Compound prism design principles, II: triplet andJanssen prisms
Nathan Hagen1,2 and Tomasz S. Tkaczyk1,3
1

Department of Bioengineering, Rice University, Houston Texas 77005, USA2e-mail: nhagen@optics.arizona.edu
3e-mail: ttkaczyk@rice.edu

Received 26 May 2011; accepted 5 July 2011;
posted 12 July 2011 (Doc. ID 148164); published 30 August 2011

Continuing the work of the first paper in this series [Appl. Opt. 50, 4998–5011 (2011)], we extend ourdesign methods to compound prisms composed of three independent elements. The increased degrees of
freedom of these asymmetric prisms allow designers to achieve greatly improved dispersion linearity.
They also, however, require a more careful tailoring of the merit function to achieve design targets,and so we present several new operands for manipulating the compound prisms’ design algorithm.
We show that with asymmetric triplet prisms, one can linearize the angular dispersion such that the
spectral sampling rate varies by no more than 4% across the entire visible spectral range. Doing this,however, requires large prisms and causes beam compression. By adding a beam compression penalty to
the merit function, we show that one can compromise between dispersion linearity and beam compression in order to produce practical systems. For prisms that do not deviate the beam, we show that Janssen
prisms provide a form that maintains the degrees of freedom of the triplet and that are capable of up to
32° of dispersion across the visible spectral range. Finally, in order to showcase some of the design flexibility of three-element prisms, we also show how to design for higher-order spectral dispersion to create a
two-dimensional spectrum. © 2011 Optical Society of America
OCIS codes: 230.5480, 080.2740, 260.2030, 300.6190.

1. Introduction

Compound prisms consist of multiple glass wedges
placed in contact with one another as a single opticalelement. The first paper of this series [1] introduced
basic design principles for doublet and double Amici
compound prisms, concentrating on compact andlinear-in-wavelength dispersive prisms. There weshowed that it is possible to achieve a degree of dispersion previously only thought possible with gratings while maintaining direct-vision geometry. This
characteristic is made possible by the underappreciated advantage that prisms have in design flexibility: with multiple surfaces and multiple glass choices
available, an optical designer has several parameters
to adjust for optimizing prism performance.
0003-6935/11/255012-11$15.00/0© 2011 Optical Society of America
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APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

Here we remove the symmetry constraints of the
double Amici design by designing “triplet” prismsof three independent elements, or maintaining symmetry while adding elements (Janssen prisms) [2].
The additional degrees of freedom in these designs
allow for large improvements in dispersion linearity.
In order to design these prisms, however, additional
constraints need to be added to the design algorithm’s merit function in order to produce practicaldesigns. For example, we show that there is a direct
trade-off between prism symmetry and dispersion
linearity—a trade-off that also causes an unwanted
compression of the transmitted beam. Balancing thisbeam compression against linearity thus requires
adding a beam compression penalty term to the merit function. Other requirements produce additional
penalty terms. Together, these tools for modifying
the merit function allow a user to tailor our custom
software or any commercial optical design softwareto produce practical triplet and Janssen dispersive
prisms.
In order to show the power of these triplet andJanssen prisms, we show prism designs for the sametarget values as in Paper I [1], for which we can directly compare the dispersion linearity with that
achieved by doublet and double Amici systems. We
also modify the design algorithm for achromatic
beam steering prisms to produce higher-order dispersion, in which the spectrum can be made to wrapback on itself one or more times. This property canbe used to real advantage in dual-band spectrometers, in which the spectrum from two different
spectral ranges are simultaneously measured on
the same detector array. One can also use higherorder dispersion together with a cross-dispersing
element to achieve an echellelike two-dimensional
dispersion pattern for increased spectral sampling.
We show some example designs for use across thevisible and near-IR spectrum.
In the third paper of this series [3], we bring together all of the methods discussed below to develop
practical direct-view prism designs for the demanding application of spectral-domain optical coherence
tomography systems.
2. Triplet Prisms

