Seidel aberrations in grating pulse stretchers

Štěpán Vyhlídka; Daniel Kramer; Galina Kalinchenko; Bedřich Rus

doi:10.1364/OE.24.030421

1. Introduction

The development of ultra-high power laser sources plays an important role in modern physics. Delivering the highest possible pulse peak powers to laser targets is vital in a range of experiments in fields such as attosecond science, particle acceleration, laser plasma interaction and relativistic physics [1]. Currently there are several laser systems under construction which aim to reach the 10 PW-regime [2–5]. The generation of laser pulses in the terawatt and petawatt regimes has been achieved through the use of the (CPA) chirped pulse amplification technique [6]. With this technique, pulses are temporally stretched in a device with positive dispersion. Typically, it is a pair of diffraction gratings separated by a 1:1 telescope. The pulses can be subsequently amplified as their peak intensity is reduced, which helps to avoid nonlinear effects and damage to amplifiers. Finally, the pulses are compressed in a device with opposite dispersion, which is usually a pair of diffraction gratings for stretched pulse durations on the order of nanoseconds. In order to retrieve the original pulse duration, it is necessary that the sum of the spectral phases acquired by the pulse at each stage is zero:

φ_{stretcher} (ω) + φ_{material} (ω) + φ_{compressor} (ω) = 0 .

The condition (1) can be reached by optimization of both stretcher and compressor parameters. While the analytical formula for φ_compressor has been proposed by Treacy [7] and φ_material can be calculated from the dispersion of the materials and optical coatings [8], φ_stretcher is usually computed by numerical ray-tracing algorithms [9] for grating stretchers because the aberrations present in the stretcher imaging system introduce a deviation from the ideal.

To simplify following discussion, it is convenient to expand the spectral phase of an ultrashort pulse φ(ω) into a Taylor series around the central frequency of the pulse ω₀:

\begin{array}{l} φ (ω) = φ_{0} + {\frac{d φ}{d ω} |}_{ω_{0}} (ω - ω_{0}) + {\frac{d^{2} φ}{d ω^{2}} |}_{ω_{0}} \frac{{(ω - ω_{0})}^{2}}{2} + {\frac{d^{3} φ}{d ω^{3}} |}_{ω_{0}} \frac{{(ω - ω_{0})}^{3}}{3!} + {\frac{d^{4} φ}{d ω^{4}} |}_{ω_{0}} \frac{{(ω - ω_{0})}^{4}}{4!} + \dots \\ = φ_{0} + GD (ω - ω_{0}) + \frac{GDD}{2} {(ω - ω_{0})}^{2} + \frac{TOD}{3!} {(ω - ω_{0})}^{3} + \frac{FOD}{4!} {(ω - ω_{0})}^{4} + \dots . \end{array}

Whereas residual group delay dispersion (GDD) mainly increases pulse duration, third order dispersion (TOD), fourth order dispersion (FOD) and higher dispersion orders also influence the temporal contrast. The constant phase offset φ₀ and the group delay (GD) do not affect the shape of the pulse and so they are not considered in the rest of the paper.

So far, several steps in finding the analytic solution of the grating stretcher phase φ_Stretcher have been undertaken. First, Martinez calculated the spectral phase of a perfect stretcher [10]. Later, an extended ABCD formalism was introduced by Martinez [11] and expanded by Kostenbauder [12]. However, the linear nature of this formalism only allows for the calculation of GDD and not the determination of higher-order terms. Druon et al. proposed another extension of the ABCD formalism by using nonlinear operators instead of linear matrices [13], which allows for the calculation of the spectral phase in the general case. The spectral dependence of the group delay was also calculated by Zhang [14] using ray-tracing. Even though the spectral phase and higher order dispersion terms for grating stretchers can be analytically obtained by the usage of the last two methods, the resulting formulae are not illustrative without numerical substitution and do not offer much physical insight.

The effect of the aberrations present in the imaging system of the stretcher has been explored more thoroughly with numerical ray-tracing algorithms. A design allowing TOD and FOD compensation by modification of chromatic and spherical aberrations of doublet lenses was presented by White et al. [15]. A cylindrical mirror stretcher design eliminating phase errors up to fifth order term was presented by Lemoff and Barty [16]. Later, the impact of the spherical aberration was discussed for the Banks stretcher [17]. The influence of aberrations on the spectral phase was also discussed in papers [18], [19] and others. However, no direct relation between aberrations and induced changes of the spectral phase has been presented up to now.

