The immersed echelle ± I. Basic properties

High-resolution spectrographs for very large telescopes lead to new challenges in grating technology to achieve optimum matching of the slit width to the seeing disc. This has led elsewhere to the development of long echelles of high blaze angle produced by mosaicking technology. An alternative approach, pursued in this paper and adopted for the Gemini HighResolution Optical Spectrograph, is to immerse a monolithic echelle in a medium of refractive index greater than unity. This paper explores the consequences of this. It will be followed by Paper II, which will report on a major technology development programme in progress. Key words: instrumentation: spectrographs ± techniques: spectroscopic. 1 I N T R O D U C T I O N Consideration of the immersed echelle as a possible enhancement of the standard echelle stems from the need to optimize the resolving power slit-width product of astronomical spectrographs for very large telescopes. The enhanced performance resulting from the immersion of a grating in a high-index (n > 1) dielectric medium has been described by Longhurst (1973), and more recently in greater detail by Dekker (1987) and Wynne (1991). A research and development programme being undertaken by the Optical Science Laboratory (OSL) at University College London (Walker, Radley & Diego 1993) is currently addressing the performance bene®ts and design trade-offs that the immersed echelle presents. This is being driven by the scienti®c requirements for the Gemini High-Resolution Optical Spectrograph (HROS) that it should provide the maximum throughput, particularly in the UV, combined with the practical requirement of a compact instrument that will ®t within the space constraints of the Gemini Cassegrain environment. Whilst the immediate context is Gemini, the methodology developed clearly has wider relevance. The underlying basis for this work is presented here, with the goal of general applicability to existing and new echelle spectrographs. The principal difference between the immersed and nonimmersed echelles is the presence of the enclosing high-refractiveindex medium in the form of a prism. When light of wavelength l enters the prism, the effective wavlength of the radiation that the grating `sees', le, is actually smaller by a factor of n, the refractive index of the prism, i.e. le ˆ l=n. Therefore, the ratio le=j (where j is the grating constant or groove spacing) also decreases. It is this ratio that primarily determines the overall photometric ef®ciency of the echelle/prism combination. In addition, we may also interpret this to mean that the effective grating constant of the echelle has been increased by the factor n, and since the number of rulings remains unchanged, then this also leads to the result that the effective length of the echelle has increased by n. Consequently, if the grating constant has increased, then the order of interference will also increase by n, resulting in a similar increase of resolving power R. Thus intuitively it appears that the immersed echelle operates identically to an echelle, the effective grating constant and length of which have been multiplied by the immersing refractive index. The following sections explore the consequences of immersion in a more quantitative manner. 2 I N T E R F E R E N C E C H A R AC T E R I S T I C S The idealized operation of an immersed echelle is shown in Fig. 1. The entrance/exit face of the prism, F, is parallel to the facets of the echelle or grating, which is itself optically coupled to the prism, either by a suitable cement or oil. In this mode the acute angle denoted by vB is equal to the blaze angle. For the purpose of determining the angular position of outgoing rays the blaze angle is not required, as its sole purpose is to modify the diffracted energy distribution. This will be the subject of the next section. Note that when the input and output beams are symmetrically disposed about the normal to the face F, then residual dispersions at the air/glass interface are cancelled. This is not the case when an asymmetry exists, but in real spectrographs this effect will be very small when compared with the much greater dispersion of the echelle. 2.1 The diffraction equation Fig. 2 shows a grating immersed in a medium of refractive index n. A ray of wavelength l in air (l=n in the medium) is incident upon the rulings at an incidence angle of a with respect to the grating normal, or v with respect to the grating facet normal, i.e. a ˆ vB ‡ v. The ray is then diffracted at an angle of b, or vB ÿ v at the blaze peak. Constructive interference with an adjacent ray occurs when the following condition is satis®ed: mnl ˆ nj cos g…sin b ‡ sin a†; mn ˆ 61; 62; . . . ; etc.; …1† Mon. Not. R. Astron. Soc. 302, 139±144 (1999)