Spatial sampling and spectral sampling of a grating spectrograph

Daigo Tomono

January 22, 2003

Abstract:

Design steps of a seeing limited spectrograph is described in this report.

Postscript version (sampling.ps.gz) and source tar ball (sampling.tar.gz) are also available.

Introduction

In the KMOS meeting held in Durham on January 13-14, 2003, it is pointed out that we have to sample a spatial element with one pixel while sampling a spectral element with two pixels. Generally, we have to use a cylindrical lens on the optical train to make the spatial element in an elliptical shape. Although, Tomono felt that we can do so adjusting dispersion of the grating and spatial magnification of the camera and collimator. In this paper, possibility to do such a thing is evaluated. Also, a sample design of the cylindrical lens to be used with the fibre IFU is tested.

Calculations on a grating spectrograph

Following discussions are based on Tomono's former report (Tomono, 2002).

Interferometric calculations

Figure 1 shows geometry of a blazed grating. Collimated light from the direction P1 with angle $\alpha$ is diffracted by the grating with grating constant $d$ and groove angle $\varepsilon$ into the direction P2 with angle $\beta$.

Figure 1: Geometry of light and blazed grating

With the following parameters:

$$A \equiv \sin \alpha + \sin \beta$$ (1)

and

$$E \equiv \frac{1}{\cos \varepsilon} (\sin [\alpha - \varepsilon] + \sin [\beta - \varepsilon]),$$ (2)

intensity of the light with wavelength $\lambda$ detected at P2 is

$$I(\beta) = \frac{1}{N^2}\frac{1-\cos[kNdA]}{1-\cos[kdA]} \times \frac{2}{(kd'E)^2} (1-\cos[kd'E])$$ (3)

with $k \equiv 2\pi/\lambda$ and number of grooves on the illuminated area of the grating $N$.

Equation 3 has peaks at $kdA = 2\pi m$ for the $m$th order of diffraction. Following the definition of $A$,

$$m\lambda = d(\sin\alpha + \sin\beta).$$ (4)

Thus, when the center of the incident angle $\alpha_0$ is fixed, central emergent angle $\beta_0$ of the $m$th order for the primary wavelength $\lambda_0$ is

$$\beta_0 = \arcsin[\frac{m\lambda_0}{d} - \sin\alpha_0].$$ (5)

On the other hand, from difference of incident angle and emergent angles:

$$\Delta \equiv \alpha_0 - \beta_0,$$ (6)

the incident angle and the emergent angle for the primary wavelength $\lambda_0$ can be calculated as

$$\alpha_0 \mbox{,} \beta_0 = \arcsin\left[ \frac{ \frac{m\... ...Delta)-(\frac{m\lambda_0}{d})^2}} {2(1+\cos\Delta)} \right].$$ (7)

Spectral resolution of a narrow slit spectrograph

For $N\gg 1$, spectral resolution of the grating is calculated as follows. First, we have a peak at $A_1 = \lambda_1 m / d$ for a wavelength $\lambda_1$. Near the peak, there are local minima at $A_2 = A_1 \pm \lambda_1 / (Nd)$. We define spectral resolution when the next spectral element of $\lambda_2$ is at the local minima $A_2$. Namely, for the $m$th order, we have a peak for $\lambda_2$ at $A_2 = \lambda_2 m / d$. Thus, the wavelength difference $\Delta\lambda = \vert\lambda_2 - \lambda_1\vert$ or

$$\Delta\lambda = \frac{\lambda}{mN}$$ (8)

or

$$R \equiv \frac{\lambda}{\Delta\lambda} = mN$$ (9)

Spectral resolution of a wide slit spectrograph

When the entrance slit is wide, Equation 3 has to be integrated over the entrance slit, assuming that the light coming to different locations on the entrance slit is not coherent. This results in spectral resolution not being geometrically limited rather than being diffraction limited as for a narrow entrance slit.