In the first paper of this series [1], we presented amethod for designing doublet and double Amici
prisms. While the double Amici design uses three elements, the first and third elements are identical, andthus it does not have as many degrees of freedom as a
general three-element prism (such as the one shownin Fig. 1) in which all three glasses and all three apex
angles are allowed to differ.
A three-glass prism has sufficient degrees of freedom that the central deviation δ ¼ δðλÞ, angular
spectrum width Δ ¼ δðλmin Þ − δðλmax Þ, and a third
constraint can all (in principle) be satisfied exactly

independent of glass choice. One can, for example,
attempt a design with zero beam displacement(Δyðλref Þ ¼ 0) at some reference wavelength λref . Alternatively, the extra degrees of freedom can be used
simply to further improve on the dispersion linearityor the overall system thinness.
As with the doublet and double Amici designs, one
can use the small angle approximations to develop
linear design equations for triplet compound prisms.
If dispersion linearity is the third design target (afterδ and Δ ), then one method for improving dispersion
linearity is to force the angular separation from thecenter to long wavelengths to be half that of the total
spectrum angular width [4]. This allows for a linear
algebra approach to a solution, as shown below. For athree-element prism, this half-angular width constraint can be represented by the equationδ − δC ¼ ðδ1 þ δ2 þ δ3 Þ − ðδC;1 þ δC;2 þ δC;3 Þ
 
 
 
P1
P2
P3
¼ δ1
þ δ2
þ δ3
;
V1
V2
V3
where P is the relative partial dispersion, δC ¼ δðλC Þ,
and λC ¼ λmax is longest wavelength. The subscripts
1, 2, and 3 represent the three elements comprising
the triplet. If we write the linear approximation to~~ and set δ − δ ¼ Δ=2,
thenthe dispersion Δ as Δ
C
~
Δ
P
P
¼ δ1 1 þ δ2 2 :2V1V2
At this point, if we include the linear equations forthe central deviation and the total dispersion (Eqs. 3
and 4 in Paper I), we have three equations and three
unknowns, namely the center wavelength deviations
δðλÞ of each prism. Writing out the system in terms oflinear equations gives0 δ 
1
1~
1=V 2¼ @ 1=V 1
Δ~
P1 =V 1 P2 =V 2
Δ=2
0 1
δ1
@
¼ H3 δ2 A;
δ3

10 1
δ1
1
1=V 3 A@ δ2 A
δ3
P3 =V 3so that an estimate of the central deviations of theindividual prism elements is obtained by
0

1
0
1
^δ
δ
1
B ^ C~ A;@ Δ@ δ2 A ¼ H−1
3
~
^δ
Δ=2
3

Fig. 1. Ray trace through an example triplet compound prism
where the second element is assumed to be oriented symmetricallywith respect to the normal to the optical axis. The system shown
here has prism apex angles ðα1 ; α2 ; α3 Þ ¼ ð20°; −70°; 60°Þ, indices ofrefraction ðn1 ; n2 ; n3 Þ ¼ ð1:45; 1:80; 1:60Þ, overall height h, beamdisplacement Δy, and axial thicknesses t1 , t2 , and t3 . The input
ray has angle θ0 ¼ −10° such that δ ¼ 15°.

giving the prism apex angles
1
1 0 ^
δ1 =ð
n1 − 1Þ
α1
C@ α2 A ¼ B@ ^δ2 =ðn2 − 1Þ A:
α3^δ =ð3 n3 − 1Þ
0

1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5013

While the linear equations can provide a goodstarting point when the system requires only weak
dispersion (and thus small prism angles αi ), theabove approach works well only when the target
spectral dispersion is quite small. Specifying thisquantitatively requires one to also define thespectral range, but one can say that, for designs
using the full visible spectrum, Δ < 1° satisfies thisrequirement.
The nonlinear equation for the deviation angle δ ofa general three-element prism is given by concatenating the refraction equations
9>
θ1 ¼ θ0 þ β1 ;θ03 ¼ arcsin nn23 sin θ3 ; >
>
>
>


