WIYN / Bench Rebuild:
VPH High-Density Grating Design

1. Resolution (R = lambda/dlambda) vs incidence angle (alpha): R = 2 * (fl/w)) * tan(alpha), where fl is the collimator focal length and w is the effective slit width.

2. Grating size (D_grating) vs incidence angle (alpha): D_grating = D_beam/cos(alpha), where D_beam is the effective collimated beam diameter.

3. Grating size (D_grating) vs resolution (R), with quantities defined as above.

In all cases we have considered the two relevant collimator focal lengths: 1021mm (current) and 800mm (adopted upgrade); and 3 effective slit-widths for 200, 300, and 500 micron diameter fibers. The effective slit width is the FWHM, which for a circular fiber is cos(30) = 0.866 of the diameter. This table summarizes the effective slit widths for WIYN Bench fiber cables, and gives the reimaged effective slit widths (at the CCD) for de-magnifications of 3.58 and 2.81 corresponding to a camera effective focal length of 285mm and collimator focal lengths of 1021mm and 800mm, respectively:

fiber diameter (microns)	200	300	500
effective slit-width (microns)	173	260	433
image size with collimator fl=1021mm (microns)	48	73	121
image size with collimator fl=800mm (microns)	62	92	154

1. Higher resolutions are possible for the same size grating with the shorter focal length collimator because the smaller effective collimated beam allows for larger unvignetted angles and higher dispersion that more than compensates for the increased magnification. At a given resolution, the wavelength coverage will be greater with the longer focal-length collimator (smaller angular dispersions yield equivalent spectral resolution with smaller magnificaion). However, this comes at the monetary cost of a larger grating and at lower throughput because of vignetting in the camera.

2. Resolutions of 28,000 are possible with a 480 mm clear aperture VPH grating and the smallest existing (200 micron) fibers.

Notes: (i) To realize the gains from the faster collimator requires good grating coatings with high efficiency at large angles. (ii) Larger dispersions may be obtained at smaller angles and gratings with 2nd order gratings or two gratings used in series. Both of these options are possible, and will be considered later, elsewhere.

This set of plots shows the relationship between nu and the grating major axis size (D_major), and the resulting relationships between nu, the maximum grating angle (alpha_max,air), wavelength, and maximum spectral resolution. The wavelength lambda_alpha-max is the central wavelength at the maximum incidence angle (alpha_max,air) for a Littrow configuration (incidence and diffracted angles relative to groove plane are equal) where the grating is unvignetted.

These relationships are for Littrow gratings choosing maxima to be the largest unvignetted configuration for a given line frequency (hence, these are a function of beam size). Spectral resolution also depends on the ratio of collimator focal length to slit width f1/w. The two examples shown are for the existing Bench collimator and the planned, shorter off-axis collimator.

CONCLUSIONS:

1. The highest spectral resolutions with VPH gratings are obtainable at blue/visible wavelengths for unvignetted systems. A confluence of three (3) facts lead to this: The highest dispersions are achievad at the largest grating angles; grating size increases with line density (l/mm); and wavelength decreases with increasing line density for a given incidence angle. For the Bench, resolutions above 10,000 are achievable below 650nm for 500 micron diameter fibers; resolutions above 20,000 are achievable below 550nm for 300 micron diameter fibers; resolutions above 30,000 are achievable below 550nm for 200 micron diameter fibers.

2. Note again that higher spectral resolutions are achievable in unvignetted (i.e., high throughput) configurations with a shorter focal-length collimator: smaller beam size allows for larger incidence angles which trumps increased magnification.

Example: A 500mm x 230mm CSL grating would have a maximum unvignetted incidence angles (alpha) of 72.5 deg for a 150mm collimated beam, or 66.4 deg for a 200mm collimated beam. For typical gelatin indices (1.43) these angles correspond to grating incidence angles in the gelatin of 41.8 and 39.8 degrees. On the Bench Spectrograph with a 800mm effective focal-length collimator the 150mm and 200mm beams enclose 87% and 97% of the total light emergent from the fibers.

Reality: The cost of substrates is a strong function of size. Substrates over 400mm are very expensive, with few companies willing to bid. Nonetheless we have settled on a final high-density grating size of 480 x 210 mm clear aperture (500 x 230 x 30 mm substrate), and contracted with Zygo for delivery in July 2004. The maximum unvignetted angle with 480mm clear aperture is 71.8 deg.

The above excerise shows that higher resolutions are achievable if the higher cost is acceptable. Note reflection loss and coating issues below.

