1. Introduction

The invention of the grating compressor [1], the prism pair, and the grating stretcher [2] enabled the rapid growth in chirped pulse amplification and ultrafast optics that followed. In early systems, dispersive material (either bulk glass or optical fiber) was used to stretch the pulse, which would be compressed by a grating pair after amplification. With long pulse durations (>100 fs), higher-order spectral phase can often be neglected, but with wide gain bandwidth materials such as Ti:sapphire, compensation of higher-order spectral phase is necessary to obtain the minimum pulse duration. As Kane and Squier [3] have observed, most optical materials possess a ratio of third- to second-order phase (ϕ3/ϕ2) that is around +0.7fs. However, both prism pairs and grating pairs have the wrong magnitude (prisms) or sign (gratings). In the now-conventional approach to chirped-pulse amplification, gratings are used in both the stretcher and compressor, and adjustment of the the incident angle and grating separation allows simultaneous correction for ϕ₂ and ϕ₃, including the optical material in the system. The simplicity of stretching or compressing with material remains attractive for a compact, efficient, and easy-to-use system.

Grism pairs offer a solution to this design problem. A grism is a combination of a prism and a diffraction grating. The grating can be either mounted on one facet (reflection or transmission) or inside the bulk (transmission only). In addition to dispersion compensation, grisms have been used to produce constant-dispersion in wavenumber [4] for interferometric spectroscopy. In this paper, we are concerned with the use of grisms in pairs for dispersion compensation and control. The use of a grism pair was first proposed for dispersion control by Tournois [5] to give second order dispersion without third-order. In a Tournois grism the light enters at normal incidence into a prism, then passes through a tilted transmission grating (see Fig. 4(a) below). As we will see below, the dispersion of a grism pair is primarily modified by the fact that one or both of the incident and diffracted sides of the grating are in a higher refractive index. Subsequently, Kane and Squier [3] [6] showed that a grism pair compressor can simultaneously compensate for both ϕ₂ and ϕ₃. This made it possible to directly compensate large amounts of material phase, as might be found in a fiber stretcher. However, widespread use of grisms in ultrafast optics was inhibited until recently by the lack of high efficiency, wide-bandwidth reflection grisms. Now that these are available [7] [8], they have been used to compress pulses to 36fs in a down-chirped pulse amplifier [9]. Here the pulse is amplified with a negative stretch from the grism pair, then compressed with material to obtain low compression loss. Wise et al has used them for compression of pulses from a Yb fiber laser [10]. Grisms have also been used as the dispersive element of a pulse shaper [11], allowing the shaper to passively compensate for some second and third-order phase.

Up to the present, the primary means for calculating the dispersion of grism pairs is by computer raytrace. Analytic calculation of the spectral phase of Tournois grisms is straighforward, since there is no refraction at the prism facet. The more general case, where there is refraction at multiple interfaces, is more complicated. This article addresses an even more general problem of analytically calculating the spectral phase in structures with refractive and diffractive interfaces. We propose a modular method that can be understood intuitively and can be applied without raytracing. The method takes advantage of the fact that prism, grating and grism structures are made up of pairs of interfaces, nested so that the outgoing spatially-dispersed rays are all parallel to the input. In our construction, each pair contributes a spectral phase to the system. Our calculations assume that the system is well-aligned - other techniques (such as raytracing) can be used to assess the effects of misalignment and angular spatial chirp.

Our article is organized as follows: In Section 2, we present our method for calculating spectral phase, showing that it yields the expected results for grating and prism pairs. In Section 3, we apply the method to grism pairs. Section 4 addresses the calculation of higher order phase, with an analysis of the spectral phase for an example system, where we show that the method agrees with a raytrace and we compare the exact calculation with an approximate form. We also discuss the residual fourth-order phase of a grism compressor.

2.1. Phase method vs. group delay method

In the literature there have been two approaches to the calculation of the dispersion of optical systems. The first is based on a direct calculation of the spectral phase (for example, see [2] [12]),

The right hand side shows explicitly the refractive index and angular contributions to the dispersion. The second approach is through a calculation of the group delay ϕ ₁ (for example, see refs [1] and [5]),

Here the path length L_j of each ray segment is divided by the group velocity through that medium, defined through

The ray path must be calculated through to a plane normal to the output rays of the structure. We will use the phase method for our analytical calculations and the group-delay method for numerical calculations of dispersion from a raytrace. By direct calculation of the group delay from the expression derived with the phase method, it can be shown that the two methods give the same result for a monolithic grating compressor (see Section 2.2 below).

