Enhanced dispersion compensation capability of angular elements based on beam expansion

Rui Du; Runhua Jiang; Ling Fu

doi:10.1364/OE.17.016415

1. Introduction

Group delay dispersion (GDD) compensation plays an important role in ultrashort pulse generation [1–3], chirped pulse amplification [4,5], laser micromaching [6], multiphoton imaging [7–15], etc. The conventional dispersion compensation methods employs four prisms [16] or four gratings [17] to provide negative GDD to compensate for positive GDD induced by the propagation of ultrashort pulses through dispersive material, such as fibers and acousto-optical deflectors (AODs) [18]. In some systems [19,20], only two angular elements are used to compensate for positive GDD. This configuration has the advantage of higher transmission efficiency and stability compared with the four angular elements configuration, because there is inevitably energy loss when a laser passes through the angular elements. However, GDD provided by prism pair is not always increased linearly with dispersion compensation distance and will be frozen to its maximal value [21,22]. Thus, in the case that the absolute value of maximal GDD provided by prism pair is smaller than that of positive GDD, one cannot get enough negative GDD to compensate for the positive GDD. Considering the advantage of higher transmission efficiency and stability with the prism pair configuration, it is preferable to enhance the GDD of prism pair.

As is known to all, the amount of negative GDD can be controlled by adjusting the angular elements’ angular dispersion or changing the dispersion compensation distance. Actually, the beam size may also have effect on the GDD. Martinez has discussed the effect of beam size on the spectral lateral walk-off and found that larger beam size can weaken the spectral lateral walk-off and thus improve the compression ratio (see point 4 in his conclusion) [23]. However, the effect of beam size on GDD has not been studied due to the well-collimation assumption applied in Martinez’s work [23].

In this paper, we have explored the beam size’s effect on the GDD and found that a modest 2 × and 4 × beam expansion can improve the dispersion compensation capability of angular elements: decreasing the available minimal pulse width and shortening the dispersion compensation distance. This will make the pulse compressor more compact, stable and of higher transmission efficiency, which are particularly important in multiphoton microscopic imaging and micromachining, where high photon densities and high stability is required.

2. Theoretical analysis

We first take prism pair as an example to describe the effects of beam sizes on the GDD compensation. As shown in Fig. 1 , the original Gaussian beam is expanded by the beam expander consisting of lenses F₁ and F₂. BW₁ and BW₂ are the beam waists of the original beam and the expanded beam, respectively. The position of BW₂ is in the vicinity of the back focal plane of the lens F₂ and the diameter of BW₂ is:

w_{2} = M w_{1} \sqrt{1 + {(\frac{L λ}{π w_{1}^{2}})}^{2}},

where w ₁ is the diameter of BW₁ and λ is the wavelength.

Fig. 1 Schematic diagram of the prism pair with the configuration of the expanded beam for GDD compensation. The lenses F₁ and F₂, having the focal lengths of f ₁ and f ₂ respectively, constitute a beam expander with the magnification coefficient M = f ₂ / f ₁. L is the distance between BW₁ and F₁ .d is the distance between BW₂ and the first prism. z is the inter-prism distance.

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From Eq. (5) in Ref. 21, the GDD for prism pair can be analytical expressed as [21]

G D D_{a} = - k β^{2} z \frac{(d + α^{2} z) d + {(π w_{2}^{2} / λ)}^{2}}{{(d + α^{2} z)}^{2} + {(π w_{2}^{2} / λ)}^{2}},

where k is the wave number. The parameters α and β describe the dispersive characters of the prism and can be derived from geometrical optics [23] (See more details in the appendix). It is noted that the negative GDD caused by the angular dispersion is related to the beam size and the change of beam size by the beam expander can affect the GDD_a.

Based on Eqs. (1) and (2), the GDD provided by the prism pair as a function of z in case of different magnification coefficients can be analyzed. For simplicity, the ratio of the GDD_a after the beam expansion to the GDD_a with the original beam size is used to analyze how the beam size affects the negative GDD caused by angular elements. According actual situation of our experiments, the following parameters are used in the calculation: w ₁ = 0.5 mm, λ = 800 nm; The incident angle upon the prism is 49°, the top angle of the prism is 60°, the refraction index of SF10 prism n = 1.7112 and dn /dλ = 4.98 × 10⁻⁵ nm⁻¹; In the case of expanded beams, L = 1.745 m and d = 2.1 m; The distance between the beam waist and the prism is 2.1 m while using the original beam (The lenses F₁ and F₂ are moved out.).