In this article we employ the Seidel aberration theory to analytically describe the effect of aberrations present in the imaging system of a pulse stretcher on the spectral phase φ_Stretcher. The effect of Seidel aberrations can be expressed with an aberration function that represents an optical path difference as a function of entrance pupil coordinates. In general, the aberration function changes the relative phase between different spatial frequency components of the object depending on the position at which the individual spatial frequencies are incident to the entrance pupil. In a pulse stretcher, the object is usually a gaussian beam diffracted by a grating. The entrance pupil coordinates then depend mostly on the chromatic dispersion of the grating [see Fig. 1]. The aberration function then represents the optical path difference between distinct spectral components of the beam and hence distorts the spectral phase of the stretched pulse.

Fig. 1 Imaging vs grating pulse stretcher case. ν represent spatial frequencies, which can be calculated from the Fourier transform of the object spatial profile, whereas ω represent angular frequencies, which can be calculated from the Fourier transform of the object temporal profile. η is the entrance pupil coordinate.

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2. Theory

2.1. General theory

According to Goodman [20], we can express the relation between the object U(x, y, ω) and its image U_img (x_img, y_img, ω) using the amplitude transfer function H(ν_x, ν_y), which acts as a spatial frequency filter and modifies the amplitude and phase of the input beam spatial spectrum as:

FT [U_{img} (x_{img}, y_{img}, ω)] (ν_{x}, ν_{y}, ω) + H (ν_{x}, ν_{y}) FT [U (x, y, ω)] (ν_{x}, ν_{y}, ω),

where x, y (x_img, y_img) are Cartesian coordinates in the object (image) plane, ν_x and ν_y are spatial frequencies and FT stands for a Fourier transform. The amplitude transfer function is a complex function and it can be expanded to:

H (ν_{x}, ν_{y}, ω) = P (ξ (ν_{x}), η (ν_{y})) e^{- i k Φ (ξ (ν_{x}), η (ν_{y}), x, y)},

where P describes the amplitude transmittance and Φ is the aberration function, which represents a wavefront deviation with respect to a Gaussian reference sphere [see Fig. 2]. For a grating pulse stretcher, the dependence of the pupil coordinates on ν_x, ν_y and ω can be expressed in the paraxial approximation as:

ξ = x_{0} + s λ ν_{x}; η = y_{0} + s θ (ω) + s λ ν_{y},

where s is the distance between the first grating and the entrance pupil, λ is the wavelength, θ(ω) is the ray angular deviation coming from angular dispersion. We can expect that the position of a ray on the entrance pupil in grating stretchers is mostly given by its frequency, which comes from the Fourier transform of the temporal profile of the pulse and not the transform of its spatial profile [see Fig. 3]. In the following, we will thus neglect the effect of a finite beam size (sλν_x ≈ 0, sλν_y ≈ 0) and proceed with the calculation for a plane wave. Pupil coordinates can then be expressed as:

ξ = x_{0}; η = y_{0} + s θ (ω) .

Assuming that the entrance pupil of the stretcher is large enough so that the spectrum of the pulse is not truncated, we can put P(ξ, η) = 1. If we substitute pupil coordinates Eq. (6) into Eq. (3), we get:

FT [U (x_{img}, y_{img}, ω)] (ν_{x}, ν_{y}, ω) = e^{- i k Φ (ξ (x_{0}), η (ω, y_{0}, s), x_{0}, y_{0})} FT [U (x, y, ω)] (ν_{x}, ν_{y}, ω) .

According to Eq. (7), the aberration function can be used to calculate the deviation of the spectral phase of the pulse φ(ω) at the image plane with respect to the ideal case without aberrations:

φ (ω, s_{img}) = \frac{ω}{c} Φ (ξ (x_{0}), η (ω, y_{0}, s), x_{0}, y_{0}) .

