Chapter 4 Atmospheric Dispersion Corrector

Chapter 4

Atmospheric Dispersion

Corrector

4.1 Introduction

Apart from astronomical seeing, the other deleterious effect of the Earth’s atmo-

sphere on the light that passes through it is caused by atmospheric dispersion.

Atmospheric dispersion smears out the light into different wavelengths due to

differential refraction as light passes through the atmosphere. Atmospheric dis-

persion is a cumulative effect of pressure, temperature, humidity and location of

the observatory, but the altitude of the object has maximum effect on dispersion;

lower the object (farther away from zenith) more is the dispersion, as light passes

through a thicker layer of the atmosphere.

The net effect of dispersion is elongation of the object’s image on the camera,

leading to loss of resolution thereby defeating the purpose of AO. Atmospheric

Dispersion Correctors (ADCs) therefore play a very important role in AO systems.

Among the many types of ADC designs, the rotating double Amici prism design

is most widely used in astronomy [96, 97].

110

Chapter 4. ADC 111

4.2 Optical Design

A celestial object’s zenith angle continuously changes as a function of time. Hence

atmospheric dispersion also varies continuously and hence a varying dispersion

compensation is required during tracking. Two same prisms in the oppositely

rotating condition introduce varying dispersion[98, 99]. Resultant dispersion (Fig.

4.1) of two same type of prism at θ off from parallel is

DR−θ = 2DP cos(θ) (4.1)

where, DP is the prism dispersion vector; DR−θ is the resultant dispersion vector

at θ[86]. Each prism should have angular dispersion over wavelength range equal

to half of the dispersion due to the atmosphere at maximum zenith angle[97] up

to which the ADC is intend to work.

As shown in Fig.4.2, 4.3 the Atmospheric Dispersion Corrector (ADC) for iRobo-

AO consists of two identical prism assemblies, each capable of rotating indepen-

dently about the optic axis. Each of the prism assemblies consists of three ce-

mented wedge shaped glass plates (Fig. 4.4), made of N-FK51A, N-LASF31A,

and N-FK51A in sequence. These glasses have good transmission over the working

wavelength range. The relative orientations of the prisms for maximum, interme-

diate and minimum dispersions, are shown in Fig.4.6. The beam diameter at ADC

is ≈10mm. To minimize reflection losses the exposed surfaces were anti-reflection

coated. The optical design of the ADC was done using Zemax and to take atmo-

spheric parameters into account we used the Zemax subroutine “Atmospheric”1.

It is designed to work across the wavelength range of 0.4 - 2.2 µm.

The ADC is placed in between TTM and fold mirror in the collimated beam (Fig.

4.5).

4.2.1 Performance

The performance of the design over the entire working wavelength range is shown

in Fig.4.7. The polychromatic rms spot size is well within the Airy disc radius of

12 µm (defined at an intermediate wavelength of 1µm) for various zenith angles (z)

1This subroutine simulates the effects of the refraction through the Earth’s atmosphere whenviewing a point source or a star.

Chapter 4. ADC 112

Figure 4.1: Prism Rotation and its resultant dispersion

Figure 4.2: ADC prismassembly (dimensions are in

mm). Figure 4.3: ADC Prism.

of the object. Similar behaviour is also obtained in the individual B, V, R, I, J, H

and K bands. From the figure we also infer that at z ≈ 65◦ (maximum dispersion

case), the prisms get aligned with prism angle (θ) ∼ 0◦ and the dispersion of

individual prisms adds up in a manner nullifying the atmospheric dispersion.

A plot for the deviation of monochromatic spot position from the reference position

(i.e. the position of the primary wavelength at 1µm) vs wavelength at the optimal

prism angle for various zenith angles are given in the Fig. 4.8. All spots sizes are

well within the Airy Disc.

The analysis was done with IGO specific parameters for observatory height, pres-

sure, humidity and latitude of 1005 meter, 993 mbar, 50%, 19.0883◦N and three

different temperatures.

The centroid of the polychromatic PSF shifts with respect to the reference position

of the PSF at 1 micron, as shown in Fig. 4.9. However, the entire shift is well

within the Airy disk.