>
>
1
0
0
>
=
θ1 ¼ arcsin n1 sin θ1 ; θ4 ¼ θ3 − α3 ;
0
0
; ð1Þ
θ2 ¼ θ1 − α1 ;
 θ4 ¼ arcsinðn3 sin θ4 Þ; >
>
>
n
>
>
θ02 ¼ arcsin n12 sin θ2 ; θ5 ¼ θ04 þ γ 3 ;
>
>
>
;
0
θ ¼θ −α ;


3

2

2

where β1 ¼ α1 þ 12α2 , γ 3 ¼ α3 þ 12α2 , and δ ¼ θ0 − θ5 .
(See Fig. 1 for an illustration defining the angles
used here.)
In order to optimize a triplet prism design, we use
the same nonlinear optimization algorithm, but now
perform a search over all three-glass combinations in
the glass catalog. The total number of glass combinations to search is thus T 3 ¼ ðN − 1ÞðN − 1ÞðN − 2Þ, aformula derived from the fact that the first element
can be any glass except AIR, the second element canbe any glass except that of the first element, and thefinal element can be any glass except that of the former element or AIR. For the Schott glass catalog,
N ¼ 101 (including air as a glass type), so that a fullsearch requires T 3 ¼ 990 000 designs. On a standard
2 GHz desktop computer, this currently requires 18 h
to perform, which can be intolerably long in manycases. If we wish to speed up the search process, we
can cull the glass catalog of “similar” glasses by deleting glasses that are within a given radius in theglass chart (that is, glasses that have both similarcentral refractive index nðλÞ and similar dispersion
Δn ¼ nðλmin Þ − nðλmax Þ). This sacrifices optimalityof the design but retains most of its performanceand greatly reduces the time needed to obtain aset of designs. If we select the parameters of theglass-culling algorithm to reduce the Schott catalog
to 23 “unique” glasses, then T 3 ¼ 10; 164, and thesearch time is reduced to 30 min.
Up to this point, the merit functions we have beenusing share three basic terms, and so we can call thisthe “base” merit function M 0 , where
M 0 ¼ ðδ − δ Þ2 þ ðΔ − Δ Þ2 þ Θ;
which involves a quadratic penalty term for errors in
the central deviation δ and the overall dispersion Δ,
plus a term Θ penalizing designs with interface angles that are too large:
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APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

Θ¼

I 
X
0
i¼1

ðθlimit − θi Þ2

θi < θlimit:
θi > θlimit

A triplet prism contains enough degrees of freedomthat one can locate a continuum of design parameters for which M 0 ¼ 0. In order to minimize the nonlinearity in the dispersion, we can thus add anotherpenalty term that selects from among the M 0 ¼ 0 solutions the one that minimizes the dispersion nonlinearity. This gives the new merit function
M nl ¼ M 0 þ wnl NL;
where wnl ¼ 25 is the weight of the nonlinearity term
relative to those of M 0, and the nonlinearity operand
NL is defined asZ  2 d δ 
NL ¼  2 dλ:dλTable 1 shows several triplet prism designs achieving 1° and 4° of dispersion—values chosen so that theprism designs can be compared directly with the double Amici designs of Paper I. Although the algorithm
allows the first and last elements to share the same
apex angle and material, as in a double Amici design,none of the best-performing designs use a symmetriclayout, indicating that substantial performance improvement can be achieved by deviating from thesymmetry of the double Amici. Compared with thecorresponding double Amici designs of Paper I (seeTable 2 in Paper I [1]), the 1° and 4° designs show
a twofold and fivefold improvement in λ linearity.
However, if we look at the layout for these designs[see the layout for design 1b in Fig. 2(b)], we cansee that there is a trade-off in prism size as wellas the introduction of large beam displacement
and compression.
Note that in all tabulated data, we remove the prefixes on the glass names (i.e., N-BK7 is written as
BK7) in order to keep the tables compact. In somecases, we have also abbreviated names, such as
LITHOSIL-Q/LITHQ and LITHOTEC-CAF2/CAF2.
Unlike all of the prism designs up to this point, thetriplet designs listed in Table 1 use different merit
functions for each section of the table: the designsof sections (a) and (b) are optimized for best dispersion linearity, while those of section (c) are optimized
for minimum thickness.
The asymmetric geometry of the layouts in section(b) is remarkably successful in reducing dispersion
nonlinearity, to the extent that the spectral sampling
rate of design 1b varies by only 13% across the entire
spectrum. It may seem counterintuitive that the
spectral sampling ratio (SSR) [1] achieved by the
4° designs is closer to 1 than that of the 1° designs,
but due to nonlinearities in the dispersion, the apex
angles of the 4° prisms cannot be simply scaled down
by a factor of 4 to achieve 1° dispersion prisms of comparable SSR. In section (c) of the table, we see that