2.2. Spectral phase of tilted window and monolithic grating compressor

With this perspective, we may now outline our construction for the calculation of the dispersion of an optical structure consisting of refractive and diffractive planar components. We first calculate the spectral phase (Eq. 1) of a tilted plane parallel window. This expression will be shown to be valid even when there is diffraction at the interfaces. Suppose the tilted window is inserted into an optical system (for example, between a pair of gratings). We can calculate the change in the spectral phase of an optical system by adding the spectral phase of the window and subtracting the phase of the space removed from the system when the window component is introduced. More complicated structures can then be thought of as superpositions of various tilted window elements.

The starting point for our method is the expression for the spectral phase change ϕ_w (ω) caused by the introduction of a tilted window of transparent material of refractive index n 2 into a medium of index n1. Referring to the schematic in Fig. 1(a), we choose the vector r to be along the z-direction, normal to the planar surfaces, we can directly write the spectral phase using Eq. 1:

Here, k ₀=ω/c, L_w is the window thickness and θ₁ and θ₂ are the angles of incidence and refraction into the window, respectively. The second term in parentheses accounts for the removal of the material of index n1.

 

Fig. 1. Schematics showing geometry for (a) a tilted window and (b) a tilted transmission grating pair.

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As there is no explicit form for the method of directional change at the interfaces, this expression can be applied directly to obtain the spectral phase of a pair of parallel gratings. Using the grating equation

to calculate the diffracted angle θ₂, Eq. 4 gives the spectral phase of a grating compressor. In the limit where n ₁=n ₂=1 (non-dispersive vacuum), m=-1, and θ ₁ is independent of ω, we can safely drop the second term in Eq. 4 since it has no effect on the dispersion of the group delay. In this case, the spectral phase reduces to the conventional Treacy form [1]:

The more general expression (Eq. 4) correctly accounts for the dispersion of the media on both sides of the gratings. It can be shown by direct calculation of the group delay for this monolithic grating compressor starting from Eq. 4, that the result is equivalent to that calculated by group-velocity method used in raytracing.

2.3. Simple applications of the tilted window as the basic phase element

Equation 4 can be used as a building block for more complicated systems. The method can be illustrated, by considering the case where a tilted window (thickness L_w) is introduced between a pair of air-spaced parallel gratings (considered by Braun et al [13]). In this case, the combined phase can be written as

If the window is tilted with an angle α to the grating normal, then the angle of incidence to the window is θ ₃=α+θ ₂. The variable angle of incidence onto the tilted window affects not only the calculation of the internal angle θ ₄ (through Snell’s Law), but also in the air-space that is subtracted from the overall phase. This latter correction was omitted from the expression in Ref. [13].

 

Fig. 2. Construction for calculating the phase of a prism pair. A tilted glass slab of thickness L ₁ forms the outer boundaries of the prism pair. A slab of air, with thickness L ₂, placed at an angle α to the glass slab defines the inner boundaries of the glass prisms. A ray propagates at angles θ ₁-θ ₄ through the first prism as shown. The line joining the prism tips has a length L_p and an angle θ_ref with respect to the normal to the prism exit face. β is the angle between the tip-to-tip line and the wavelength-dependent ray direction at the first prism exit.

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Another important example of the method is the calculation of the phase of a prism pair. Consider the construction in Fig. 2. A prism pair can be thought of as a thick tilted glass slab with a second tilted slab of air removed, leaving the two wedges of glass for the prisms. The spectral phase is then

To put this into a more useful and conventional form, we represent the distances L ₁ and L ₂ in terms of the tip-to-tip prism separation L_p:L ₁=L_pcos(θ_ref-α) and L ₂=L_p cosθ_ref. We make use of the geometric relation θ ₃=α-θ ₂ and define β=θ_ref-θ ₄, which represents the frequency-dependent deviation of the refracted rays from the tip-to-tip reference line. In Appendix 1 it is shown that the prism phase can then be expressed as

where for later use we have allowed for an ω-independent phase term, ϕ₀, that is zero in this case. The first term in the square brackets corresponds to the conventional form derived by Fork [14], allowing for the prisms to be embedded in a material of index n ₁. As with Eq. 4 there is a second term subtracted from the first, that accounts for an effective optical path (L_pcos(θ ₁+θ_ref-α)) that is removed from the system by the insertion of the prism pair. This term gives only a constant group delay shift if θ ₁ is frequency-independent.