Chapter 4. ADC 113

Figure 4.4: ADC unit has two identical cemented prisms. Each of which hasthree components as shown (Dimensions are in mm).

Figure 4.5: ADC position in iRobo-AO system.

From Fig. 4.10, it is clear that about 90% energy is encircled with in the Airy

Disk.

4.3 Optical Alignment

The ADC prism assemblies are placed in the collimated beam produced by OAP3

at the pupil plane. The pupil plane and an image of this pupil shows the structure

of the DM clearly. Each ADC triplet is attached to a rotating stage. The stage is

screwed to a bracket and put on a Four-Axis (X, Y, θx, θy ) mount. This mount

Chapter 4. ADC 114

Figure 4.6: Illustration of the relation of prism angle to zenith angles for threecases, A, B and C with prism angles 0◦, 30◦, 90◦ and zenith angles 65◦, 61.8◦

and 0◦ in sequence.

� ��

��

�

��

��

��

��

��

��

��

��

��

��

��

��

��

�

�

�

�

�

��

��

��

��

��

��

Figure 4.7: The optimal prism angle and the rms spot size for three workingtemperatures, as function of zenith angle.

is identical to the one used for Deformable Mirror too. A back reflection test is

carried out to ensure that each ADC triplet’s surface is normal to the optic axis.

4.4 Relation between prism angle and dispersion

The atmospheric angular dispersion between wavelenghts λmax and λmin at a par-

ticular location and object position can be defined as

Chapter 4. ADC 115

0.5 1.0 1.5 2.0

Wavelength (µm)

0

2

4

6

8

10

12

14

Dev

iati

on(µm

)

z=0o

z=20o

z=40o

z=60o

z=65o

Figure 4.8: Deviation of monochromatic spot position over the working wave-length band as a function of different zenith angle (z) at T:280 K

��

��

��

��

��

��

��

��

��

��

Figure 4.9: ADC PSF at z : 65o (Zemax). It is well within the Airy Disc(Diameter: 24.52 µm, over the entire working wavelength range) but has asym-

metry due to the dispersion.

Datm = ξλmax − ξλmin(4.2)

where ξλ is the net bending of light from upper atmosphere to the observer at

wavelength λ. Assuming p to be the dispersion of individual prisms which are

rotated by ±θ (one prism rotates by +θ and the other rotates by −θ and the

differential angle between the two prisms is 2θ) from a nominal position, it can be

shown using vector addition formula [86] that the dispersion of the ADC is,

Chapter 4. ADC 116

� ��

��

��

��

��

��

��

��

��

��

�

��

��

��

Figure 4.10: ADC Encircled Energy at z : 65o (Zemax), 90% energy is encir-cled with in the Airy Disk marked by the bold line.

Dadc = 2p cos(θ) (4.3)

This expression can be inverted to estimate the prism rotation angle θ for a

particular D, as

θ = cos−1�Dadc

2p

�(4.4)

The analytical method of estimating Datm is discussed in section 4.4.2. To nullify

the atmospheric dispersion, Dadc must be equal to Datm but opposite in sign.

4.4.1 Laboratory performance of ADC

OAP3 produces collimated light of the telescope simulator (given in Section 2.3.6.1

of Chapter 2) which transmits through the ADC. A convex lens of focal length

50 mm is placed in the light path to focus on an SBIG ST9 CCD. A neutral

density filter is used in front of the convex lens to decrease the beam intensity,

which avoids the CCD saturation. The experimental setup is given in Fig. 4.11.

The pixel size of the ST9 is 20 µm. The plate scale of the system at ST9 is

0.361��/pixel, as estimated from Zemax. The centroids and their measurement

errors of the UV and visible spots at different prism angles were calculated using

IRAF. The dispersion is calculated from the separation of the UV and the visible

spots at each prism angle. Then the distance between the two (UV and visible)

Chapter 4. ADC 117

Figure 4.11: Experimental setup for laboratory performance measurement ofthe ADC.

spots are multiplied by the plate scale to get the dispersion in angular units. The

experimental dispersion and the measurement error are calculated as described

below.