Table 1.

Glass 1Glass 2

1a
2a
3a
4a
5a

LASF41
LAF34
LASF31ALASF31ALASF31A

KZFS4
KZFS2
KZFS4
KZFS4
KZFS11

1b
2b
3b
4b
5b

LAK34
LAK33ALASF43
LASF43
LASF9

F5
BASF2
SF1
SF1
SF57

1c
2c
3c
4c
5c

CAF2
CAF2
LITHQ
LITHQ
LITHQ

SF56A
SF4
SF11
SF56A
SF14

Best-Performing Prism Triplets for θ0  0, Optimized Over the Schott Glass Cataloga

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)merit func ¼ linearity, λ range ¼ 400−700 nm, Δ ¼ 1°
LITHQ
46.11
−149:52
100.18
0.000
1.000
CAF2
44.63
−146:85
98.09
0.000
1.000
LITHQ
42.28
−145:71
107.20
0.000
1.000
CAF2
43.79
−141:5097.84−0:012
0.988
K7
44.11
−150:57
90.21
0.000
1.000
merit func ¼ linearity, λ range ¼ 400−700 nm, Δ ¼ 4°
CAF2
−80:90
33.52
68.39
0.005
3.992
CAF2
−79:70
30.10
70.11
0.000
4.000
CAF2
−81:8340.17
63.90−0:015
3.992
PK52A−84:18
38.35
64.80
0.004
3.997
CAF2
−85:55
41.09
63.86
0.003
4.000
merit func ¼ thinness, λ range ¼ 400−700 nm, Δ ¼ 4°c
LITHQ
57.29
−61:66
57.28
0.014
4.085
CAF2
57.29
−62:3457.35−0:005
3.907
LITHQ
57.29
−63:22
57.29
0.012
4.070
LITHQ
57.29
−63:2557.29−0:003
3.929
CAF2
57.29
−63:40
57.29
0.006
4.017

NL (×104 )

SSR

Kb

0.095
0.158
0.164
0.232
0.242

1.04
1.08
1.09
1.13
1.12

0.04
0.11
0.06
0.12
0.07

0.281
0.283
0.303
0.317
0.412

1.13
1.48
1.37
1.40
1.36

0.20
0.21
0.24
0.25
0.27

2.519
2.271
2.710
2.503
2.556

9.08
8.26
10.05
9.74
9.26

1.00
1.00
1.00
1.00
1.00

a
All of the designs are nondeviating (δ ¼ 0) and constrained to interface angles of 65° or less. Figure 2 shows the dispersions of all fifteen
prisms listed below.
b
For an explanation of column K, see Sec. 3 below.
cNote that section (c) of the table uses a different merit function than the previous two sections, as indicated.

despite the added degrees of freedom available to thetriplet, when searching for the most compact prisms
the algorithm has selected a double Amici configuration in each case. (Designs 1c, 2c, and 5c are not exact
double Amici prisms, but they differ from the latter

only in selecting a neighboring glass or by adjustingslightly the prism apex angle.)
In order to see how to obtain the values reported
in Table 1 from the curves in Fig. 2, one can selecta curve in the figure and choose the value of theFig. 2. (Color online) Dispersion gradients dδ=dλ and example layouts of the triplet designs of Table 1. Note that an ideal linear dispersion
would produce a horizontal line on these plots and that the three layouts are drawn to the same relative scale.
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5015