There are several things to note about this calculation that apply equally to the calculation for the grism pairs that follows. First, while prism pairs are typically used at the incidence angle that results in minimum deviation, Eq. 9 is valid for any angle of incidence. In fact, any path through prism glass is directly accounted by Eq. 9. This is verified by taking the limit where the prism faces are touching, such that θ_ref=π/2 and L ₁=L_p sinα. In this case Eq. 9 correctly reduces to that of a tilted window (Eq. 4). A change of the insertion of one prism or the other into the beam changes θ_ref and L_p, while movement of both prisms in the same direction has no effect on the phase. The entire beam experiences the same spectral phase shift; extra insertion of the first prism to accommodate the beam size must be accounted by correct evalation of θ_ref. A second issue is the experimental determination of θ _ref. One method (shown schematically in Fig. 3) is to position each prism at minimum deviation and inserted so that its tip allows some portion of the beam to pass. Observation of the log of the transmitted (double-pass) spectrum should show a hard cutoff wavelength (λ _ref). This wavelength corresponds to a ray that passes through both prism tips, allowing the calculation of θ_ref. From that position, changes to θ_ref can be calculated from measurements of displacements of the prisms from that calibrated position.

 

Fig. 3. Schematic of a method to determine the value of θ_ref. The first prism is pulled out of the beam to place an edge of the refracted beam. The second prism is also pulled out to pass a portion of the beam. A plot of the log of the transmitted spectrum shows a hard cutoff at the wavelength that passes from tip to tip.

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3. Spectral phase of grism pairs

The most common form of a grism can be described as a transmission grating embedded in a prism whose faces are at apex angles α ₁ and α ₂ with respect to the grating surface. We will discuss below how reflection grisms can be represented by equivalent transmission grisms. The calculation of spectral phase is particularly straightforward for two limiting cases (applicable only to transmission grisms): where the transmission gratings are mounted on the outside surfaces (α ₁=0) or the inside surfaces (α ₂=0) of the prism pair. (A special case of the latter is the Tournois grism pair, shown in Fig. 4(a).) In these cases, the module construction is identical to that of a standard prism pair, described above, except that the grating equation (Eq. 5) is used to calculate the ray angles at the grating surface. With some algebra, it can be shown that the form of the grating phase for these cases is identical to that of the prism pair (Eq. 9), with the additional constant grating phase of ϕ ₀=-2πmL_psin(α-θ_ref)/d (for α ₁=0) or ϕ ₀=-2πmL_psin(θ_ref)/d (for α ₂=0). In the typical case where the incident angle is independent of frequency (θ ₁ constant), the frequency-dependent spectral phase reduces to a form that is formally the same as for a prism pair:

This grism pair actually has the same form of spectral phase as the prism pair except that the angular dependence must include the diffraction at the grating. This result can also be seen in the context of the method of Fork [14], in which phase continuity of the plane waves is applied at the tips of the prisms: what is important is the angular change at the prisms, not the mechanism (refraction or diffraction) for the change.

 

Fig. 4. a) A Tournois grism pair (no refraction at prism entrance) b) A reflection grism pair (dark). The lower grism is unfolded, showing the equivalent transmission grism (α ₁=α ₂) c) Modular construction to calculate the phase of the grism pair shown in panel (b). A pair of transmission gratings is at an angle α ₁ to a tilted glass slab. A slab of air, with thickness L_air, is placed at an angle α ₂ to the glass slab, to define the inner boundaries of the glass prisms. d) A reflection grism (and the equivalent transmission grism) where the beams pass through different prism faces.