To check laboratory performance of the ADC, dispersion data points were ob-

tained both with the experimental set up described above and also with Zemax

between wavelengths 0.355 and 0.556 µm by rotating the prisms from 0◦ to 90◦.

Atmospheric dispersion effects were not considered here. In Fig.4.12 the disper-

sion of the ADC at various prism angles obtained from Zemax (Zemax Data) are

superposed with dispersion data obtained from Equation 4.3 (model, Zemax data)

with p ∼1.9039�� which was again estimated from Zemax. To fit the laboratory

data a dispersion equation in more general form was written as

Dadc = dc+ 2pexp cos(θ + φ) (4.5)

where a dc term ‘dc’ and epoch ‘φ’ were introduced in Equation 4.3. This model

was fitted (model, Exp. Data) to the (Experimental data) from which we obtain

Chapter 4. ADC 118

0 20 40 60 80

Prism Angle (degree)

0

1

2

3

4

5

AD

CD

ispe

rsio

n(a

rcse

c)Zemax datamodel, Zemax Datamodel, Exp. DataExperimental data

Figure 4.12: Comparison of laboratory performance of ADC with Zemax.

pexp ∼ 1.548543�� ± 0.057332��, φ ∼ 0.557698◦ ± 2.525664◦ and dc ∼ 0.936802�� ±0.155637��.

The difference between the experimental data and Zemax prediction as seen from

Fig.4.12 can be attributed to alignment errors, minor difference in refractive indices

of the melt glasses with that of the indices of Zemax glass catalogue and also due

to minor fabrication error particularly in the wedge angles of the prisms.

Error Calculation The centroid and their measurement error of all the UV

and visible spots at different prism angle are done using IRAF ‘apphot’ package,

‘phot’ task. Let x1, δx1, y1, δy1, are the x and y centroid with error of UV spot and

x2, δx2, y2, δy2, are the x and y centroid with error of visible spot. The dispersion

(R) of the two lasers through the ADC is the distance between the UV and the

visible spot. Therefore,

R =�

(x1 − x2)2 + (y1 − y2)2 (4.6)

The measurement error (δR) is given by [100]

δR =

�� ∂R

∂x1

δx1

�2

+� ∂R

∂x2

δx2

�2

+� ∂R

∂y1δy1

�2

+� ∂R

∂y2δy2

�2(4.7)

Now from Equation 4.6,

Chapter 4. ADC 119

∂R

∂x1

δx1 = −(x1 − x2)√R

δx1

∂R

∂x2

δx2 =(x1 − x2)√

Rδx2

∂R

∂y1δy1 = −(y1 − y2)√

Rδy1

∂R

∂y2δy2 =

(y1 − y2)√R

δy2

(4.8)

δR =

��(x1 − x2)√R

δx1

�2

+�(x1 − x2)√

Rδx2

�2

+�(y1 − y2)√

Rδy1

�2

+�(y1 − y2)√

Rδy2

�2

(4.9)

Equation 4.9 is used to estimate the dispersion measurement error in Fig. 4.12.

The centroid

4.4.2 Estimation of atmospheric dispersion

Atmospheric dispersion is calculated in two ways and given below.

4.4.2.1 Fixed atmospheric parameters

Following Smart (1977)[101] we assume an atmospheric dispersion model

Datm ∝ tan(z) (4.10)

Equating Equation 4.10 to Equation 4.3, we get

cos(θ)

tan(z)= k (4.11)

where k is the proportionality constant. Rearranging,

θ = cos−1[k tan(z)] (4.12)

Chapter 4. ADC 120

� ��

��

��

��

��

��

��

��

Figure 4.13: Mean value of the constant k at 280 K.

� ��

��

��

��

��

��

��

��

��

��

��

��

Figure 4.14: Difference of prism angle obtained from Zemax and model atT=280 K, using k=0.466.