gradient corresponding to its maximum and minimum points. The ratio of these two gives the SSR
value. Since the optimization algorithm is better behaved when given quadratic error terms, the designcode uses the NL value rather than SSR inside themerit function. The NL value can be obtained by firstlocating a least-squares fit line to the curve, and thensumming the squared deviations of the curve fromthat line. The result is proportional to the tabulated
NL values.
From Table 1 we see that the NL and SSR values do
not follow one another monotonically. Since the NL
operand measures the quadratic distance while the
SSR operand measures the absolute deviation—
quantities that are nonlinearly related—the two tend
to increase and decrease together, but not monotonically. For the nonlinearity-penalized merit function
M nl , we use the NL operand rather than SSR because
the optimization code has better convergence behavior for a quadratic penalty than a nonquadratic one.
3. Constraining Beam Compression

The triplet prism layout shown in design 1b of Fig. 2
indicates a design that requires the prism to be substantially larger than the beam width and that compresses the beam to ∼20% its original width. Beamcompression is generally unwanted because it can
produce a loss in spectral resolution due to reductionin the beam diameter, and therefore a loss of NA in
the spectral spread dimension. Thus, a designer will
want to compromise the dispersion linearizationachieved by the asymmetric layout with a reductionin the beam compression. This can be accomplishedthrough the addition of a quadratic penalty term to
the merit function, so that the new merit function becomes M nl−K ¼ M nl þ wk ðK − 1Þ2, where K is thebeam compression calculated by [5]Table 2.

Glass 1Glass 2

K¼

N
Y
j cos θn j
n¼1

ð2Þ

j cos θ0n j

and wk ¼ 0:01 is the weight of the additional term
relative to those of M nl. In the formula for K, N is
the number of prism elements, and the angles θnand θ0n are defined as in Fig. 1. The ðK − 1Þ2 term penalizes any beam compression that deviates from thetarget value of K ¼ 1.
Using this modified merit function and rerunning the algorithm for the same target values as in
Table 1, one gets the prisms listed in Table 2. We see
the effect of the additional term in the merit functionin the improvement of K, but we also see that thishas slightly weakened the improvement in the linearity, as we would expect. The designs here thus
show improved performance over the equivalent
double Amici (see Table 2 in Paper I [1]), but reduced
linearity in comparison to the equivalent unconstrained triplet designs of Table 1. The additional
constraint on K also increases the algorithm’s calculation time—a product of the fact that tighter constraints on the design space force the optimization
iteration to make smaller steps.
For the Δ ¼ 1° designs, we see that the new
constraint on beam compression has improved the
K value, and that the designs’ NL values remain better than those of the corresponding double Amici
prism designs. However, for the Δ ¼ 4° systems,
the new constraint has actually caused the algorithm
to select a double Amici layout (or something very
near to one) as the best available. This implies thatwhile the Δ ¼ 1° design space has enough degrees of
freedom to prevent severe trade-offs, the confined design space available to a Δ ¼ 4° prism does not overthis spectral range. For the larger dispersion prisms,Best-Performing Prism Triplets for θ0  0, Optimized Over the Schott Glass Cataloga

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)NL (×104 )

