Resolving Powers of Immersed Transmission Gratings

I. K. Baldry, J. Bland-Hawthorn & J. G. Robertson

Appendix of Publ. Astron. Soc. Pacific, Vol. 116, pp. 403-414; published in May 2004.

Keywords: spectral resolution diffraction grating prism
VPH gratings can be sandwiched between glass prisms. This reduces the total beam deviation and reduces the air-to-glass incidence angle ($\alpha_0$), for a given grating and wavelength. This can be useful because the total beam deviation is limited by the physical sizes of the camera and collimator and because higher incidence angles on air-glass boundaries give higher reflection losses (for unpolarized light). Here, we give the equations for calculating the resolving power of a transmission grating immersed between two prisms.

  

Figure: Diagram of a prism model for an immersed transmission grating. In Littrow configuration, with both prism angles equal to $\gamma$, the resolving power is approximately proportional to $n_1 \tan\alpha_1 \cos(\alpha_1-\gamma) / \cos\alpha_0$.

Figure 1 shows the prism model that we are using with the appropriate angles defined. Light passing through the prism and the immersed grating, with a total beam deviation of $\alpha_0 + \beta_0 + \gamma_a + \gamma_b$, obeys the following equations:

 

$$n_1 \sin (\alpha_1 - \gamma_a) \: = \: n_0 \sin \alpha_0$$ (1)

$$\sin \beta_1 \: = \: \frac{m \lambda}{\Lambda_g n_1} - \sin \alpha_1$$ (2)

$$n_0 \sin \beta_0 \: = \: n_1 \sin (\beta_1 - \gamma_b) \:.$$ (3)

The resolution can be determined by solving

$$\beta_0(\alpha_0, \lambda + \Delta \lambda) = \beta_0(\alpha_0 - \Delta \alpha, \lambda) \:.$$ (4)

Here, the output angle is regarded as a function of the input angle and the wavelength. This expression represents the condition that incrementing the wavelength by $\Delta \lambda$ shifts the output image by the same amount as does the change in the incidence angle across the slit width. $\Delta \alpha$ is the angular size of the slit in the collimated beam and is given by

$$\Delta \alpha = \theta_s \frac{f_{\rm tel}}{f_{\rm coll}}$$ (5)

where $\theta_s$ is the angular size of the slit on the sky, and $f_{\rm tel}$ and $f_{\rm coll}$ are the effective focal lengths of the telescope and collimator. If n₁ and n₀ are independent of wavelength (i.e., ignoring differential refraction), then Eqn. 4 can be solved analytically to give a resolving power of

$$\frac{\lambda}{\Delta \lambda} = \frac{f_{\rm coll}}{\theta_s... ...tan \alpha_1 + \frac{\sin \beta_1}{\cos \alpha_1} \right) \:.$$ (6)

Note that, with $\gamma_a = \gamma_b = 0$, this reduces to the well known equation for the resolution of an unimmersed grating:

$$\frac{\lambda}{\Delta \lambda} = \frac{f_{\rm coll}}{\theta_s... ...tan \alpha_0 + \frac{\sin \beta_0}{\cos \alpha_0} \right) \:.$$ (7)

To include the dispersive effects of glass (n₁ varying with $\lambda$), Eqn. 4 can be solved numerically. Differential refraction marginally increases the resolving power for typical VPH grism designs. In Littrow configuration, $\gamma_a=\gamma_b\,(=\gamma)$ and $\alpha_i=\beta_i$, with a total beam deviation of $2\alpha_0 + 2\gamma$, the resolving power (n₀ and n₁ constant) is given by

$$\frac{\lambda}{\Delta \lambda} = \frac{f_{\rm coll}}{\theta... ...\alpha_1 - \gamma)}{\cos (\alpha_0)} \, 2 \tan \alpha_1 \:.$$ (8)

The usefulness of Littrow configuration is three fold: (i) VPH unslanted fringes can be used (slanted fringes may curve during DCG processing); (ii) the beam size remains about the same, which keeps the camera optics smaller and simpler; and (iii) the angular size of the slit remains nearly the same. With unslanted fringes, the important Bragg angle is given by $n_2\sin\alpha_{2b} = n_1\sin\alpha_1$. Figure 2 shows resolving powers at Littrow versus total beam deviation with the Bragg angle annotated. Note that for a given grating (fixed diffraction order, wavelength and lines ${\rm mm}^{-1}$), prisms typically reduce resolving power, but for a given total beam deviation, prisms typically increase resolving power .

  

Figure: Resolving powers of VPH transmission gratings versus total beam deviation. The top line represents a grating immersed between two $40^{\circ }$ prisms (with n₁=1.5), the dashed line between two $20^{\circ }$ prisms and the lower line represents a grating with no prisms attached. The crosses are set at $10^{\circ }$ intervals in Bragg angle (with n₂=1.3). The resolving powers are normalized to unity for the zero deviation $40^{\circ }$ prism model. See Fig. 1 for the prism model. Note that the dispersion caused by differential refraction is not included.