To calculate the spectral phase at some other point, we can proceed the same way as for the ideal stretcher with a grating distance L [8]. To our knowledge, the additional residual angular dispersion originating from the aberrations at the image point does not induce additional phase changes when the pulse is propagated from the image point to the position of the second grating. The total spectral phase of the pulse at the arbitrary position L is then given by the sum of the spectral phase deviation coming from aberrations and the spectral phase of the ideal stretcher:

φ (ω, L) = \frac{ω}{c} Φ (ξ (x_{0}), η (ω, y_{0}, s), x_{0}, y_{0}) - L \frac{ω}{c} cos (θ (ω)) .

This theory can also be used to calculate the residual angular dispersion of the stretcher, which can generally lead to additional pulse broadening and reduces peak power in the focus [21]. The residual angular dispersion is present because rays with different frequencies propagate through the stretcher with different angles and acquire different angular deviation due to aberrations. In other words, the derivative of the wavefront at a given point corresponds to the propagation direction of the ray coming from that point. The aberration function represents a deviation of the real wavefront from a perfect wavefront. The derivative of the aberration function with respect to the pupil coordinate then expresses a variation in the propagation direction of aberrated rays. Since each pupil coordinate is assigned to a ray of different frequency, this variation of propagation direction is spectrally dependent. Residual angular dispersion can then be calculated as:

\begin{array}{l} d θ_{y} (ω) = \frac{cos (β)}{cos (α)} \frac{d Φ (ξ (x_{0}), η (ω, y_{0}, s), x_{0}, y_{0})}{d η} = \frac{cos (β)}{cos (α)} \frac{1}{θ (ω)} \frac{d Φ (x_{0}, y_{0}, s, ω)}{d s} \\ d θ_{x} (ω) = \frac{d Φ (ξ (x_{0}), η (ω, y_{0}, s), x_{0}, y_{0})}{d ξ} = \frac{d Φ (x_{0}, y_{0}, s, ω)}{d x_{0}}, \end{array}

where the

\frac{cos (β)}{cos (α)}

factor comes from the grating equation and rescales the angular dispersion in the diffraction plane as the pulse hits the second grating. In the rest of the article, the residual angular dispersion dθ(ω) is expressed as a function of wavelength, as it is more common, rather than frequency.

Fig. 2 Schematic of a grating pulse stretcher. s is the distance between the first grating and the entrance pupil (the first lens), s_img is the distance between the last lens and the image of the first grating, L is the distance between the image plane and the second grating.

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Fig. 3 Coordinate system definition. We define the object as a place, where the laser pulse hits the diffraction grating with coordinates (x₀, y₀). To simplify calculation, it is convenient to choose coordinate-axis so that diffraction occurs in the horizontal yz plane (diffraction plane). We assume that the optical axis of the stretcher telescope is parallel with z-axis and the propagation direction of central wavelength of the pulse and intersects the object plane at the point (0,0).

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2.2. Aberration function

Following Welford [22], we can define a wave aberration function:

\begin{array}{l} Φ (ξ, η, x, y) & = B {(ξ^{2} + η^{2})}^{2} + C {(ξ x + y η)}^{2} + D (x^{2} + y^{2}) (ξ^{2} + η^{2}) \\ + E (x^{2} + y^{2}) (ξ x + y η) + F (ξ^{2} + η^{2}) (ξ x + y η), \end{array}

where B is the spherical aberration, C is the astigmatism, D is the field curvature, E is the field distortion and F is the coma coefficient. In the following calculations, we deal with the impact of the fourth order aberrations only. The aberration function can be easily extended to include higher order aberrations as well, but in general their analytic calculation is very complicated. We also neglect tilt and defocus terms as they represent a shift of the center of the Gaussian reference sphere with respect to the image point and the aberrations are usually measured with respect to the sphere originating from the Gaussian image point.