Spectral Resolution and Coverage vs Angle for 3550 l/mm VPH Grating1
alpha (deg)	Central Wavelength (nm)	Blue Wavelength (nm)	Red Wavelength (nm)	Wavelength Coverage (nm)	Spectral Resolution (Dfiber = 200 mu)	Spectral Resolution (Dfiber = 300 mu)	Spectral Resolution (Dfiber = 500 mu)
55	461.5	446.7	474.6	27.9	13190	8795	5277
60	487.9	474.8	499.2	24.4	16000	10670	6400
65	510.6	499.3	519.9	20.6	19810	13210	7924
70	529.4	520.1	536.7	16.7	25380	16920	10150
72	535.8	527.3	542.3	15.1	28430	18950	11370
Spectral Resolution and Coverage vs Angle for 316@63.4 Echelle (order 11)
69	513.1	500.0	526.2	26.2	200502,3 200501,4	197002 144401	117472 86651

       ¹ Collimator focal-length of 800mm is adopted for all resolution values.
       ² Collimator focal-length of 1021mm is adopted for all resolution values.
       ³ 2-pixel resolution is 20050; higher resolutions of 29560 achievable with new CCD.
       ⁴ 2-pixel resolution is 20050; higher resolutions of 21660 achievable with new CCD.

Efficiency:

Assuming the highest index-modulation is achieved, compared to the current echelle:

• (+) Peak throughput will be up from 55% to 90%.
• (+) No order-blocking filter is needed since 2nd-order is in the UV: X14 has at best 90% throughput.
• (o) No fold-flat is needed (in both cases).
• (-) Reflection losses of 4-10% will be incurred.

Detailed Information on Diffraction Efficiency: Below are RCWA predicted 1st-order grating diffraction efficiencies for the three pairs of values of gelatin thickness and index modulation, assuming n1 = n2 = 1.462 and n3 = 1.5: d = (6, 8, 10) microns, dn = (0.10, 0.078, 0.063), and dlogn = (0.042, 0.052, 0.067). These values are what CSL has settled on for the WP4200 grating, with their current goal to obtain d = 6 microns and dn = 0.1 (dlogn = 0.0667). What is currently shown are efficiency plots of incidence angle (alpha) vs wavelength in BW and color, as well as color line plots of efficiency vs wavelength for a series of angles. As before, these plots do not include reflection losses.

• As seen in the RCWA D-DLOGN maps, these choices closely follow a ridge-line in peak efficiency.
  • At higher d and lower dn, the ridge-lines in peak efficiency fall in the troughs of the bandwidth maps, and hence these gratings have lower band-widths while retaining peak efficiency. I.e., they were optimized for peak efficiency.
  • At low d and high dn, the ridge-lines in peak efficiency and band-width have closer overlap, thus giving the d = 6 micron grating the best overall performance.
• We have explored small changes in d,dn that further optimize the bandbass as per the RCWA D-DLOGN maps shown here. Further optimization is indeed possible, as shown here. Small trade-offs in peak effieciency yield significantly broader band-widths. Typically this is possible by lowering dn slightly for a given thickness d to be on the ``back side'' of the efficiency peak, which is closest to the nearest band-width peak.
• Note that the ridge lines (dashed curves) in the BW efficiency maps below are significantly NON-LITTROW (Littrow condition is shown for dotted line), and becomes increasingly non-Littrow for larger angles.
• Also note that the thinner gelatin / higher modulation index increases the non-Littrow condition of the blaze peak, and moves the blaze and super-blaze to bluer wavelengths. This will become more obvious when the lines plots are available. This is not desirable, although it may be necessary to achieve a high-efficiency grating. Now knowing what CSL is able to achieve, a slightly altered grating design with lower l/mm would have been more optimal.

	d = 6 microns dn = 0.100 n1=n2=1.462, n3=1.50	d = 8 microns dn = 0.078 n1=n2=1.462, n3=1.50	d = 10 microns dn = 0.063 n1=n2=1.462, n3=1.50
Efficiency Maps: incidence angle vs wavelength (BW version)
Efficiency Maps: incidence angle vs wavelength (color version)
Efficiency Maps: incidence angle vs wavelength (color version)

For comparison:

These are RCWA predicted grating diffraction efficiency in first order for a 3300 l/mm VPH grating with unknown parameters (Barden). Our design for the 3550 grating has shifted the superblaze from about 62.5 deg and 525 nm seen in this figures to the proposed values of 65 deg and 511 nm.
These are the RCWA predicted efficiency of 3300 l/mm VPH grating WP3200 parameters from CSL (Bershady estimates): d = 12 microns (effective) dn = 0.048 n1=n3=1.43, n2=1.4 Horizontal, colored lines span sampled Bench band-pass at given grating angle at minimum efficiency. Thin, dotted vertical lines are existing Bench-echelle half-order wavelengths for 11-deg camera-collimator angle.