When the direction of the collimated light spreads from $\alpha_1 \equiv \alpha_0-\delta\alpha$ to $\alpha_2 \equiv \alpha_0+\delta\alpha$, Equation 3 has to be integrated from $\alpha_1$ to $\alpha_2$ and profile of the spectra can be assessed.

$$J(P_2) \equiv \int_{\alpha_1}^{\alpha_2} I(P_2) \mbox{ d}\alpha$$ (10)

$$= \int_{A_1}^{A_2} I(P_2) \cos\alpha \mbox{ d}A$$ (11)

$$= \int_{A_1}^{A_2} \frac{1}{N^2}\frac{1-\cos[kNdA]}{1-\cos[kdA]} \frac{2}{(kd'E)^2} (1-\cos[kd'E]) \cos\alpha \mbox{ d}A$$ (12)

with $A_{1,2} \equiv \sin\alpha_{1,2} + \sin\beta$. We can assume $\delta\alpha\ll 1$ (slit width is small, if not narrow) and $kd'E\ll 1$ (the spectrometer is used around the Bragg's condition). In the case, $\cos\alpha \sim \cos\alpha_0$ and $1-\cos[kd'E]\sim(kd'E)^2/2$. When the grating is used in the $m$th order, $A$ can be replaced with $A' \equiv A - 2\pi m/(kd)$. Near the $\alpha$ and $\beta$ of spectra we are observing, $A'\ll 1$. Therefore, $1-\cos[kdA] = 1-\cos[kdA'] \sim (kdA)^2/2$. Here, we define $A'_{1,2} = A_{1,2} - 2\pi m/(kd) \sim \sin \alpha_0 + \sin\beta - 2\pi m/kd \pm \cos[\alpha_0] \delta\alpha$ and the integration becomes

$$J(P_2) \sim \cos\alpha_0 \int_{A'_1}^{A'_2} \frac{2(1-\cos[kNdA'])}{(kNdA')^2} \mbox{ d}A'$$ (13)

$$= \frac{\cos\alpha_0}{kNd} \int_{x_0-\delta x}^{x_0+\delta x} \frac{2(1-\cos x)}{x^2} \mbox{ d}x$$ (14)

$$\propto \int_{x_0-\delta x}^{x_0+\delta x} \frac{1-\cos x}{x^2} \mbox{ d}x$$ (15)

with

$$x_0\equiv kNd\times(\sin\alpha_0 + \sin\beta - 2\pi m/kd)$$ (16)

and

$$\delta x\equiv kNd\times\cos[\alpha_0] \delta\alpha.$$ (17)

The equation cannot be integrated analytically.

Figure 2 shows results from numerical integration of Equation 16 for different $\delta x$. For $\delta x$ smaller than $\sim \pi$, width of the spectra is comparable with that for $\delta x = 0$. On the other hand, for $\delta x > \pi$, the profile gets half of the center at $x_0 \sim \delta x$.

Figure 2: Numerical integration of Equation 16. From left to right, curves show results for $\delta x$ of 0, 2, 4, 6, 8, 10, and 12. Results are normalized at $x_0 = 0$.

Here we define the critical angular half width of the entrance slit $\alpha_{crit}$ corresponding to $\delta x = \pi$ as

$$\delta\alpha_{crit} \equiv \frac{\pi}{kNd\cos\alpha_0} = \frac{\lambda}{2Nd\cos\alpha_0}.$$ (18)

If the half width of the direction of the incident light onto the grating is more than $\delta\alpha_{crit}$, spectral resolution of the spectrometer is not limited by diffraction of the grating, but limited by geometrical width of the slit.

For $\delta\alpha$ larger than $\delta\alpha_{crit}$, half width of the slit image corresponds to

From Equation 17, change of $x_0$ corresponds to change of $\beta$ as

$$\delta x_0 = kNd \delta\beta \times \cos\beta_0$$ (19)

or

$$\delta\beta = \frac{\delta x_0}{kNd \cos\beta_0} = \frac{\cos\alpha_0}{\cos\beta_0} \times \delta\alpha$$ (20)

with central emergent light direction $\beta_0$.