If we substitute pupil coordinates ξ and η from Eq. (6) to Eq. (11) and put x = x₀ and y = y₀, we get:

\begin{array}{l} Φ (x_{0}, y_{0}, s, ω) = & B s^{4} θ^{4} (ω) + y_{0} (4 B + F) s^{3} θ^{3} (ω) + y_{0}^{2} (6 B + C + D + 3 F) s^{2} θ^{2} (ω) \\ + x_{0}^{2} (2 B + D + F) s^{2} θ^{2} (ω) + \dots . \end{array}

In a grating stretcher, θ(ω) is defined by the grating equation and can be linearly approximated by:

θ (ω) = arcsin (m N \frac{2 π c}{ω} - sin (α)) - arcsin (m N \frac{2 π c}{ω_{0}} - sin (α)) ≐ \frac{m N (2 π c)}{ω_{0}^{2} cos (β_{0})} (ω - ω_{0}),

where ω₀ is the central frequency of the pulse, α is the angle of incidence on the grating, β₀ is the diffraction angle of the central frequency, m is the diffraction order and N is the line density of the grating. The linear approximation of θ(ω) ∼ (ω − ω₀) allows for an estimation of the impact of each term from Eq. (12) on the spectral phase dispersion terms [Eq. (2)]. The first term from Eq. (12) is quartic in θ(ω) and it deviates the FOD. The second term is cubic and deviates the TOD. The third and the fourth terms then deviate the GDD. In the following calculations, the terms that are linear or constant in θ(ω) are neglected because they mostly introduce additional group delay or a constant phase offset respectively, and do not change the temporal profile of a pulse. The odd power terms in θ(ω) are present only when the object has a y₀ offset as they originate from the middle term of

η^{2} = {(y_{0} + s θ (ω))}^{2} = y_{0}^{2} + 2 y_{0} s θ (ω) + {(s θ (ω))}^{2}

.

The linear approximation of θ(ω) limits the accuracy of the method because it neglects the contribution of the terms of the lower powers to the given dispersion order. For example, the quadratic term of Eq. (12) also contributes to TOD through the nonlinear part of θ(ω). In general, the contribution of terms derived within the linear approximation to the corresponding dispersion orders is a few orders of magnitude bigger than the contribution of these nonlinear corrections. These corrections are still required for systems with a high stretch ratio or broad bandwidth.

In general, x₀ and y₀ offsets are required for the separation of input and output beams in multi-pass stretchers. Usually, the x₀ offset is used for separation of the beams between subsequent passes while y₀ offset is required for the separation of the beams only in 4 and higher pass designs. In stretchers using only one grating and reflective optics, the x₀ offset is usually much bigger than y₀ because it also separates the beam between subsequent grating hits. In the following two sections, two possible setups will be discussed. First, we consider a situation when the object (i.e. the place where the pulse hits the grating) is on the optical axis of the stretcher telescope. This case is important for 2-pass stretchers and stretchers using cylindrical optics, where the x₀ offset does not matter because the beam is imaged only in the yz plane. Second, we consider a general situation when the pulse hits a grating at the arbitrary point (x₀, y₀), which is relevant for the majority of stretcher designs.

2.3. On-Axis object

First, let’s assume that the object lies on the optical axis of the stretcher telescope (x₀ = 0 and y₀ = 0). Only spherical aberration is present because all the other coefficients from Eq. (12) vanish. The spectral phase deviation at the image plane $φ_{on}^{a b}$ is then:

φ_{on}^{a b} (ω, s_{img}) = \frac{ω}{c} Φ (s, ω) = \frac{ω}{c} B s^{4} θ^{4} (ω) ≐ B s^{4} \frac{m^{4} N^{4} {(2 π)}^{4} c^{3}}{ω_{0}^{8} {cos}^{4} (β_{0})} ω {(ω - ω_{0})}^{4} .

Because the first three derivatives with respect to frequency at the central frequency of the spectral phase defined by Eq. (14) are zero, the spherical aberration induces FOD deviation FOD^ab:

{\frac{d^{4} φ_{on}^{a b}}{d ω^{4}} |}_{ω_{0}} ≐ 24 B s^{4} \frac{m^{4} N^{4} {(2 π)}^{4} c^{3}}{ω_{0}^{7} {cos}^{4} (β_{0})} = {FOD}^{a b} .

Higher order dispersion terms can be calculated when the linear approximation of θ(ω) is not used.