SSR

K

1.000
1.000
1.000
1.000
1.000

0.177
0.179
0.181
0.192
0.192

2.53
2.63
2.60
2.81
2.82

1.00
1.00
1.00
1.00
1.00

3.984
3.903
3.982
3.963
3.983

1.762
1.766
1.785
1.792
1.806

5.74
5.91
6.40
5.93
5.97

1.00
0.99
1.00
1.00
1.00

3.967
3.995
3.994
3.938
3.966

0.736
0.768
0.773
0.777
0.780

2.56
2.63
2.65
2.69
2.69

1.00
1.00
1.00
0.99
1.00



1a
2a
3a
4a
5a

CAF2
CAF2
CAF2
CAF2
CAF2

LASF47
LASF44
LASF46A
LAF33
LAF35

1b
2b
3b
4b
5b

CAF2
CAF2
CAF2
CAF2
CAF2

LAK33ALAK33ALASF31A
LAK33A
LAF34

1c
2c
3c
4c
5c

CAF2
CAF2
CAF2
CAF2
CAF2

LASF40LASF46ALASF9LASF45
LASF45

merit func ¼ M nl-K , λ range ¼ 400−700 nm, Δ ¼ 1°
87.03
−103:46
92.01
0.000
91.03
−101:29
93.27
0.000
84.61
−104:96
93.17
0.000
89.08
−104:36
93.11
0.000
86.18
−107:24
92.22
0.000
merit func ¼ M nl-K , λ range ¼ 400−700 nm, Δ ¼ 4°
CAF2
94.94
−94:30
94.94
0.000
PK52A
94.14
−99:1794.92−0:004BAF4
92.84−93:87
92.91
0.000
FK51A
94.58
−98:4394.90−0:001
CAF2
91.94−90:4391.94
0.000
merit func ¼ M nl-K , λ range ¼ 475−625 nm, Δ ¼ 4°
CAF2
99.51
−89:9899.62−0:001
CAF2
92.60−80:2392.60−0:000
CAF2
94.28−85:24
94.28
0.000
PK52A
96.57
−95:5596.61−0:001
FK51A
96.92−94:9396.80−0:001
SF2
F5
SF56A
F2
BAF4

a
All of the designs are nondeviating (δ ¼ 0) and are constrained to interface angles of 65° or less. Figure 3 shows the dispersions of all 15
prisms.

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APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

Fig. 3. (Color online) Dispersion gradients dδ=dλ and example layouts of the K-constrained triplet designs in Table 2, together with three
example layouts. An ideal linear dispersion would produce a horizontal line on these plots. The layout figures were generated by replicating the design parameters in Zemax [6] and selecting the thicknesses of each prism.

the only practical means of maintaining beam size isthus to choose a symmetric layout.
Another curious feature of the 4° designs in Table 2is the prominence of CAF2 as a glass choice. This is
the least dispersive glass in the Schott glass catalog,
allowing for the partial cancellation of glass dispersion nonlinearity with nonlinearity in the surface incidence angle to have maximum effect.
When designing a dispersive prism, one may want
to place more emphasis on dispersion linearity versus beam compression than we have here, in whichcase adjusting the weight parameter wk to a smaller
value will have the desired effect, producing a compromise between the designs of Table 2 and those of
Table 1.

avoiding this complexity while minimizing beam displacement is to use a symmetric prism system. A
double Amici is the simplest example. If we wish totake advantage of the extra degrees of freedom available to a general triplet prism, however, we can moveto a five-element Janssen prism, in which two pairsof identical wedge elements surround a central element (see Fig. 4).[2]As in the double Amici, one achieves zero beam deviation in a Janssen prism by ensuring that rays in
the central element travel parallel to the optical axis.
Another advantage of the increased number of elements in a Janssen prism is that dispersion can beincreased beyond that achievable by a double Amici.

4. Janssen-Type Five-Element Three-Glass SymmetricPrisms

A prism designer may want to have a zero displacement system, which requires adding a penalty term
proportional to ðΔyÞ2 to the merit function. Unfortunately, this introduces a large increase in complexity
to the design algorithm, as obtaining the beam displacement value requires specifying the pupil sizeand location, the marginal ray angle at the pupil,
and the individual thicknesses of each prism element. Rather than restricting the prism model to
one angular dimension, one thus needs most of theapparatus of a full ray tracing engine in order to complete the calculation. A straightforward means ofFig. 4. Ray trace through an example Janssen prism, where thecentral element is assumed to be oriented symmetrically withrespect to the optical axis normal. The diagram shown here has
input ray angle θ0 ¼ 0° and a ray deviation angle δ ¼ 0° for clarity.
The beam displacement is zero, Δy ¼ 0.
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5017

To illustrate this type of design and its potential, we
once again establish the nonlinear equation for thedeviation angle δ by concatenating the refraction
equations9>
θ04 ¼ arcsin nn32 sin θ4 ; >
>
>


>
>
>
1
0
0
>
θ1 ¼ arcsin n1 sin θ1 ; θ5 ¼ θ4 − α4 ;
>
>

>
>
>
n2
0
0
=
θ5 ¼ arcsin n1 sin θ5 ; >θ 2 ¼ θ 1 − α1 ;