In the case, spectral resolution is estimated comparing half width of the slit image (Equation 21) with dispersion of the spectrograph. Following the definition of spectral resolution in Section 2.2, we define a spectral element of the spectrograph to be the wavelength difference $\Delta\lambda$ corresponding to the full width of the slit image $2\delta\beta$. From Equation 4, spectral resolution of the spectrograph becomes

$$\Delta\lambda = \frac{d\cos\beta_0}{m} \times 2\delta\beta = \frac{d\cos\alpha_0}{m} \times 2\delta\alpha$$ (21)

or

$$R \equiv \frac{\lambda}{\Delta\lambda} = \frac{m\lambda}{d\cos\alpha_0 \times 2\delta\alpha} = \frac{mN}{r}$$ (22)

for $\delta\alpha \equiv r \times \delta\alpha_{crit} = r \times \lambda/(2Nd\cos\alpha_0)$ with $r > 1$.

Critical width of the slit

We assume to have a collimator with focal length in the dispersion direction $f_{col}$. When we obtain spectra with a telescope of focal ratio $F_{tel}$, full width of the beam on the grating as seen from the entrance slit $W_{col}$ becomes

$$W_{col} = \frac{f_{col}}{F_{tel}}.$$ (23)

In the case, width of the grating is calculated geometrically:

$$W_{gra} = \frac{W_{col}}{\cos\alpha_0}.$$ (24)

Therefore, number of grating grooves on the grating illuminated by the incident light becomes

$$N = \frac{W_{gra}}{\cos\alpha_0} = \frac{f_{col}}{F_{tel}\times d\cos\alpha_0}$$ (25)

In this case, the critical full width of the entrance slit is

$$2\alpha_{crit} = \frac{F_{tel}\times\lambda}{f_{col}}$$ (26)

or physical full width of the slit $w_{crit}$

$$w_{crit} = 2\alpha_{crit} \times f_{col} = F_{tel}\times\lambda$$ (27)

On a telescope with entrance pupil diameter $D_{tel}$, the slit width corresponds to

$$\theta_{crit} = \frac{w_{crit}}{D_{tel}\times F_{tel}} = \frac{\lambda}{D_{tel}} \sim \theta_{diff}.$$ (28)

where $\theta_{diff}$ is the diffraction limit spatial resolution of the telescope. Therefore, a spectrograph with a seeing limited spectral resolution (i.e. spatial resolution more than the diffraction limit) has its spectral resolution limited by Equation 23, not by Equation 9.

Design of a seeing limited spectrograph

Boundary conditions

Following parameters must be defined from a scientific point of view and from physical limitations to define a spectrograph:

• Grating groove interval $d$,
• Diffraction order $m$,
• Primary wavelength $\lambda_0$,
• Width of a spectral element $\Delta\lambda$,
• Difference of central incident angle and central emergent angle $\Delta$,
• Entrance slit width $w$, and
• Slit image width on the detector $i$.

We have to note that $d$ has to be at least five times bigger than the maximum wavelengths to avoid polarization.

From Equation 7, central incident angle $\alpha_0$ and central emergent angle $\beta_0$ can be calculated. Selection of $\alpha_0$ and $\beta_0$ from the solutions is rather arbitrary but might affect performance of the spectrograph (Tomono, 2002).

Focal lengths of the collimator and the camera

At first, we consider focal lengths of the optics in the dispersion direction. From Equation 22, the image of a spectral element has a full width of

$$2\delta\beta = \frac{m\Delta\lambda}{d\cos\beta_0}$$ (29)

which is imaged on the detector by the camera optics with focal length of $f_{\lambda,cam}$:

$$i = f_{\lambda,cam}\times2\delta\beta = \frac{f_{\lambda,cam}m\Delta\lambda}{d\cos\beta_0}.$$ (30)

Therefore, $f_{\lambda,cam}$ should be:

$$f_{\lambda,cam} = \frac{i d \cos\beta_0}{m\Delta\lambda}.$$ (31)

and for the incident angle width, same derivation can be applied for the collimator focal length in the dispersion direction $f_{\lambda,col}$ as

$$f_{\lambda,col} = \frac{w d \cos\alpha_0}{m\Delta\lambda}.$$ (32)

Along the spatial direction, focal lengths of the collimator and the camera can be defined independently. Nevertheless, it is practically easier to make the focal length same for both of the directions. In this case, spatial magnification $M_x$ of the spectrograph is

$$M_x \equiv \frac{f_{x,cam}}{f_{x,col}} = \frac{f_{\lambda,cam}}{f_{\lambda,col}} = \frac{i \cos\beta_0}{w \cos\alpha_0}$$ (33)