2.4. Off-axis object

When the object does not lie on the optical axis, all terms from Eq. (12) are relevant. Combining Eq. (8) and Eq. (12), we can calculate the spectral phase deviation $φ_{off}^{a b}$ for any object position in the object plane as:

φ_{off}^{a b} (ω, s_{img}) = \frac{ω}{c} Φ (x_{0}, y_{0}, s, ω) = φ_{off}^{a b 1} + φ_{off}^{a b 2} + φ_{off}^{a b 3} + φ_{off}^{a b 4} + \dots,

where the individual terms

φ_{off}^{a b 1}

,

φ_{off}^{a b 2}

,

φ_{off}^{a b 3}

and

φ_{off}^{a b 4}

originate from the terms on the right of Eq. (12) with preserved order.

The first term $φ_{off}^{a b 1}$ represents the “on-axis” term $φ_{on}^{a b}$ from the previous section [Eq. (15)]. The second term $φ_{off}^{a b 2}$ varies linearly with the y₀ offset and it has an impact on TOD and higher order terms. If we substitute the linear approximation of θ(ω) [Eq. (13)] into Eq. (16), we can derive TOD deviation TOD^ab at the image position:

{\frac{d^{3} φ_{off}^{a b 2}}{d ω^{3}} |}_{ω_{0}} = \frac{y_{0} (4 B + F) s^{3}}{c} \frac{d^{3} (ω θ^{3})}{d ω^{3}} ≐ 6 y_{0} (4 B + F) s^{3} \frac{m^{3} N^{3} {(2 π)}^{3} c^{2}}{ω_{0}^{5} {cos}^{3} (β_{0})} = {TOD}^{a b} .

The contribution of

φ_{off}^{a b 2}

to FOD deviation

{FOD}_{TOD}^{a b}

can be obtained if we leave out the linear approximation of θ(ω) from Eq. (13) and calculate the fourth derivative of

φ_{off}^{a b 2}

with respect to frequency numerically:

{\frac{d^{4} φ_{off}^{a b 2}}{d ω^{4}} |}_{ω_{0}} = \frac{y_{0} (4 B + F) s^{3}}{c} \frac{d^{4} (ω θ^{3})}{d ω^{4}} = {FOD}_{TOD}^{a b} .

The last two terms $φ_{off}^{a b 3}$ and $φ_{off}^{a b 4}$ from Eq. (16) are quadratic in both θ(ω) and either y₀ or x₀ and thus mainly introduce GDD deviation GDD^ab:

{\frac{d^{2} (φ_{off}^{a b 3} + φ_{off}^{a b 4})}{d ω^{2}} |}_{ω_{0}} ≐ 2 (y_{0}^{2} (6 B + C + D + 3 F) + x_{0}^{2} (2 B + D + F)) s^{2} \frac{m^{2} N^{2} {(2 π)}^{2} c}{ω_{0}^{3} {cos}^{2} (β_{0})} = {GDD}^{a b} .

The contribution of the

φ_{off}^{a b 3}

and

φ_{off}^{a b 4}

terms to TOD deviation

{TOD}_{GDD}^{a b}

again cannot be calculated in the scope of the linear aproximation of θ(ω), but it be analytically derived with the help of the outcome of Eq. (19) and the known ratio between the TOD and GDD of the ideal stretcher [8]. The formula for the ratio between the GDD, TOD and FOD depends on the power of θ(ω).

φ_{off}^{a b 3}

,

φ_{off}^{a b 4}

and the ideal stretcher phase (the second term of taylor series of cosine in Eq. (9)) are all quadratic in θ(ω) and they thus retain the same ratio between the GDD, TOD and FOD. Because we know the ratio between the individual dispersion orders for a perfect stretcher, we can easily calculate the induced TOD deviation from GDD^ab:

{\frac{d^{3} (φ_{off}^{a b 3} + φ_{off}^{a b 4})}{d ω^{3}} |}_{ω_{0}} = - G D D^{a b} \frac{3 λ}{2 π c} (1 + \frac{λ}{d} \frac{sin (α)}{{cos}^{2} (α)}) = {TOD}_{GDD}^{a b} .