The values of k at different zenith angles are calculated using the optimised data

points ( θ for various z) of Fig.4.7 in Equation 4.11. The mean value of k was

estimated to be 0.466 for T=280 K (Fig. 4.13). Fig. 4.14 shows a good match

between Equation 4.12 and the Zemax data points of Fig. 4.7.

Similar behaviour is noticed at other temperatures but with different values of k

(Fig. 4.15 and 4.16). As the value of k changes (Fig. 4.17, 4.18) with varying

temperature throughout night, this method makes it difficult to accurately esti-

mate Datm and therefore, it is necessary to look for a more generic method of

Chapter 4. ADC 121

� ��

��

��

��

��

��

��

��

��

��

��

��

��

��


� ��

��

��

��

��

��

��

��

��

��

��

��

��

��


estimating Datm using atmospheric parameters only and also taking into account

their real-time variation.

4.4.2.2 Varying atmospheric parameters

Here we estimate Datm following the atmospheric model proposed by Sinclair

(1982)[102] and C.Y. Hohenkerk, A.T. Sinclair, NAO technical note [103]. The

assumptions of the atmospheric model are as follows.

Chapter 4. ADC 122

� ��

��

��

��

��

��

��

��

��

Figure 4.17: Mean value of the constant k=0.445 at T=293 K.

� ��

��

��

��

��

��

��

��

��

��

Figure 4.18: Mean value of the constant k=0.430 at T=303 K.

There is a constant rate of decrease of the temperature up to the tropopause, which

is about 11 km height. The temperature remains constant above the tropopause

(in the stratosphere). The perfect gas law holds for the combined mixture of dry

air and water vapour, as well as separately, i.e. the dry air and water vapour. The

atmosphere holds the hydrostatic equilibrium. There is constant relative humidity

throughout the troposphere and equal to its value at the observer.

The mathematical expression for describing the bending of a light ray through the

Earth’s atmosphere is given below. The light ray path through the atmosphere, to

an observer at O, at r0 distance from the centre of the Earth is shown in Fig. 4.19.

Let us consider a general point P along the light path which is at zenith angle z

Chapter 4. ADC 123

Figure 4.19: Ray path through atmosphere[103].

and r radial distance from the centre of the Earth. The refractive index of two

layers of the atmosphere at distances r + dr and r are n and n+dn respectively.

The Snell’s law of refraction predicts the total bending of the ray relative to a

fixed direction as the ray passes though the r+dr to r layer of atmosphere is tan

z dn/n. Thus the total bending of the light ray is given by,

ξ =

� n0

1

tan(z)

ndn =

� r0

∞

tan(z)

n

dn

drdr (4.13)

where n0 is refractive index at observer’s site at a distance r0 from the centre of

Earth. The refractive index outside the atmosphere is one at infinite distance from

the Earth. The atmospheric dispersion can be calculated from Equation 4.2 using

Equation 4.13.

All the atmospheric and object parameters are stored in a file. These can be up-

dated at regular intervals and are called by the computer program, that calculates

Chapter 4. ADC 124

� ��

��

�

��

��

��

��

��

��

��

��

��

��

��

��

Figure 4.20: Comparison of computed data following NAO technical Notewith Zemax at T=280 K.

the required prism rotation angle. The program can also accommodate all ob-

servatory related parameters, thus making it versatile enough to be used at any

observatory. Computing Datm for a set of atmospheric parameters (as in section

4.1) and using Equation 4.4 with p ∼2.266�� for the entire working wavelength

range we obtain θ for various z as shown in Fig. 4.20. To accommodate for the

slight mismatch 2 at higher values of z, we slightly modified the expression for A3

in the troposphere region [103] with a factor 0.92257. The final expression can be

written as

A = 0.92257�287.604 +

1.6288

λ2+

0.0136

λ4

� 273.15

1013.25(4.14)

Fig.4.21 now shows a good match between the computed values and Zemax data

points. We adopted this algorithm to drive the iRobo-AO ADC unit.

2 This mismatch is unrelated to the results of the laboratory measurements mentioned inLaboratory performance of ADC paragraph.

3The variable A is part of the expression (Cauchy’s equation) of the refractive index of air inthe troposphere.