As shown in Fig. 2 and its inset, -GDD_a for 2 × and 4 × expanded beams are enhanced in comparison with the case of the original beam, implying that angular elements can provide more negative GDD at the same z after beam expansions. This is due to the fact that -GDD_a is multiplied by two parts, in which one is kβ ² z and the other is $η = \frac{(d + α^{2} z) d + {(π w_{2}^{2} / λ)}^{2}}{{(d + α^{2} z)}^{2} + {(π w_{2}^{2} / λ)}^{2}}$ . At the same z, the value of η increases as the beam size becomes larger. For example, at z = 2 m, the values of η are 0.22, 0.77 and 0.98 for original, 2 × and 4 × expanded beams, respectively, while the values of kβ ² z are the same for different beam sizes. As a result, the larger beam size leads to the increase of the -GDD_a amount and the enhanced compensation capability of angular elements. It is noted that under the plane wave approximation, by setting w ₂ → ∞ in Eq. (2), the negative GDD caused by angular elements reduces to -kβ ² z, which is consistent with the demonstrated Eq. (11) in Ref. 16.

Fig. 2 Enhancement of GDD as a function of z with different beam sizes. Enhancement of GDD is defined as the ratio of the GDD_a after the beam expansion to the GDD_a with the original beam size. The inset shows the actual GDD_a with different beam sizes.

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It is also observed from Fig. 2 that the ratio of -GDD_a for 2 × and 4 × expanded beams monotonically increased with z before they reach their maximal values, which are 4 and 16 times higher compared that with the original beam, respectively. In terms of quantity, the maximal GDD _a provided by prism pair with original, 2 × and 4 × expanded beam can be calculated with Eqs. (1) and (2), which is −10000 fs², −40000 fs² and −160000 fs², respectively. To illustrate this, analysis of Eq. (2) is necessary. The part -kβ ² z, which describes the negative GDD caused by angular elements under the plane wave approximation [16,17], increases linearly with z, while η monotonically decreases as a function of z from its maximal value 1 at z = 0. Therefore, the multiplication of the increasing function kβ ² z and decreasing function η constitutes the formula that has a peak value. It is noted that the decrease of η with z turns slower for the larger beam size, resulting in the higher maximal value of -GDD_a compared with the lower magnification coefficient. As shown in the inset of Fig. 2, the maximal negative GDD provided by the prism is approximately −10000 fs² and the increasing of inter-prism distance cannot provide more negative GDD. This is due to the fact that η tends to zero as z increases [21,22].

3. Experimental results

To verify the effectiveness of beam expansion on dispersion compensation capability described by Eq. (2), pulse widths are measured for different beam sizes when a prism pair with various dispersion compensation distances is used to compensate for the positive GDD of the Tellurium oxide (TeO₂) material with total thickness 36 mm. The experimental setup is the same as Fig. 1 except that the TeO₂ material is inserted between BW₁ and F₁. The femtosecond laser generated from the Ti: Sapphire laser (Mai Tai, Spectra Physics) at wavelength λ = 800 nm has a pulse width of τ ₀ = 82 fs, radius of original beam waist of w ₁ = 0.5 mm, and a repetition rate of 80 MHz. The TeO₂ material introduces the total positive GDD of approximately 16000 fs². The beam is expanded by the combination of two lenses. For the 2 × beam expansion, the expander is composed of an f ₁ = 38.1 mm and an f ₂ = 75 mm lenses; An f ₁ = 38.1 mm and an f ₂ = 150 mm lenses are used for the 4 × expansion. The material dispersion of the beam expander is about 1500 fs². When using expanded beams, L is 1.745 m and d is 0.22 m. The distance between BW₁ and the prism is 2.1 m for the original beam. The expanded beam passed through the prism pair with an incident angle of 49°. As mentioned in above paragraph, the refraction index of SF10 prism n = 1.7112 and dn /dλ = 4.98 × 10⁻⁵ nm⁻¹. The pulse widths of the emerged beams after the prisms are measured with the autocorrelator based on two-photon absorption (Carpe, APE, Germany).