3. Martinez stretcher

As an example of how to apply this method, we can look into the impact of Seidel aberrations in the Martinez-type stretcher with spherical optics [Fig. 4]. The aberrations of this design are essentially the same as those of the Banks stretcher (depending on the conic constant of the optics used), which has been used, for example, for both 30 fs [23] and 150 fs [24] TW/PW CPA chains. Seidel aberration coefficients for a telescope consisting of 2 spherical mirrors at a 2f distance are presented in Table 1. For the following calculations, it is convenient to introduce dimensionless variables which scale with the radius of curvature R = 2 f:

s^{'} = \frac{s}{R}; x_{0}^{'} = \frac{x_{0}}{R}; y_{0}^{'} = \frac{y_{0}}{R} .

Fig. 4 Martinez stretcher scheme

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Table 1. Aberration coefficients of Martinez-type stretcher. Coefficients were derived using theory presented in [22].

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3.1. On-axis

Using Eq. (15) and the spherical aberration coefficient from Table 1, the dependence of the FOD deviation on the distance of the first grating from the first lens of the stretcher can be calculated:

{FOD}^{a b} = 12 \frac{{(R - s)}^{2}}{R^{3} s^{2}} s^{4} \frac{m^{4} N^{4} {(2 π)}^{4} c^{3}}{ω_{0}^{7} {cos}^{4} (β_{0})} ~ R (s^{' 2} - {2 s}^{' 3} + s^{' 4}) .

The FOD deviation linearly increases with R and decreases as the object approaches the first lens. It reaches maximum when the object lies in the focal plane. It seems logical to put the object close to the first lens to minimize the FOD deviation. However, this can increase the residual angular dispersion dθ_y (λ). Using Eq. (10) and the spherical aberration coefficient from Table 1, this can be expressed as:

d θ_{y} (λ) = \frac{cos (β)}{cos (α)} \frac{1}{θ (λ)} \frac{d (B s^{4} θ^{4} (λ))}{d s} = \frac{cos (β)}{cos (α)} (s^{'} - {3 s}^{' 2} + 2 s^{' 3}) θ^{3} (λ) ≐ K_{1} \cdot {(λ - λ_{0})}^{3},

where K₁ is a function of s′. Within the linear approximation of θ(ω), the residual angular dispersion in the yz plane is cubic in (λ − λ₀). Its value is given by the relative position of the grating within the stretcher s′ regardless of the absolute value of R. Figure 5 shows a comparison of FOD and dθ_y (λ) dependencies on s′. The results were verified by independent numerical ray-tracing calculations in Zemax OpticStudio.

Fig. 5 Comparison of FOD^Ab [Eq. (22)] and the residual angular dispersion [Eq. (23)] for different relative grating positions. Analytically calculated data are in solid lines, results obtained by numerical ray-tracing are marked with dots. The calculation was done for N = 1480 ln/mm, α = 48.19, λ₀ = 0.82 μm (parameters taken from [17]).

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3.2. Off-axis

When a laser pulse hits the grating at a point which is not on the optical axis of the telescope, aberrations introduce GDD and TOD deviations and increase residual angular dispersion in the xy and xz planes. By substitution of the respective aberration coefficients from Table 1 into Eq. (17), TOD deviation TOD^ab can be calculated as follows:

{TOD}^{a b} = 6 y_{0} (4 \frac{{(R - s)}^{2}}{2 R^{3} s^{2}} - \frac{(R - s)}{R^{2} s^{2}}) \frac{m^{3} N^{3} {(2 π)}^{3} c^{2}}{ω_{0}^{5} {cos}^{3} (β_{0})} ~ y_{0}^{'} R (s^{'} - {3 s}^{' 2} + {2 s}^{' 3}) .

Analogously, GDD deviation

{GDD}_{PB}^{a b}

can be derived with the help of Eq. (19) and Table 1 :

{GDD}^{a b} = R (x_{0}^{' 2} (- 1 - 2 s^{'} + {2 s}^{' 2}) + y_{0}^{' 2} (1 - 6 s^{'} + {6 s}^{' 2})) \frac{m^{2} N^{2} {(2 π)}^{2} c}{ω_{0}^{3} {cos}^{2} (β_{0})} .

