Compact orthogonal-dispersion device using a prism and a transmission grating

Spectroscopy was one of the earliest applications of gratings, and it remains one of the main grating uses [1]. More particularly, the echelle grating suggested first by Harrison [2] is preferred in a number of spectrometers. The echelle grating is usually used in high diffraction orders that will overlap. To solve this problem, traditional echelle spectrometers usually employ cut-off filters or a cross disperser in the form of low-dispersion gratings or prisms. An early echelle spectrometer uses two planar reflection gratings, i.e., an echelle grating and a reflection grating, operating in two perpendicular planes [3]. The resulting dispersion produces a two-dimensional spectrogram including several lines side by side that correspond to different diffraction orders, namely, several diffraction-order lines side by side with the wavelength varying with its position along each line. As an improved echelle spectrometer [4], the second grating is replaced by a combination of a reflection grating with a prism so that the light from the echelle grating is refracted into the prism, diffracted by the grating, then refracted back out of the prism. The diffraction orders can be spread out almost evenly by choosing the prism parameters, allowing more of the detector area to record useful information. For the above referenced echelle spectrometers, both the incident and diffracted light rays are perpendicular to the grating grooves.

The diffraction grating can also be used in the conical diffraction mode [5–10] in which the incident light beam is not perpendicular to the grating grooves, thus the diffracted light rays lie on a circular cone rather than in a plane. Typically, the conical diffraction spectrometers reported in [11] uses a Littrow configuration coupled with a shared focusing lens and a linear detector array, and they are compact with a substantially reduced size. Nevertheless, these two designs are unable to separate the overlapping diffraction orders of the echelle grating. On the contrary, the Littrow conical diffraction spectrometer proposed by [12] includes an additional prism for separating the overlapping diffraction orders of the echelle grating to produce two-dimensional high dispersion without requiring a large field of view.

Most spectrometers today are based on reflection grating designs [13–15], since high quality transmission grating technologies have only matured over the last several years. However, it is currently possible to produce high quality transmission gratings that rival reflection gratings in all aspects. Accordingly, transmission gratings can be used as dispersive devices in the infrared, visible or near-ultraviolet light [16–21]. Transmission gratings offer low alignment sensitivity, which helps minimize alignment errors. A transmission grating offers a basic simplicity for optical designs that can be beneficial in spectral separation. In certain types of instruments, transmission gratings are much more convenient to use than reflection gratings. For example, a configuration inserts a transmission grating in front of the lens, which can be used for studying the composition of falling meteors or the re-entry of space vehicles, where the distant luminous streak becomes the entrance slit.

The direct-vision grism combines a low-dispersion transmission grating with a beam-bending prism to provide in-line viewing for a single wavelength [22]. The purpose of the prism is simply to change the direction of a given diffracted wavelength, for zero net deviation. The constant-dispersion grism consists of a prism and a transmission grating to disperse the spectrum so that the spectrum fringes are equally spaced across the detector. Namely, for the constant-dispersion grism the angular dispersion with respect to wave number is approximately constant. Both the transmission grating and prism contribute to the dispersion, and the net deviation angles are not constrained to be zero. For the direct-vision or constant-dispersion grism, both the prism and transmission grating have the same dispersion direction.

This paper is a theoretical study of a compact, small-sized and broadband orthogonal-dispersion prism-grating device used for spectrometers. Section 2 presents the basic principle and derives the analytical expressions for the dispersion of the device. Section 3 shows the numerical results and ray-tracing simulations by ZEMAX software. The last section gives the conclusion.

Figure 1a shows the optical layout of the orthogonal-dispersion prism-grating device (PGD) that combines a prism with a transmission grating. The dispersion directions of the prism and transmission grating are orthogonal. The transmission grating is superimposed on the hypotenuse surface of an optical wedge, and the right-angle surface of the optical wedge is directly attached to one surface of the prism. The grating normal is parallel to y-axis, the grating grooves are parallel to z-axis, and the y´-axis is perpendicular to the right-angle surface of the optical wedge. The prism disperses the incident light along the vertical direction (i.e., z-axis), then the transmission grating disperses the separated beams along the horizontal direction (i.e., x-axis). The spectrum is spread out in two dimensions.

Suppose that γ denotes the vertex angle of the prism, γ₁ denotes the vertex angle of the optical wedge, n _p (λ _i ) is the refractive index of the prism for wavelength λ _i , and n _g (λ _i ) is the refractive index of the grating material for wavelength λ _i .

Figure 1b shows the ray tracing in the sagittal plane (i.e., x´-y´ plane), in which δ is the angle between the inward normal vector of the grating and the projection of the incident wave vector \( \overrightarrow{k} \) onto the normal plane of the grating (i.e., x-y plane). It is easily obtained that δ=γ₁.

where m is the diffraction order, λ _i is the wavelength of light, d is the groove spacing of grating, ε is the angle between the incident light path and the plane perpendicular to the grating grooves, α is the angle of incidence measured from the grating normal, β _m is the diffraction angle measured from the grating normal. The angle sign convention is that angles measured counter-clockwise from the normal are positive and angles measured clockwise from the normal are negative.

Figure 2(a) shows the conical diffraction by a plane transmission grating, in which the grating lies in the x-z plane, the grating grooves are parallel to z-axis, and the grating normal is parallel to y-axis. Both the incident ray and the reflected rays lie in the y < 0 half-space; the transmitted diffracted rays lie in the y > 0 half-spaceThat is, the incident light is dispersed on the opposite side of the transmission grating. All diffracted rays fall on the surface of a half-cone whose cone axis is parallel to the grating grooves.

Based on Eqs. (3), (5), (7) and (8), the azimuth angle ρ _m , for the PGD, is given by

Eqs. (9) and (10) completely specify the diffracted ray angles in terms of the incident ray angles. For an incident ray of a given wavelength, the diffracted rays will all have the same value of φ _m given in Eq. (9) and will therefore all lie in a common conical surface, as shown in Fig. 2(a). For reference, if ε = 0, the conical diffraction grating equation [Eq. (3)] simplifies to yield the classical in-plane diffraction grating equation, mλ _i  = d[n _g (λ _i ) sin α − sin β _m ], for a plane transmission grating.

An orthogonal-dispersion PGD was investigated in theory. The ray-tracing simulations by ZEMAX indicated the accuracy of the theoretical analysis and mathematical formulas. In order to improve the spectral resolution, the dispersive power of the PGD needs to be improved, which can be realized by either appropriately increasing the vertex angle of the prism, properly reducing the groove spacing of grating, or both. In addition, the dispersive power of the PGD can also be changed by choosing different materials of the prism and grating. There are two main features to the PGD. First, the orthogonal-dispersion PGD is based on the transmission grating, whereas traditional cross-dispersion systems are based on the reflection grating. Second, the transmission grating is superimposed on the hypotenuse surface of an optical wedge with a right-angle surface directly attached to the prism, which not only makes the PGD compact, but also makes the air-glass interfaces fewer and therefore makes the reflection loss less. The unique design makes the PGD capable of providing a compact, small-sized and broadband orthogonal-dispersion device that is applicable to the medium resolution spectrometers.