Efficiency measurements at CSL in a small (1 cm) beam indicate peak diffraction efficiencies at 510 nm, corresponding to an incidence angle of rougly 57.25 deg. Note that in the Figure here (left), an attempt has been made to correct the measured transmissivity (transmission efficiency) for surface (reflection) losses. Diffraction efficiency is expected to be "1 - transmission efficiency."

This is a combined measurement of the corrected TE and TM polarizations. The absolute calibration of this diagram is TBD. Laser-line measurements in both polarizations indicate a somewhat higher diffraction efficiency. (A detailed exchange between MAB and PAB at CSL is in an email record.) Measurements at CSL in various grating locations indicate some non-uniformities in the diffraction efficiency over the grating. In general, the center of the grating appears to perform best.

The measured >70% peak diffraction efficiency in this plot corresponds to incidence angles alpha between 50 and 65 degrees or central waves of 465 nm and 545 nm. The >80% peak diffraction efficiency is expected between alpha of 53 and 60 degrees or 485 nm and 525 nm.

• Central wavelengths, wavelength ranges and spectral resolution for 200, 300, and 500 micron fibers are given here for the current (long) collimator:
  • 200 microns
  • 300 microns
  • 500 microns

The detailed procedure must take into account the fact that the current camera mount constrains setups to camera-collimator (&theta_cc) angles smaller than 70 degrees (see this email from Di). This means we are restricted to testing grating incidence angles &alpha = (180-70)/2 = 55 deg, which corresponds to cwl of 496.5 nm or larger. This means we can match the central wavelengths of the following VPH and echelle setups as per this table:

The specific test-measurements for this April 2006 T&E, in order of priority:

• [1] Confirm the image-quality of the grating is good over the range of accessible &alpha where the grating is likely to be useful, specificaly: &alpha = 55, 60, 65, 70 deg. Refer to the table below. A series of suitable line-lamp and dome flat exposurs should be taken at each of these angles used the Red Hydra fibers after a suitable focus and dewar azimuth angle ahas been found for each angle.
• [2] For each VPH configuration (&alpha), measure the grating camera distance as defined by the distance between the grating center and the front surface of the first elements of the camera. Determine whether this distance can be decreased, and what the minimum distance would be to keep the camera from vignetting the 160 mm incoming collimated beam onto the grating. Can this minimum distance be achieved or does a mechanical collision prevent this distance from being achieved? What is the minimum distance constrained by the collision? What is the nature of the mechanical collision, and how might it be mitigated?
• [3] Compare the relative throughput of the VPH at the different angles using dome flats and any cable. The above data should be sufficient.
• [4] Estimate the absolute throughput of the system with the VPH at any one angle with on-sky measurements using spectrophotometric standards and the SparsePak cable. I suggest &alpha in the 55-60 deg range is reasonable, but any angle will do as time and setup constraints require.
• [5] Explore aperture effects: As time allows, it would be interesting to compare the image quality and throughput for the inner and outer EE50 power-points. About half the energy in the unobstructed output beam from the fibers is within f/7, although a good amount of the light outside of f/7 will not make it through the system. An f-ratio of 7 corresponds to a collimated beam size of 146 mm in diameter for the current collimator. The upgraded system is designed to put f/5 within 160 mm collimatored beam diameter.

  If it is possible, create two grating masks that (1) maks-off the elliptical area of the inner region of roughly 150-160 mm diameter in the slit dimension (appropriately elongated in the dispersion dimension for the grating inclination), and (2) mask off the annular area outside the 150-160 mm diameter.

  Choose the one setup (&alpha) appropriate for the elliptical masks (I would recommend 55-60 deg), and take line-lamp and dome-flat exposures in a sequence starting with no mask, mask-1, mask-2, and then no mask again.

• [6] compare the relative throughput at the same cwl with the echelle (using dome flats with wavelength calibration from line-lamps). Any cable is fine as long as it is the same. See tables above and below for specific setup and cwl.

As always: Comments and discussion more than welcome. If we need to depart from this plan I'd like to hear about it in advance.

-MAB, 09 Apr 06