Both GDD^ab and TOD^ab linearly increase with R, their dependencies on the grating position s′ are shown in Fig. 6. If we substitute the aberration coefficients from Table 1 into Eq. (10) we can derive formulae for the residual angular dispersion:

\begin{array}{l} d θ_{x} (λ) & = \frac{x_{0}^{'}}{2} (1 + 2 s^{'} - {2 s}^{' 2}) θ^{2} (λ) ≐ K_{2} \cdot {(λ - λ_{0})}^{2} \\ d θ_{y} (λ) & = \frac{cos (β)}{cos (α)} ((s^{'} - {3 s}^{' 2} + {2 s}^{' 3}) θ^{3} (λ) + y_{0}^{'} (1 - 6 s^{'} + {6 s}^{' 2}) θ^{2} (λ) + 3 y_{0}^{' 2} (1 - 2 s^{'}) θ (λ)) \\ ≐ K_{1} \cdot {(λ - λ_{0})}^{3} + K_{3} \cdot {(λ - λ_{0})}^{2} + K_{4} \cdot (λ - λ_{0}), \end{array}

where K₂, K₃ and K₄ are functions of s′, x′₀ and y′₀. The residual angular dispersion in the xz-plane is quadratic in (λ − λ₀) and it linearly increases as we put the object further from the optical axis. The angular dispersion in the yz plane constitutes three terms. The first term K₁, which is cubic in (λ − λ₀), originates from the on-axis spherical aberration [Eq. ((23)]. The second term K₃ linearly increases with the object distance from the optical axis in diffraction plane and is quadratic in (λ − λ₀). The third term is quadratic in y₀, while it is linear in (λ − λ₀). Total angular dispersion in the yz plane is then given by the interplay of these terms. The dependencies of K₂, K₃ and K₄ on s′ and the angular deviation of individual spectral components are shown in Figs. 7(a) and 7(b) respectively. For example, when a grating lies in the focal plane of the first lens, only the K₃ term is present which results in quadratic residual angular dispersion in the yz-plane. If we place the grating close to s′ = R/4 in the given configuration then we can minimize residual angular dispersion because the individual terms from Eq. (26) almost cancel each other out.

Fig. 6 a) Comparison of GDD^Ab [Eq. (25)] and angular dispersion coefficient K₂ [Eq. (26)] for various grating positions with x′₀ = 1/20. b) TOD^Ab [Eq. 24]) and GDD^Ab deviation (Eq. (25)) for various relative grating positions with y′₀ = 1/40. The contribution of the $φ_{off}^{a b 3}$ phase term [Eq. 20] was included in the calculation of TOD^Ab. The calculation was done for N = 1480 ln/mm, α = 48.19, λ₀ = 0.82 μm (parameters taken from [17]) and R = 2 m. Analytically calculated data are in solid lines, results obtained by numerical ray-tracing are marked with dots.

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Fig. 7 a) Comparison of normalized K₁, K₃ and K₄ dependencies on grating position s′, their relative scaling depends mostly on y′₀ and θ(ω). b) Angular dispersion dθ_x (λ) in the yz-plane for different s′. In this example, θ(ω) was obtained directly from the grating equation without linear approximation [Eq. (10)] due to broad bandwidth. Analytically calculated data are in solid lines, results obtained by numerical ray-tracing are marked with dots. The calculation was done for y′₀ = 1/20, N = 1480 ln/mm, α = 48.19, λ₀ = 0.82 μm and R = 2 m.

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3.3. Discussion

GDD, TOD and FOD deviations in a Martinez stretcher depend on the position of the first grating s′ and linearly scale with R, while the residual angular dispersion depends on s′ only. Spectral phase distortion tend to decrease if the grating is very close to the first lens which allows to reach very high stretch ratios. Nevertheless, the distortion which depend on y′₀ increase. As the x′₀ offset is usually much bigger than y′₀ offset (which can even be zero for the 2-pass design), the impact of the y′₀ offset is for much smaller then the impact of the x′₀ offset most cases.

Residual angular dispersion can be suppressed in multi-pass setups by employing roof mirrors in the design. Roof mirrors flip the beam over either xz or yz plane which reverses the sign of the angular dispersion. Unfortunately, total compensation is not possible because of the fact that roof mirrors also change either the x′₀ or the y′₀ coordinate for the next pass, which according to Eq. (23) leads to uneven absolute values of residual angular dispersion between subsequent passes.