This is different from spectral magnification $M_\lambda$ defined as the ratio of the width of slit image and that of the entrance slit:

$$M_\lambda \equiv \frac{i}{w}.$$ (34)

Sizes of the optical elements

Using the focal lengths defined in the previous section, minimum sizes of the optical elements can be estimated. We have to note that the sizes calculated here are for each spatial and spectral element. In practice, they are bigger because the optics has to handle more spatial and spectral elements. For calculations, following parameters are assumed:

$F_x$

input focal ratio in the spatial direction

$F_{\lambda}$

input focal ratio in the dispersion direction

Physical sizes of a spatial and spectral element on the entrance slit is ignored in this section.

From above parameters, Sizes of the collimated beam in the dispersion direction $W_{\lambda,col}$ and the spatial direction $W_{x,col}$ are

$$W_{\lambda,col} = \frac{f_{\lambda,col}}{F_{\lambda}} = \frac{w d \cos\alpha_0}{F_{\lambda}m\Delta\lambda}$$ (35)

The grating is illuminated by the beam with the incident angle $\alpha_0$, therefore its sizes are

$$W_{\lambda,gra} = \frac{W_{\lambda,col}}{\cos\alpha_0} = \frac{w d}{F_{\lambda}m\Delta\lambda}.$$ (36)

``Reflected'' by the grating, the beam goes through to the camera optics:

$$W_{\lambda,cam} = W_{\lambda,gra} \times \cos\beta_0 = \frac{w d \cos\beta_0}{F_{\lambda}m\Delta\lambda}.$$ (37)

From the above, monochromatic focal ratio of the camera in the dispersion direction is

$$F_{\lambda,cam} = \frac{f_{\lambda,cam}}{W_{\lambda,cam}} = \frac{i}{w} \times F_{\lambda}.$$ (38)

On the other direction, widths of the optics are

$$W_{x,col} = W_{x,gra} = W_{x,cam} = \frac{f_{x,col}}{F_x}.$$ (39)

Sample design

A sample design of a K-band spectrograph directly attached to the Nasmyth focus of the VLT is shown in Table 1 and Figure 3.

Table 1: A sample design of a K-band spectrograph

Item	Parameter
Primary wavelength	2.2 $\mu$m
Spectral resolution	0.00055 $\mu$m
Slit width	116.4 $\mu$m
Slit image width	37.0 $\mu$m
Input focal ratio	15.00
Collimator focal length	1145.8839 mm
Collimator beam width	76.39 mm
Grating groove interval	11.00 $\mu$m
Grating width	77.57 mm
Primary incident angle	-9.9977$^{\circ}$
Primary emergent angle	35.0023$^{\circ}$
Camera focal length	303.0778 mm
Monochromatic camera focal ratio	4.77
Monochromatic cemera beam width	63.54 mm
Spatial magnification	0.2645

Figure 3: K-band spectrograph with paraxial lenses

.

Calculation script

Following is a ruby script to execute the described calculations.

#!/usr/bin/ruby
=begin
= grating.rb
calculates geometries of a spectrograph as described in
tomono:~/Presen02/030120_Grating/sampling.tex

(C) 2003 Daigo Tomono <tomono at mpe.mpg.de>
 
Permission is granted for use, copying, modification, distribution,
and distribution of modified versions of this work as long as the
above copyright notice is included.

== Methods and Classes
=== Usefule functions
=end

=begin
--- Math.asin (x)
      returns arcsin of x in radians
=end
module Math
  def Math.asin (x)
    Math.atan2 (x, Math.sqrt(1-x*x))
  end
end

=begin
--- Numeric#to_rad
      returns the receiver (in degrees) converted into radians
--- Numeric#to_deg
      returns the receiver (in radians) converted into radians
=end
class Numeric
  def to_rad
    self*Math::PI/180.0
  end
  def to_deg
    self*180.0/Math::PI
  end
end

=begin
=== class GratingSpectrograph
GratingSpectrograph class is used to describe a spectrograph with a
collimator, a grating, and a camera.