Chapter 4. ADC 125

� ��

��

�

��

��

��

��

��

��

��

��

��

��

��

��

Figure 4.21: Comarision of computed data following NAO technical Note withZemax at T=280 K after modification.

4.5 ADC Tracking

4.5.1 Effect of telescope derotator

When an altitude azimuth telescope (as in IGO) is pointed exactly towards the

North (South) direction (azimuth angle 0 or π), the elevation axis moves along a

line of constant RA and a change in telescope elevation produces only a change

in declination. For any other azimuth orientation there is some angular difference

between a line of constant RA and the line traced in the sky by moving the

telescope in elevation; this angle is known as the parallactic angle. Thus the

parallactic angle is the angle between a line of constant azimuth and a line of

constant RA. Lines of constant azimuth converge at the zenith and lines of constant

RA converge at the projection of the Earth’s axis on the sky.

Thus, when an alt-azimuth telescope tracks an object field, the image of that field

at the telescope focal plane rotates with time. The object field orientation on the

focal plane is kept steady by moving the Cassegrain derotator by an angle equal

to the change in the parallactic angle [104].

At the Cassegrain focus of an alt-azimuth telescope, the mid-angle of the two

prisms has to remain fixed relative to the telescope tube, as the dispersion direction

is always perpendicular to the elevation axis. As the ADC unit is located behind

the derotator the entire unit will rotate relative to the telescope tube. Prism 1

Chapter 4. ADC 126

and the Prism 2 are on mounted on two independent rotation stages. So to keep

the mid angle of the prisms fixed relative to the dispersion axis, we have to rotate

both the prisms in exactly the opposite direction as the derotator by an amount

equal to the parallactic angle. This is in addition to the differential prism angles

required for dispersion correction. The prism angle (θ) is the theoretical angle of

the prism calculated from equation 4.12. The net mechanical rotation angle of the

stages of Prism 1 and Prism 2 would then be [105]

Ω1 = θ − ωpa + offset1 (4.15)

Ω2 = −θ − ωpa + offset2 (4.16)

respectively, where ωpa is the parallactic angle and offsets (offset1, offset2 of the two

prisms stages) are the difference between the actual mechanical and theoretical

estimated prism angle (Ω − θ) corresponds to, when ωpa = 0, that is when the

object crosses the meridian. The signs of the angles in Equation 4.15, 4.16 are

best estimated on-sky. A computer program estimates ωpa and θ and updates Ω

at a rate discussed in the next section.

4.5.2 Rate of rotation

As it is not desirable for the prisms to lag behind in time from the stipulated

positions an estimate of the optimal rotation rate needs to be done. For k = 0.466,

from Equation 4.12 and Smart [101] we obtain4

dΩ1

dt=

dθ

dt− dωpa

dt(4.17)

dθ

dt=

dθ

dz

dz

dtdθ

dz= − k sec2(z)�

1− (k tan(z))2

dz

dt=

15

3600

�1 +

1

365.2422

�sinA cosφ ◦/sec

dθ

dt= − 15

3600

�1 +

1

365.2422

� � k sec2(z)�1− (k tan(z))2

�sinA cosφ ◦/sec

(4.18)

4 The rate of change of hour angle is 3600

23h56m04.0905s= 15

3600 (1 + 1365.2422 )

◦/sec, neglectinghigher order terms.

Chapter 4. ADC 127

� ��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

Figure 4.22: Rate of rotation of the ADC prism for various azimuth and zenithangles.

anddωpa

dt= − 15

3600

�1 +

1

365.2422

�cosφ cosA cosec(z) ◦/sec (4.19)

dΩ1

dt=

1

240

�1 +

1

365.2422

��− k sec2(z)�

1− (k tan(z))2sinA + cosA cosec(z)

�cosφ ◦/sec

(4.20)

where A and φ are the azimuth angle and latitude. Using Equations 4.15, 4.18

and 4.19, an estimate of the rate of rotation for one prism stage (Equation 4.20) is

shown in Fig. 4.22 for T=280 K, where the mean value of k is larger as compared

to T=293 K and T=303 K. The other prism angle (Ω2) changes in a similar fashion

but with opposite sign. It is seen that the required maximum rate of rotation is

∼ 0.28◦/sec, while the default speed of the ADC rotational stages have been set

to 2.0◦/sec which is much above the estimated maximum value.