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3.1. The Tournois grism limit

While the reasoning behind Eq. 10 is straightforward, it leads to an apparently different conclusion from what is expected for the still more restricted limiting case of the Tournois grisms (Fig. 4(a), θ ₁=0,α ₂=0), where there is no refraction at the prism entrance. With no refraction at the entrance face, one expects that the spectral phase would be the same for a grating compressor with the addition of spectral phase from a constant thickness of prism glass(see Fig. 2(b), with n ₁=n_glass and n ₂≈1). In fact we can show that Eq. 9 reduces to this form. Working in the limit where θ ₁=0, θ ₃=θ_i and θ ₄=θ_d, we expand the first cosine term with β=θ_ref-θ_d. We then use the grating equation to represent the grating phase term in terms of the incident and diffracted angles and selectively represent the prism tip separation L _p in terms of the grating separation measured along the normal, L_g=L_p cosθ_ref. After reducing the equation, we find that the spectral phase is

After expanding the cosine of the last term, we find

Since the incident angle on the grating is α, the first two terms on the right-hand side are simply the phase of the grating compressor with the incident surface in glass (see Eq 4). The last term represents the contribution from the material, with an effective glass slab thickness of L_mat=L_psinθ_ref sinα, as shown in Fig. 4(a). In the limit where the grating separation (L_g) goes to zero (by translating the grisms along the grating normal), θ_ref goes to π/2, and the remaining phase is just k ₀(n-1)L_p sinα.

3.2. Grisms with refractive surfaces

The application of the modular approach to calculating the dispersion of reflection grism pairs is easier if we unfold the grism around the grating plane (see Fig. 4(b)) to represent the grating in transmission. If the entrance and exit rays pass through the same facet of the reflection grism, the equivalent transmission grism is an isoceles prism with a grating along the normal to the base, so that α ₁=α ₂. The case of a reflection grism where the diffracted beam exits a different facet from the input is discussed below. We will treat the reflection grism geometry more generally by considering a situation where the grating is embedded in a glass prism, with non-zero α ₁ and α ₂.

We can use the modular approach by considering a transmission grating pair embedded at an angle α ₁ inside a tilted glass slab (Fig. 4(c)). This object could be thought of as a grism pair with gratings mounted on the internal faces of the prisms, but with glass in the region separating the two grisms. The spectral phase for this object would thus be given by Eq. 9, where we use the constant grating phase for the case where α ₂=0. Next we introduce a tilted air gap of thickness L_air between these grisms at an angle α ₂ to the gratings. With some work to reduce the equation, the spectral phase again reduces to the form shown in Eq. 9, with α=α ₁ +α ₂ and a different constant grating phase of ϕ ₀=-2πmL_p sin(α ₂-θ_ref)/d.

Not all grism pairs can be represented in the simple prism form. Consider the case where the beam in a reflection grism exits from a different facet, as shown in Fig. 4(d). When the prism is unfolded around the grating, it is seen that the second prism apex angle α ₂ is negative with respect to the grating line. The approach to constructing the expression for the spectral phase is essentially the same as in the previous paragraph, but the tilted air gap is rotated such that α ₂<0. Since there all paths pass through glass, and the spectral phase is no longer a function of a single length.

4. Dispersion analysis of grism pairs

4.1. Calculation and estimation of spectral phase orders

Armed with an analytic expression for the spectral phase of a given structure, we can calculate the terms of a Taylor expansion of the phase around the central frequency ω ₀:

Here Δω=ω-ω ₀ and ϕ _n=∂ⁿ_ωϕ (ω)|_ω0. For systems such as the double-facet grisms that are more complicated than the simpler form shown in Eq. 10, evaluation of these derivatives to high order is cumbersome to do manually. They can be performed easily without approximation by computer using programs such as Mathematica [15] that can calculate analytic derivatives. An alternative is to use our modular method as described above to represent the system phase in terms of the form ϕ_m cosθ, where ϕ_m(ω)=L_mωn(ω)/c is the material phase for that medium and the frequency-dependence of θ varies according to the angular dispersion within that unit. The derivatives can then be evaluated for this general form:

Here the primes denote derivatives with respect to ω. The higher-order derivatives may be evaluated in a similar fashion.