Under the above experimental conditions, measured and calculated pulse widths as a function of path length between the prism pair for different beam sizes have been depicted in Fig. 3 . The theoretical curves of pulse width evolution are achieved by substituting GDD of Eq. (2) into Eq. (9) in Ref. 22. As shown in Fig. 3, the experimental results have good coincidence with the theory data. In the case of the original beam without expanding, it is found that the minimal pulse obtained is 203 fs, with the inter-prism distance of 1.5 m, and the increase of the inter-prism distance cannot lead to the shorter pulse width. This is due to the fact that the maximal available negative GDD for the original beam size is approximately −10000 fs² (see the blue line in the inset of Fig. 2), which is insufficient to completely compensate the TeO₂-induced positive GDD of 16000 fs². It is also noted that the negative GDD cannot be increased by increasing the inter-prism distance due the fact that that the η turns to zero as z increase [21,22]. In contrast, for the 2 × and 4 × expanded beams, the minimum pulse widths are 140 fs and 106 fs, with the shorter inter-prism distance of 0.7 m and 0.8 m, respectively. Additionally, as the inter-prism distance increases, the pulse widths are further broadened after reaching their minima. This fact suggests that the positive GDD induced by TeO₂ can be completely compensated by using the expanded beams. Actually, the calculated GDD _a is −16030 fs² at 0.7m for the 2 × expanded beam and −18020 fs² at 0.8 m for the 4 × expanded beam, resulting in the sufficient compensation for the positive GDD induced by TeO₂. Since the positive GDD of the prism’s material in the 4 × case is bigger than that of the 2 × case, (larger beam size experiences longer path length in the prism) its corresponding inter-prism distance for the minimal pulse width is slightly longer. Therefore, it is demonstrated that beam expansion can improve the dispersion compensation capability of the prism pair: decreasing the achievable minimum pulse width and shortening the inter-prism distance to obtain the minimal pulse width.

Fig. 3 Measured and calculated pulse widths as a function of z when to compensate for the positive GDD introduced by the Tellurium oxide (TeO₂) with different beam sizes.

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This idea also works for other angular devices, and is demonstrated with an AOD pair, which is actually a dynamic grating pair [19,20]. The experimental condition is identical as above, except that the two prisms are substituted by an AOD pair activated at 96 MHz with counter-propagating acoustic waves. As shown in Fig. 4 , both measured and calculated pulse widths exhibit the similar behavior as that in Fig. 3. The minimal pulse widths for 2 × and 4 × expanded beams are 109 fs and 96 fs respectively, which are shorter than the minimal pulse width of 120 fs achieved by using the original beam. The inter-AOD distance for the expanded beams to reach their minimal pulse widths is 0.8 m, shorter than that for the original beam. As a result, the expanded beam can also improve the dispersion capability of gratings.

Fig. 4 Measured and calculated pulse widths as a function of the distance between the acousto-optic deflector pair with different beam sizes.

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4. Discussion

We have shown that the beam expansion can enhance dispersion capability of angular elements. It is noted that, the pulse width of the expanded beam (140 fs and 106 fs) shown in Fig. 3 cannot be recovered to its original value (82 fs). It results from the spectral lateral walk-off of the emerged beam from the prisms [23-25]. This is common in such systems using a prism pair, and can be fixed with four prisms configuration. Thus, the configuration of a prism pair becomes impractical in the case that the original transform-limited pulse width is needed. However, in various applications, such as in multiphoton imaging, the original transform-limited pulse width is not strictly demanded. Thus, the prism pair with the beam expansion can be applied in many practical systems.

Beam expander and dispersion compensation system are routinely used in multiphoton microscopy and micromachining systems. This paper helps to clarify the role of beam expander in the dispersion compensation system. The beam expander is not only to manipulate beam size to fit optical aperture, to enlarge the effective numerical aperture and obtain better spatial resolution, but also to affect dispersion capability of angular elements if placed before the two-element pulse compressor. Thus, the expander’s magnification coefficient and its position in optical path should be deliberately chosen according to practical experimental conditions.