The ADC moves at the set default maximum speed quickly, stop, and wait about

3 seconds until the next update. The maximum mismatch of around 0.84o(=

0.28o × 3) between the required and achieved position occurs at the maximum

zenith angle (z) ≈ 65o in the 3 second interval. The rms and geometrical spot

radius changes are well within the Airy disc due to the mismatch up to ±2o prism

angle (Figure 4.23), which justifies the 3-second update frequency.

Chapter 4. ADC 128

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

Figure 4.23: Spot variation due to mismatch of the required and achievedprism angle at 65o zenith angle.

4.6 ADC Control Software Architecture

The source code for ADC control software is written in ‘C’, as a module. This

module can be easily integrated with Robo-AO software[85] and can communicate

with the Telescope Control System (TCS) to know the present zenith angle and the

parallactic angle. The software Architecture is given in Figures 4.24 and 4.25. The

two pauses in the flow chart for human intervention, can be removed for complete

automation. This module calculates the required prism angle and drives the high

precision rotary positioner (Fig. 4.26) to place the ADC prism sub-assemblies at

the right orientation. At first, communication is established between the stages and

the computer through a USB interface using the connector is done with vendor-

supplied APIs5. Then the atmospheric parameter sensitive ADC prism angle is

computed. One mount is rotated in clockwise and the other anti-clockwise. As

explained in Section 2.3.5.5 the half-wave plate in front of the range gate system

is also mounted on an identical rotating stage. A master controller controls both

the ADC and the half-wave plate. The master code developed by combining the

ADC and half-wave plate code by multithreading governs these two subsystem

independently as per their governing algorithms.

5Application Programming Interface

Chapter 4. ADC 129

Yes No

Retarder ADC

Calculate Maximum Correctable

Observed Zenith angle ( zmax )

Tp=Tp+3

Wait 3 sec

Get atmospheric parameter Temperature , Pressure, Humidity

|Tp-Tstart|☎�✁

Note: To begin

observation again (Wait

till user press 'yes')

Store Tp: Tstart

Get Exposure Time (from Robo-AO daemon): TExp

Show Message: Telescope Pointing to Target Position

* Get ADC Ready for Operation

Configure MCS 1. Open MCS

2. Enable Sensor 3. Set Frequency (Retain company setting, do not edit)

4. Referencing

Show Message: ADC is Ready for

Operation, Take Exposure!!

Exit ?

Yes

No

Close MCS

Get Present Time: Tp

Proceed Further, Pause1?

Note: Wait till

user press 'yes'

Proceed Further, Pause2?

Note: Sign of angles to be

finalized after calibration

on sky.

Note: ✂✄ ✆✄Exp+ ✝extra , (✞✟✠✡ ✝extra from config file, ✝extra ☛ 1 minutes, editable)

Get obs. z from TCS

Yes

No

Move stage to ☞1 =( ✌ - ✍pa + offset1)o & ☞1 =( -✌ - ✍pa + offset2)

o

Cal. prism ang. ✌0

z<zmax ? Show Message:

Outside zmax Prism ang. ✌0 =0

Get Paralectic angle from TCS: ✍pa0

Get Offset angle from config file : offset1 , offset2

Figure 4.24: ADC Flow Chart (“Get ADC Ready for Operation” sub blockand the description of the symbols used in are given in Fig. 4.25).

Chapter 4. ADC 130

Figure 4.25: “Get ADC Ready for Operation” sub block of the main ADCflow chart (Fig. 4.24), with the description of symbols.

Figure 4.26: ADC and Retarder rotational computerized mounts (P.C. Smar-act website)

Chapter 4 Atmospheric Dispersion Corrector

Documents

Transcript of Chapter 4 Atmospheric Dispersion Corrector