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The typical phase correction term introduced in a diffraction grating-pair is rediscussed. It shows that the correction causes a conceptual difficulty in geometrical optics. A study reveals that Fraunhofer diffraction explains the correction and only mean-phase light rays are allowed for diffraction analysis. Besides, an equivalent phase formulation without correction is recommended.
© 2011 Optical Society of America
1. Introduction
In ultrafast optics, angular dispersive elements like diffraction gratings wield a great influence in pulse generation, amplification, shaping and compression, which are frequently applied in the well-known chirped pulse amplification [1, 2], compression of self-phase modulated or supercontinuum-generation pulses in fibers [3, 4, 5, 6], pulse-front matched or group-velocity matched parametric processes [7, 8, 9, 10, 11]. Now it is well understood that the pulse stretch and compression by a grating-pair comes from a strong second order frequency dispersion imposed on incident ultrashort pulses [12, 13, 14, 15], which can be derived by a phase function ϕ(ω), as was first shown by Treacy [12]. To catch a right result, Treacy added a phase correction term R(ω) to an initial eikonal phase, which was claimed as “a consistent definition for the waves in the grating system”. Treacy’s model, for the second and higher order dispersions, has been verified in many experiments through optical pulse stretch and compression [1, 2, 3, 4] and theoretically confirmed by the Fresnel-Kirchhoff integration and the matrix optics [16, 17, 18]. However, Treacy’s interpretation on the correction has not convinced followers due to an inadequacy in his reasoning. The confusion therein is that the original phase term without the correction is seemingly equivalent to that with in the background of geometrical optics, nevertheless, the two terms deliver different dispersive relations. This makes one easy to misuse ϕ(ω) [19, 20]. Although Brorson and Haus tried to understand the group-delay time rendered by Treacy’s model through a general Fermat’s principle [21], their analysis did not find a way to obviate the correction if a phase function has to be introduced. It is quite necessary to clear the confusion owing to the significance of grating devices in fundamental optics and applications. In present paper, we will explore the physics beneath the phase correction and clarify the use of phase terms involving grating-pairs.
2. Treacy’s interpretation
Consider a planar ultrashort optical pulse impinging into a parallel grating-pair. The first grating will cause angular dispersion or pulse-front tilt [22, 23, 24], and the second one counteracts the tilt but creates spatial chirping and frequency dispersion [16]. A simple geometrical scheme is illustrated in Fig. 1(a) where two rays belonging respectively to angular frequency ω and neighboring frequency ω′ are sampled for a comparison. The incident and reflective angles of the ω component on the first grating are γ and γ − θ, respectively. They obey the following grating equation
where d is the grating constant, c is the light speed in vacuum, m is the order of interference. In the following part, we will consider a practical case m = 1. Accordingly, the length p of the ray path PABQ for ω can be expressed as where b is the slant distance between the two gratings. Thus, the phase through the path PABQ is expected to be However in terms of Treacy’s consideration, Eq. (3) should be revised to be where R(ω) is the phase correction, the necessity of which is illustrated in Fig. 1(b) where two rays with the same frequency are displaced transversely. DD′ and EE′ therein represent the equiphase fronts before and after diffraction by the second grating (G2), respectively. Since the path length of DBE does not match that of D′B′E′, a phase compensation is needed. When we slowly move the ray “1” towards the ray “2” without changing its propagation direction, the point B′ will thus move to B along the line BB′ on the surface of G2. If B′ crosses one groove for a distance equal to d, the relative phase change through D′B′E′ is 2π according to Eq. (1). Then, a total phase shift can be predicted between the light paths DBE and D′B′E′. Furthermore, we can fix B′ at the point O which makes AO [see Fig. 1(a)] normal to the surface of G2. With respect to the reference ray [dashed green line in Fig. 1(a)] through O, the phase shift can be expressed as with the definition . Above interpretation is the main idea shown by Treacy. One can refer to the original paper if details are required. The validity of Eqs. (4), (5) can be simply checked as follows. In general, a group-delay time is derived from ∂ϕ/∂ω which can be rewritten as p/u where u is the group velocity. Equation (3) in the configuration γ > θ will give rise to a group velocity greater than c, whereas Eq. (4) yields u = c as is easily accepted since every frequency component of an ultrashort pulse propagates in a constant speed c [25], moreover it is a result of Fermat’s principle [21].
Fig. 1 Grating-pair arrangement for pulse compression (a) and schematic diagram of Treacy’s phase correction (b). Light path calculation starts and ends at the equiphase plane Z = 0 for the incident and output rays where the frequency ω′ is slightly smaller than ω. DD′, EE′, FF′, HH′ represent symmetrically sampled equiphase planes.
However by the geometrical optics, the relative phase of the two rays in Fig. 1(b) only differ in a number of 2π induced by G2 since they start at the same equiphase (DD′) and arrive at the same one (EE′). Therefore, R(ω) can be canceled out because a couple of 2π do not contribute to the dispersion. So one is easy to doubt the validity of R(ω). On the other hand, if we add two other equiphases of FF′ and HH′ in Fig. 1(b) and make the four equiphase fronts to form a centrosymmetric configuration (symmetric point pairs are A/B, E/F, D/H and their primed ones), obviously an opposite phase compensation should be made between the two light paths F′A′H′ and FAH. The path length of F′A′BE is actually equal to that of FAB′E′, as does not require any phase correction for the the two rays! Now it seems that we are trapped in an awkward dilemma, thus R(ω) demands a right physical interpretation.