According to our calculations, there is no ideal grating position in the Martinez/Banks stretcher. However, with given space and spectral phase requirements Eqs. (22) – (26) may be used to find the ideal combination of R, s′ and θ(ω) to reach the ideal spectral phase profile with minimal angular dispersion. The calculation in the previous sections was done for a perfectly aligned stretcher, but it can be extended to cases when there is some kind of misalignment present in the stretcher telescope.

In general, the accuracy of this method can be considered to be equivalent to ray-tracing when there is a very small (y′₀ ≲ 1/100) offset and it is bounded to the accuracy of the linear approximation of θ(ω). The most severe limitation of the accuracy originates in the terms with odd powers of θ(ω) in Eq. (12). These terms occur only with the y′₀ offset and they introduce errors in the calculation of higher dispersion orders within the linear approximation. For example, the neglected term that is linear in θ(ω) mainly introduces group delay deviation but its higher derivatives, which are zero under the linear approximation, also partly contribute to GDD and higher order dispersion terms. However, the linear approximation does not restrict the usage of this method with respect to the bandwidth of the system as the individual dispersion terms are defined as local derivatives at the central wavelength of the pulse. For a precise calculation of the angular dispersion of a pulse with broad bandwidth (>30 nm), it is necessary to abandon the linear approximation and use the general grating equation. Nevertheless, this calculation is very straightforward as there are no derivatives of θ(ω) required.

The method gives accurate results even for a full round trip and multi-pass setups. However, it should be noted that the residual angular dispersion acquired after each pass may lead to errors in the calculation of the spectral phase for the next pass as it modifies θ(ω). These errors especially increase with y′₀ offset as it induces changes in the angular dispersion in the diffraction plane. They mainly distort the FOD and higher dispersion orders but they strongly depend on the profile of the residual angular dispersion and their impact is thus hard to generalize.

The deviation of the spectral phase dispersion terms obtained by this method coincided with a numerical ray-tracing algorithm with deviation less than 0.1% for “on-axis” and x′₀ “off-axis” terms and less than 5% for y′₀ “off-axis” terms. If the accuracy is not sufficient, it is possible to either expand θ(ω) into the next Taylor term or use this method numerically and directly calculate the spectral phase from Eq. (12). Also, it might be necessary to consider higher order aberrations.

4. Conclusion

In conclusion, an approach for calculation of spatio-temporal properties of ultrashort pulses propagating through aberrated stretchers is presented. In the linear approximation of grating dispersion, simple formulae for GDD, TOD, FOD deviation and residual angular dispersion are obtained and their dependencies on Seidel aberration coefficients is presented. As an example, the impact of aberrations in Martinez/Banks stretchers on the spectral phase is calculated. It is shown that the spectral phase distortion increases with the focal length of the stretcher telescope, whereas the residual angular dispersion depends only on the relative position of the grating within the stretcher.

The approach presented in this article provides a physical insight on the role of aberrations in a pulse stretcher design and may be used for the optimization of the spectral phase of CPA chains without extensive numerical ray-tracing. Even though the theoretical framework of this approach covered only Seidel aberrations, the impact of higher order aberrations can be included and it can also be extended to cases when the propagation direction of the central frequency component is not parallel with the optical axis of the imaging system. In general, it can be used when different frequency components of the beam hit the entrance pupil of the imaging system at different positions as it is also in the case of tilted-pulse-front pumping scheme used for the ultrashort THz pulse generation [25] and spatio-temporal focusing [26].

Funding

Czech Ministry of Education Youth and Sport (“CZ.1.05/1.1.00/02.0061”, “CZ.1.07/2.3.00/20.0091” and ”CZ.02.1.01/0.0/0.0/15_008/0000162”)

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Spherical	Astigmatism	Field curvature	Coma
B	C	D	F
$\frac{{(R - s)}^{2}}{2 R^{3} s^{2}}$	$\frac{1}{{Rs}^{2}}$	$- \frac{1}{2 {Rs}^{2}}$	$- \frac{R - s}{R^{2} s^{2}}$

Seidel aberrations in grating pulse stretchers

Abstract

1. Introduction

2. Theory

2.1. General theory

2.2. Aberration function

2.3. On-Axis object

2.4. Off-axis object

3. Martinez stretcher

3.1. On-axis

3.2. Off-axis

3.3. Discussion

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (26)

Optics Express