Note that in this formulation the cosθ and sinθ terms are generally not small and cannot always be expanded for small θ. However, we saw above that many grism constructions of interest have the form of Eq. 10, where β is typically small, and ϕ_m=ω/c. In this case the second- and third-order phase have the approximate forms:

We can use this approximate form to see how a grism pair allows for tunability of ϕ ₃. In order for ϕ ₃ to be able to change sign, the angular dispersion must satisfy the condition ωθ″=-θ′. This is impossible for a standard grating pair, but it is possible with immersion of the grating in a higher index. The approximate form above can be used as a check on a given design to determine the approximate incident angle for a grism system.

4.2. High-order phase analysis of an example system

To illustrate these calculations, we will use as an example a grism pair that has been used in some of our experiments [9] [11]. A reflection grating (groove density 1480 lines/mm) is mounted on one leg of a 45° right angle prism made of BK7 glass. The incident angle on the grating θ ₁ is related to the grism incident angle θ_i (see Fig. 2) through Snell’s law: sinθ ₁=nsin (α ₁+θ_i). The diffracted angle θ_d is calculated through the grating equation (5), with m=-1 and n ₁=n ₂=n. Finally, the grism exit angle θ ₄ is sinθ ₄=nsin (α ₂+θ_d). (The sign convention for the grating equation is for θ_i>0: if, as θ ₁ is varied, θ_i<0, the sign of the grating order must be changed to +1.)

Figure 5 shows the ratio ϕ ₃/ϕ ₂ as a function of the incident angle, calculated using different methods. The solid lines show the analytic calculations based on symbolic derivatives of Eq. 10, for several values of the reference angle (θ_ref), larger values of which correspond to greater prism insertion. A sampling of values calculated by raytrace are shown as dots: these lie exactly on the analytic curves, confirming the calculations. The dashed line results from the approximation based on Eq. 15. Note that this approximation includes only angular dispersion and does not include effects of prism insertion. The approximate curve lies close to the reference angle θ_ref=50°, which corresponds to the insertion that places the central wavelength (800nm) near the tips of both grisms. Although the approximation neglects the larger insertion required to accept the full bandwidth and beam size, it gives a result that is quite close to the exact formulation. (Estimating the extra insertion of the second prism to accommodate a finite beam as the beam width D, the reference angle will have to be increased from θ_ref to θ′_ref according to sinθ′_ref=sinθ_ref +D/L_p.)

Armed with a means to calculate the spectral phase, we can perform a more extensive exploration of the design phase space than would be practical with a raytrace. Figure 6 shows a plot that illustrates the performance of the grism pair of Fig. 5. Again, we plot the ratio ϕ ₃/ϕ ₂, this time against reference angle, for several values of the incident angle. This graph shows how ϕ ₃/ϕ ₂ is most easily tuned in the lab: an incident angle is selected, then the insertion of the grism is adjusted for the desired value of the ratio of ϕ ₃/ϕ ₂. The horizontal gray line is at the ratio required to compensate bulk BK7 crown glass (a wide range of glasses have the same ratio). Given this target and an incident angle, the intersection of a solid curve with the gray line gives the optimum value of the reference angle. The termination of the upper left portions of the curves is at the reference angle where a full bandwidth of 100nm cannot pass through the grism pair. The plot illustrates the degree of tolerance in the choice of incident angle.

4.3. Residual fourth-order phase

When the ϕ ₃/ϕ ₂ control of grisms is used to compensate for material dispersion, for example to compress after propagation in optical fiber [10], or to prestretch in down-chirped pulse amplification (DCPA) [9], the analysis above yields the residual fourth-order phase after compression.

 

Fig. 5. Comparison of the calculation of the ratio of third- to second-order spectral phase for a grism pair (1480 grooves/mm, BK7 glass apex angles α ₁=α ₂=45°). Solid lines: calculation based on analytic derivatives of Eq 10, for several reference angles θ_ref=50°,70°,75°,80° (from right to left in decreasing grayscale). Larger θ_ref corresponds to greater prism insertion. Dashed line: calculation based on approximation (Eq 15). Dots: calculation from raytrace for several combinations of θ_ref and θ ₁. Inset: raytrace for θ_ref=70° and θ ₁=15°, for the spectral range 750nm (blue) to 850nm (red).