--- GratingSpectrograph.new (delta, w, i, l0, dl, m, fratio_l, d = l0*5., flip = false)
      spectrograph with the following parameters:
      * delta: Difference of central incident angle and central emergent angle
        (radians)
      * w: Entrance slit width
      * i: Slit image width on the detector
      * l0: Primary wavelength
      * dl: Width of a spectral element
      * m: Diffraction order
      * fratio_l: Focal ratio of the incident light in the dispersion direction
      * d: Grating groove interval
      When flip is false, |alpha_0| < |beta_0|, otherwise, |alpha_0| >
      |beta_0|.

--- GratingSpectrograph#delta
      the parameter delta, other parameters can also be read in the same
      manner.

--- GratingSpectrograph#alpha_0
      primary incident angle in radians
--- GratingSpectrograph#beta_0
      primary emergent angle in radians
--- GratingSpectrograph#f_col
      the collimator focal length in the dispersion direction.
--- GratingSpectrograph#f_cam
      the camera focal length in the dispersion direction.
--- GratingSpectrograph#m_lambda
      the magnification in the dispersion direction.
--- GratingSpectrograph#m_x
      the magnification in the spatial direction assuming the focal
      lengths in the spatial direction is the same as those for in the
      spectral direction.
--- GratingSpectrograph#width
      Beam width in the spatial direction assuming the focal length of
      the collimator and the focal ratio of the incident light is the
      same in the spectral direction.
--- GratingSpectrograph#width_col
      Collimator beam width in the dispersion direction.
--- GratingSpectrograph#width_gra
      Grating width in the dispersion direction.
--- GratingSpectrograph#width_cam
      Camera beam width in the dispersion direction.
--- GratingSpectrograph#fratio_cam
      Camera focal ratio in the dispersion direction.

--- GratingSpectrograph#to_s
      Formats the input and derived parameters in a fancy way assuming
      that the input length parameters are in microns.
=end
class GratingSpectrograph
  attr_reader :delta, :w, :i, :l0, :dl, :m, :fratio_l, :d, :flip
  attr_reader :alpha_0, :beta_0, :f_col, :f_cam, :m_lambda, :m_x
  attr_reader :width, :width_col, :width_gra, :width_cam

  def initialize (delta, w, i, l0, dl, m, fratio_l, d = nil, flip = false)
    # input parameters
    @delta = delta.to_f
    @w = w.to_f
    @i = i.to_f
    @l0 = l0.to_f
    @dl = dl.to_f
    @m = m.to_f
    @fratio_l = fratio_l.to_f
    @d = d ? d.to_f : @l0 * 5.0
    @flip = flip
    # derivations
    ab = alphabeta
    unless flip then
      @alpha_0, @beta_0 = ab
    else
      @alpha_0, @beta_0 = ab.reverse
    end
    @cos_alpha_0 = Math.cos (@alpha_0)
    @cos_beta_0 = Math.cos (@beta_0)
    @f_col = @w*@d*@cos_alpha_0 / (@m*@dl)
    @f_cam = @i*@d*@cos_beta_0 / (@m*@dl)
    @m_lambda = @i/@w
    @m_x = @i*@cos_beta_0 / (@w*@cos_alpha_0)
    @width = @f_col/@fratio_l   # assuming f and F are same for both directions
    @width_col = @w*@d*@cos_alpha_0 / (@fratio_l*@m*@dl)
    @width_gra = @w*@d / (@fratio_l*@m*@dl)
    @width_cam = @w*@d*@cos_beta_0 / (@fratio_l*@m*@dl)
    @fratio_cam = @i*@fratio_l / @w
  end

  def to_s
   "\
== Grating spectrograph for #{@l0} um with #{@dl} um resolution
: Entrance slit and Collimator
  * #{'%.1f' % @w} um wide slit
  * #{'%.2f' % @fratio_l} focal ratio
  * #{'%.3f' % (@f_col/1000)} mm focal length
  * #{'%.2f' % (@width_col/1000)} mm wide
: Grating
  * #{'%.2f' % @d} um groove
  * #{'%.2f' % (@width_gra/1000)} mm wide
  * #{'%.3f' % @alpha_0.to_deg} deg primary incident angle
  * #{'%.3f' % @beta_0.to_deg} deg primary emergent angle
: Camera
  * #{'%.1f' % @i} um wide slit image
  * #{'%.2f' % @fratio_cam} focal ratio
  * #{'%.3f' % (@f_cam/1000)} mm focal length
  * #{'%.2f' % (@width_cam/1000)} mm wide
  * #{'%.3f' % (@m_x)} spatial magnification
  * #{'%.3f' % (@m_lambda)} spectral magnification
"
  end