; ð3Þ
>
θ02 ¼ arcsin nn12 sin θ2 ; θ6 ¼ θ05 − α5 ;
>
>
>
>
0
>
θ3 ¼ θ02 − α2 ; θ6 ¼ arcsinðn1 sin θ6 Þ; >
>
>
>
n
>
θ03 ¼ arcsin n23 sin θ3 ; θ7 ¼ θ06 þ β;
>
>
>
;
0
θ 4 ¼ θ 3 − α3 ;
θ1 ¼ θ0 þ β;

where β ¼ α1 þ α2 þ 12α3 and δ ¼ θ0 − θ7 (see Fig. 4 foran illustration defining the angles used here). With
this setup, we can search for optimal designs with thesame methods as those used to model triplets (seeSection 2): use nonlinear optimization to locate asolution together with a brute force search over all
three-glass combinations in the catalog. Table 3
shows the best-performing prisms among the resulting designs.
The prisms of Table 3 show improvements in dispersion linearity over the equivalent K-constrained
triplet designs in Table 1, but cannot achieve the linearity achieved by the unconstrained triplets. Moreover, we see that the symmetry of the Janssen prisms
has, without enforcing the beam compression constraint explicitly, maintained the beam width in allof the listed designs.
All of the designs of Table 3 use a low-dispersion
glass for the outer elements, and a high-dispersion
glass for the thin element, so that the central element lies somewhere between the two in its Abbe
number. Taking a look at the layouts for designs
Table 3.

Glass 1

1a, 1b, and 1c shown in Fig. 5, we can see an interesting feature in the low-dispersion design (1a),where elements 2 and 4 (the thin wedges placed between the central element and the outer elements)
deflect the rays by more than the outer elements
do and actually change the sense of the deflection angle. Thus, the Janssen prisms do not use the added
wedge elements to “soften” the ray deviation in theway that a good multielement lens design softens
the bending at each surface by spreading out the
ray-bending angle across multiple surfaces. Rather,
the thin wedge elements increase the overall bending
as a means of inducing a positive amount of quadratic dispersion to cancel some of the inherent negative
quadratic dispersion caused by the glasses’ refractive
index behavior.
If we look at the layouts for the prisms in this set(see Fig. 5), we see that as we go from design 1a to 1b
and 1c the dispersion requirements increase, andthat the thin elements (elements 2 and 4 in the system) have less and less effect on deviating the transmitted ray, so that the system behavior comes toapproach that of a double Amici design. For the higher dispersion systems, the constraints on the design
space have restricted how far we can use these thinelements to improve dispersion linearity. Nevertheless, if we compare the linearity achieved by the designs of section (b) of Table 3 with the equivalent
triplet designs (section (b) of Table 2), then we cansee a significant improvement in NL value. Moreover, all of the Janssen prism designs have negligible
beam displacement, which can be important in some
applications.
One feature where the Janssen prisms completely
outperform the triplet designs is in high-dispersion
systems. In Paper I we explored the dispersion limitsof double Amici prisms and found that we canachieve angular dispersions of up to ∼23° across thevisible spectrum. Performing a similar procedure for

Bst-Performing Janssen Prisms for δ  0° and θ1  0 Optimized Over the Schott Glass Cataloga

Glass 2

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)NL (×104 )