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Fig. 6. Calculation of higher-order phase of a grism pair (1480 lines/mm, BK7, apex angles 45o) vs. reference angle θ_ref. Solid lines: ϕ ₃/ϕ ₂ (fs) for incident angles θ ₁=10° (blue), θ ₁=15° (red), and θ ₁=20° (green). Lines terminate on the left when θ_ref is too small to accommodate the full 750–850nm bandwidth. Horizontal gray line indicates the value of ϕ ₃/ϕ ₂ required to compensate BK7 glass. Dashed lines: ϕ ₄/ϕ ₂ (fs ², scaled down by 2). Vertical lines guide the eye to indicate the value of residual ϕ ₄/ϕ ₂ when θ_ref is set to compensate BK7 glass.

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The dashed lines in Fig. 6 show the ratio ϕ ₄/ϕ ₂ for two of the incident angles. While the ratio of ϕ ₃/ϕ ₂ can be controlled from negative to positive values, the ratio ϕ ₄/ϕ ₂ is observed to always be positive. This has been seen in a wide range of grism designs that we have tested. From the graph we can see that the residual fourth-order phase can be lessened by working at an incident angle that is as small as possible while still achieving the desired ratio ϕ ₃/ϕ ₂.

Designs for specific systems will vary, but more details about this sample system will give a sense of scale. Let us pick an incident angle of 15° (red curves) and a reference angle that matches the ϕ ₃/ϕ ₂ ratio for BK7, and an edge-to-edge bandwidth of 100nm (but a FWHM bandwidth of 50nm). If we choose a grism separation of 10cm, the pulse will be stretched (or compressed) by 80ps. The spectral width will be approximately 2.5cm at the second prism, and there would be sufficient room for a beam size of 2cm.

A full analysis of the ϕ ₄ scaling of grism pairs is outside the scope of this paper, but it is useful to estimate how much the residual ϕ ₄ affects compression. Suppose the residual higherorder phase is only limited by ϕ ₄. Let us define Δω and τ as the full-width half-maxima of the spectral and temporal intensity profiles. The excess group delay at ω=±Δω/2 is τ₄= ϕ ₄ (Δω/2)³, where ϕ ₄ is evaluated at the central frequency. When τ ₄≈τ, the excess group delay will noticeably broaden the pulse. For a transform-limited Gaussian pulse (and the FWHM widths), the time-bandwidth product is Δωτ=4ln2, so an estimate for the value of ϕ ₄ that is important is ϕ ₄=8τ ⁴ (4 ln2)^-3≈0.375τ ⁴.

Noting that the values for all phase orders for the single-facet grism pairs are linearly proportional to the tip-to-tip spacing (as with prisms), the value of the residual ϕ ₄ will depend on the stretch duration, which scales as T_str≈ϕ ₂Δω. When the grism alignment is adjusted to obtain a desired value of ϕ ₃/ϕ ₂, there will be a corresponding ratio of ϕ ₄/ϕ ₂, which for the parameters considered in Fig. 6 is approximately 8 fs². This ratio is much greater than the contribution from the optical material, which is in the range -0.3 fs² (fused silica) to 0.15 fs² (SF18). Putting this together, there is a maximum value of stretch that can be tolerated for a starting pulse duration: T_str≈aτ ³, where the proportionality constant is a=8(4ln2)^-2/(ϕ ₄/ϕ ₂). For a pulse duration of τ=80fs, the maximum stretch by this scaling is approximately 67ps. By applying the spectral phase in the frequency domain and taking the Fourier transform, we find that this estimate is somewhat conservative, so that an extra 2-3x larger stretch is possible without a strong distortion of the compressed pulse. Nevertheless, the scaling clearly shows that residual fourth-order phase should be taken into account for pulse durations much less than 100fs.

5. Summary

In this article, we have described a method for analytically constructing the spectral phase which can be applied to a wide variety of systems: prisms, gratings and grism structures. All of these can be represented as a series of nested pairs of planar interfaces that either refract or diffract the light. As a new pair of interfaces is inserted, spectral phase is added for the new element while it is subtracted for the space taken up by the new element. We have also applied the method to the analytic calculation of the spectral phase of grism pairs for the first time to our knowledge. For the case of a reflection grism in which the light enters and exits the same facet, the form of the phase is of the same form as prism pairs, except that the calculation of the angles of dispersion must account for the gratings. Other more complicated grism structures can easily be treated by the method: e.g. double-facet grisms, or grisms that have a gap between the grating and the prism face.