  def to_TeX
   "\
\\begin{tabular}{cc}
\\hline
  Item & Parameter \\\\
\\hline
\\hline
  Primary wavelength & #{@l0} $\\mu$m \\\\
  Spectral resolution & #{@dl} $\\mu$m \\\\
  Slit width & #{'%.1f' % @w} $\\mu$m \\\\
  Slit image width & #{'%.1f' % @i} $\\mu$m \\\\
  Input focal ratio & #{'%.2f' % @fratio_l} \\\\
  Collimator focal length & #{'%.4f' % (@f_col/1000)} mm \\\\
  Collimator beam width & #{'%.2f' % (@width_col/1000)} mm  \\\\
  Grating groove interval & #{'%.2f' % @d} $\\mu$m \\\\
  Grating width & #{'%.2f' % (@width_gra/1000)} mm \\\\
  Primary incident angle & #{'%.4f' % @alpha_0.to_deg}$^{\\circ}$ \\\\
  Primary emergent angle & #{'%.4f' % @beta_0.to_deg}$^{\\circ}$ \\\\
  Camera focal length & #{'%.4f' % (@f_cam/1000)} mm \\\\
  Monochromatic camera focal ratio & #{'%.2f' % @fratio_cam} \\\\
  Monochromatic cemera beam width & #{'%.2f' % (@width_cam/1000)} mm \\\\
  Spatial magnification & #{'%.4f' % (@m_x)} \\\\
\\hline
\\end{tabular}
"
  end

  private
=begin
==== Private methods
--- GratingSpectrograph#alphabeta
      Returns an array [alpha_0, beta_0] calculated from Delta with
      |alpha_0| always less than |beta_0|.
=end
  def alphabeta
    x = @m * @l0 / @d   # \frac{m\lambda_0}{d}
    c = 1.0 + Math.cos (delta)
    k1 = x/2.0
    k2 = Math.sin (delta) * Math.sqrt (2.0*c - x*x) / (2.0*c)
    [Math.asin(k1+k2), Math.asin(k1-k2)].sort {|a, b| a.abs <=> b.abs }
  end
end

=begin
== Assumptions
: Telescope
  VLT Nasmyth
  * Focal ratio of 15
  * Entrance pupil diameter of 8 m
  * Spatial element of 0.2 arcsec
: Spectrograph
  * Incident and emergent angle difference of 45 degrees
  * 2.2 um optimized
  * R=4000 at 2.2 um, 2 pixel for each spectral resolution
  * 5 times 2.2 um groove
  * 18.5 um pixel
  * diffraction order of 2
=end

spectrograph = GratingSpectrograph.new (
  45.to_rad,                    # incident angle difference
  (0.2/3600).to_rad * 15 * 8e6, # entrance slit width (um)
  18.5*2,                       # slit image width (um)
  2.2,                          # primary wavelength (um)
  2.2/4000.0,                   # spectral element (um)
  2,                            # diffraction order
  15,                           # focal ratio
  2.2*5,                        # groove interval (um)
  false                         # alpha and beta flip
)

puts spectrograph.to_TeX

File locations

Original source of this document is at gin-an:~tomono/Presen02/030120_Grating/, with Zemax files at siroan:~/Presen02/0120_Grating/. HTML version of this document is posted at http://www.rzg.mpg.de/~tomono/MOS/auth/Reports/030120_Grating/. Some of the figures are copied from gin-an:~tomono/Presen01/011228_Grating/.

Revision history

2003.1.22.

-

• First release

Bibliography

Tomono, Daigo, ``Interferometry on a blazed grating,'' http://www.rzg.mpg.de/~tomono/MOS/auth/Reports/011228_Grating/, February 5, 2002.

About this document ...

Spatial sampling and spectral sampling of a grating spectrograph

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