Beam size affects both the spectral lateral walk-off and the GDD, which are two factors determine the compression ratio. However, in the case of using the prism pair compressor with expanded beam, which factor contributes more to the improvement of compression ratio is unknown. To state the question, the compression ratio in the four cases of t₀ = t (U ₄, GDD_a ₄), t₁ = t (U ₀, GDD_a ₄), t₂ = t (U ₄, GDD_a ₀), and t₃ = t (U ₀, GDD_a ₀) are depicted in Fig. 5 (parameters is the same as in Fig. 2, material dispersion GDD _m = 19000 fs²). t (U, GDD_a) is the compression ratio as a function of the spectral lateral walk-off U and the GDD of angular elements GDD_a (See the Eq. (9) in Ref. 22). U ₀ and GDD_a ₀ are the spectral lateral walk-off and GDD of prism pair with unexpanded beam, respectively. U ₄ and GDD_a ₄ are the spectral lateral walk-off and GDD of prism pair with 4 × expanded beam, respectively. t₀ and t₃ is the compression ratio with the 4 × expanded beam and unexpanded beam, respectively.

Fig. 5 Compression ratio as a function of inter-prism distance with different U and GDD_a parameters.

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To separate the effect of spectral lateral walk-off and negative GDD on compression ratio, t₁ is obtained only by substituting GDD_a ₄ with GDD_a ₀ in t₃, and t₂ is acquired only by substituting U ₄ with U ₀ in t₃. As shown in Fig. 5, t₁ indicates the improvement of compression ratio, which is similar as t₀, while t₂ still suffers larger compression ratio. Please note that the significant improvement of compensation ratio can be achieved, as indicated by t₁, only by the enhancement of negative GDD (-GDD_a ₄ is bigger than-GDD_a ₀ as shown in Fig. 2). As a result, the main cause for compression ratio improvement with expanded beam is the enhancement of negative GDD. The negligible difference between t₀ and t₁ is due to the spectral lateral walk-off. When the GDD is zero, due to the weakness of spectral lateral walk-off [23], larger beam size can slightly decrease the minimal pulse width, as have been shown in both Fig. 3 and Fig. 4. Therefore, the weakness of spectral lateral walk-off is the minor cause for the compression ratio improvement.

5. Conclusion

The mechanism of how beam size affects the GDD and the dispersion compensation has been analyzed theoretically and experimentally. It is found that beam expansion could improve the dispersion compensation capability of the prism and AOD pair. These results suggest that the pair configuration pulse compressor with large beam size is able to make the compressor more compact and stable, and improve the power transmission efficiency by using less number of devices. These results are very useful for designing practical femtosecond pulse laser application systems such as multiphoton microscopy and micromachining systems.

Appendix

The calculation of α and β for prism.

The deflecting angle θ emerging from angular element for a femtosecond laser is depend on the incident angle γ and the spectral component ω . The difference between the angle of a spectral component and the angle of the central spectral component can be expressed as [23]

$Δ θ = α Δ γ + β Δ ω,$ (A1)

where α = dθ / dγ and β = dθ / dω.

For a prism, the emerging angle related to incident angle γ, apex angle ϕ and refractive index of material n is

$\sin θ = \sin φ \sqrt{n^{2} - \sin^{2} γ} - \cos φ \sin γ .$ (A2)

By taking the first derivative of Eq. (2) with respect to γ and ω, respectively, the coefficient α and β can be expressed as

$α = (\frac{\sin φ \sin γ \cos γ}{\sqrt{n^{2} - \sin^{2} γ}} + \cos φ \cos γ) / \cos θ,$ (A3)

$β = - \frac{n \sin φ}{\sqrt{1 - n^{2} \sin^{2} φ - \sin^{2} γ \cos 2 φ + \sin 2 φ \sin γ \sqrt{n^{2} - \sin^{2} γ}} \sqrt{n^{2} - \sin^{2} γ}} \frac{λ^{2}}{2 π c} \frac{d n}{d λ} .$

(A4)

In the case of Brewster angle incidence, i.e., n = tg (γ), $α = 1, β = - \frac{λ^{2}}{π c} \frac{d n}{d λ}$ , which is consistent with equations (13) and (14) in Ref. 23.

Acknowledgments

The authors thank the support by the National Natural Science Foundation of China (No.60708025, 60828009) and the National High-Tech Research and Development Program of China (No.2006AA020801).

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Enhanced dispersion compensation capability of angular elements based on beam expansion

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental results

4. Discussion

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (2)

Optics Express