} (\sum_{j = 1}^{n} e^{\sum_{k = 1}^{j - 1} g_{k} L} L_{eff}^{(j)}) .

Substituting Eq. (27) into Eq. (42), we can get the time-varying quadratic phase factor described by

Φ_{TimeLens} (τ) = - γ (\sum_{j = 1}^{n} e^{\sum_{k = 1}^{j - 1} g_{k} L} L_{eff}^{(j)}) \frac{E_{0} τ_{0}^{3}}{{| D_{1} |}^{3} \sqrt{π}} τ^{2} .

Therefore, the focal length of the time-lens is

D_{f} = - \frac{\sqrt{π}}{2 E_{0} τ_{0}^{3} γ (\sum_{j = 1}^{n} e^{\sum_{k = 1}^{j - 1} g_{k} L} L_{eff}^{(j)})} {| D_{1} |}^{3} .

C. Chirp characteristic of the output pulse from a CPA system satisfying the temporal imaging condition

From Eq. (7), it can be concluded that for a temporal imaging system, the output pulse is not transform-limited when the input pulse is transform-limited. Since the characteristic of the chirp is such that the frequency of the pulse at different times have different frequency shifts relative to the carrier frequency, we can introduce the small chirped condition as

Δ ω_{\max} ≪ 2 π ν_{0},

where Δω_max is the maximum angular frequency shift during the pulse duration and ν₀ is the carrier frequency.

From the factor $e^{i \frac{τ^{2}}{2 {MD}_{f}}}$ in Eq. (7), we can have

Δ ω (τ) = | \frac{τ}{{MD}_{f}} | .

If the FWHM of the output pulse is τ′_FWHM, then the maximum value of Δω(τ) is

Δ ω_{\max} = | \frac{τ_{FWHM}^{'}}{2 {MD}_{f}} | .

Substituting Eq. (4) and Eq. (8) into Eq. (47), the maximum value of Δω(τ) is

Δ ω_{\max} = | \frac{D_{1} + D_{2}}{2 D_{2}^{2}} | τ_{FWHM}^{'} .

We usually want pulses to be compressed, so |M| < 1, that is, |D₂| < |D₁|. Therefore, we can get

| D_{1} + D_{2} | ⩽ | D_{1} | + | D_{2} | < 2 | D_{1} | .

From this inequality and Eq. (48) we can have

Δ ω_{\max} < | \frac{D_{1}}{D_{2}^{2}} | τ_{FWHM}^{'} = \frac{τ_{FWHM}^{'}}{M^{2} | D_{1} |} .

Take designed values in Sec.4 and substitute them into the right side of Eq. (49), where τ′_FWHM = 70.0 fs, |M| = 0.18, D₁ = −89381 fs², we can get Δω_max < 24.1 THz. Taking ν₀ as 193.5 THz (the central wavelength is 1550 nm), then we can get

\frac{Δ ω_{\max}}{2 π ν_{0}} < 0.02 .

Therefore, the output pulse of our fiber CPA system that satisfies the temporal imaging condition can be considered as nearly transform-limited (small chirped).

Funding

National Key R& D Plan of China (2016YFB1102404); Technological Innovation Major Project of Hubei Province (2016AAA004); Open Fund of the State Key Laboratory of High Field Laser Physics (Shanghai Institute of Optics and Fine Mechanics).

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Time-lens perspective on fiber chirped pulse amplification systems

Abstract

1. Introduction

2. Imaging theory of time-lens

3. Time-lens in a fiber CPA system

4. Experimental results

5. Conclusion

A. The focal length of a time-lens produced by SPM without considering gain saturation effects

B. The focal length of a time-lens produced by SPM considering gain saturation effects

C. Chirp characteristic of the output pulse from a CPA system satisfying the temporal imaging condition

Funding

References

Cited By

Figures (5)

Equations (52)

Optics Express