In the next section, we will try to deal with this issue.
3. Fraunhofer correction
Let’s consider a typical diffraction of a monochromatic plane wave near a grating surface, shown in Fig. 2. According to Huygens-Fresnel principle, the diffracted field is made by wavelets emitted by infinite secondary coherent sources on the air-grating interface, for example, the sampled points R, S, T in the nth groove. The standard knowledge of diffraction gratings tell us that the reflected light field can be written as a single-unit diffraction term U multiplied by an interference factor H, which only allows for diffraction along the γ direction governed by Eq. (1). Thus only the diffracted rays pointing to γ in these secondary coherent sources contribute to U. This permits treating the reflected field as a plane wave. So, the single-unit diffraction term U at the equiphase front which is distant from the grating can be expressed as a Fraunhofer diffraction integral [26]
where [x0, x1] is the range on the equiphase front which just contains all the diffracted rays generated from the nth groove, C, a complex constant, F(x), the pupil function of a diffraction window restricted by a groove, as can also be approximated as a constant F over the integration range, z(x), the path length of a ray from the incident front to a secondary source and to the equiphase front. Equation (6) can be converted to Here, zs is a path length including a special point S on the surface of the groove, which we may call the mean-phase point (it depends on ω). It makes ωzs/c a resultant phase formed by all the diffracted rays on the groove, a result of the mean value theorem of complex integrals. The ray “2” then is a representative light ray on the nth groove. All the mean-phase (S) points on the grating are connected to make an effective grating surface. In a rare case, one groove may have two or more mean-phase points. We can arbitrarily choose one of these point as a representative. Above result implies that only the rays reflected by those representative mean-phase points are legal in geometrical optics.Fig. 2 Principle of the diffraction grating where blazed grooves are simply shown for an analysis of the phase variation of a pencil of diffracted rays in one unit. Phase difference exist between the three sampled ray pairs 1 − 1′, 2 − 2′, 3 − 3′. The incident and diffracted angles are the same as those of the second grating in Fig. 1(a).
In this point of view, the foregoing problem may be attributed to a misuse of light rays between the two gratings. Now, reconsider a ray of ω diffracted by the two gratings, as is shown in Fig. 3. We can set the first diffraction point of an incident ray to be a “S” position. However in general, the second diffraction point of this ray (B) displaces a distance δ ≤ d leftwards from a nearest mean-phase point, for example Sk. Then, PABQ should be replaced by a legal path PA + A′SkQ′ which comparatively has an advanced phase 2πδ/d. Evidently, δ depends on the frequency ω. The total phase from the equiphase front P to that of Q then becomes
Note that we need not to record an absolute phase but its parts depending on the frequency ω. Let Sn be a mean-phase location inside a groove which includes O, where AO is the same line as that in Fig. 1(a). Thus is a number of 2π and is a constant which is independent of ω. Therefore, with is equivalent to −2πδ/d, so Eq. (8) actually is the same as Eq. (4). We may call the phase correction R(ω) or −2πδ/d, referring to the analysis above, Fraunhofer correction. The correction has to be taken into account to regulate a ray tracing procedure in grating systems, if dispersion works. In terms of Fraunhofer correction, the confusion in Treacy’s model disappears at last.Fig. 3 Schematic show of a diffracted ray between two parallel gratings in which mean-phase points (denoted by a subscript S ) are illustrated as a reference for phase correction. Here, an illegal light path PABQ should be replaced by a correct representative path PA + A′SkQ′ where A locates at a mean-phase point.
Actually, there is a way to circumvent the correction issue. If we parallel shift the ray of ω to make its diffraction position on G2 in coincidence with that of a reference frequency ω0, then Fraunhofer correction will not appear. The method is shown in Fig. 4. We just make that the light path PACR of ω is substituted with the path PA + A′BQ. Then the phase becomes
Since A and B are two fixed points formed by the reference light ray, Eq. (9) does not depend on the validity of the light path AB (It will differ in a constant phase when a representative path is chosen). If we move the reference point O in Fig. 3 to B in Fig. 4, Eq. (9) will turn into Treacy’s expression, i.e. Eq. (4). We hereby recommend Eq. (9) in applications for which one needs not trying to understand Fraunhofer correction. Besides, Eq. (9) has a distinct advantage for phase estimation in a system with multiple grating-pairs. Certainly if one starts a work from the group-delay time ∂ϕ/∂ω = p/c, the correction does not have to be considered too [25].Fig. 4 Light paths of two frequencies inside a grating-pair similar to those in Fig. 1(a), where δθ is the angle between the diffracted directions of a ray ω and a referred ray ω0.
4. Conclusion
Treacy’s model for a grating-pair pulse compressor is reconsidered and one conceptual confusion in phase correction is analyzed. Analysis shows that the original interpretation can not account for the phase correction, which however, is due to Fraunhofer diffraction. It follows that only those light rays reflected at mean-phase positions on the grating surface stand to reason in optics for a coherent beam containing many frequencies. Nevertheless, the correction is negligible for a monochromatic wave since it is equivalent to a constant phase shift.
Acknowledgments
This work was supported by the tiptop-talent fund and science foundation for youth scholars (No. 2009YF032) from Harbin University of Science and Technology in China.
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