SSR

K

0.131
0.146
0.147
0.151
0.156

1.94
1.91
2.07
2.02
2.29

1.00
1.00
1.00
1.00
1.00

0.193
0.205
0.223
0.241
0.242

3.38
3.56
3.76
4.07
2.82

1.00
1.00
1.00
1.00
1.00

2.615
2.882
2.918
2.950
2.981

4.55
4.90
4.84
5.18
5.24

1.00
1.00
1.00
1.00
1.00

merit func: ¼ linearity, λ range ¼ 400−700 nm, Δ ¼ 1°

1a
2a
3a
4a
5a

CAF2
CAF2
CAF2
CAF2
PK52A

LASF40
BASF2
SF67
LASF40
LAFN7

1b
2b
3b
4b
5b

CAF2
CAF2
CAF2
CAF2
CAF2

SF6
LASF9
SF14
SF66
SF4

1c
2c
3c
4c
5c

CAF2
FK51A
CAF2
PK52A
PK51

SF8
LASF44
LASF44
LASF31ALASF31A

a

LAK8
90.57
12.48
−129:12
0.000
0.998
LAK14
88.88
10.85
−116:08−0:001
0.997
KZFS2
79.35
2.88
−122:58
0.000
0.998
LAF34
88.58
17.95
−129:78
0.000
0.998
LAK33A
89.95
10.83−122:57
0.000
0.997
merit func: ¼ linearity, λ range ¼ 400−700 nm, Δ ¼ 4°
LASF31A
101.7813.05−110:29
0.001
3.998
LASF31A
99.88
20.81−125:26
0.000
3.999
LASF4099.57
17.87−120:36
0.002
3.996
LASF41
98.86
5.18−101:17
0.000
3.998
LASF31A
103.0817.07−114:70
0.000
4.000
merit func: ¼ linearity, λ range ¼ 400−700 nm, Δ ¼ 8°
LASF31A
103.57
14.59−108:33
0.005
7.958
BAF4
102.02−101:17128.30
0.003
7.971
BAF4
102.53−90:30105.53
0.003
7.971
BASF2
98.79
−89:80106.15−0:010
7.949
BASF2
98.96
−93:75111.460.000
7.954

Dispersions and layouts of the prisms are shown in Fig. 5.

5018

APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

Fig. 5. (Color online) Dispersion gradients dδ=dλ and example layouts of the Janssen prisms shown in Table 3. An ideal linear dispersion
would produce a horizontal line on these plots.

the Janssen prisms, we use the algorithm first to
search for the best glass combinations for high dispersion. We then use this combination to attempta series of prism designs for progressively greater
dispersion, giving the results shown in Fig. 6. Here
we see that the LAK34/SF66/LITHQ glass set canachieve up to 32:1° of dispersion (roughly equivalent
to a 1400 lines=mm grating) across the 400–700 nm

Fig. 6. (Color online) Three prism apex angles α1 , α2 , and α3 , together with the central deviation δ and total angular dispersion Δ
achieved by a LAK34/SF66/LITHQ prism. Above Δ ¼ 32:1° (indicated by the greyed region of the plot), the algorithm can no longer
locate a solution that satisfies the design targets, defining the dispersion limit achievable by this glass set.

spectral range. Beyond 32:1°, the design algorithm
fails to find a solution that can simultaneously satisfy the target values of δ ¼ 0 and Δ simultaneously.
5. Higher-Order Spectra

In Paper I, we gave example designs of beamsteering prisms. From the plot of angular dispersion
for these prisms (see Fig. 9 of Paper I), we see thatthe design algorithm effectively suppresses the linear and quadratic terms to the prism dispersion, leaving a cubic shape as the dominant form. In lensdesign, spectrum shape is commonly called “secondary spectrum” (for a quadratic form) or “tertiary spectrum” (for a cubic form). If such a prism were used ina spectrometer, the quadratic shape would cause thedispersion to start in one direction, stop, and then go
back in the opposite direction, so that the spectrum
folds back on itself. While this may not at first appear
to be a useful property, Mooney et al. have shown that
one can take advantage of it to design a dual-band
spectrometer [7] or a spectral pseudoimager [8].
Figure 7 shows the layout, prescription (translated
to our notation), and dispersion of their dual-band
prism for reference.
One can also imagine inserting a cross-dispersing
element together with such a second-order-dispersing
prism, so that the dispersed spectrum travels twice
across the width of the array in a spectrometer in a
U shape, effectively doubling the sampling of the spectrum. The cubic dispersion shown in Fig. 9 of Paper I,
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5019

µ

of the dispersion while maintaining target valuesof δ and total dispersion Δ. The merit function forthis is therefore
M H ¼ M 0 þ wH ð1=NLÞ;Fig. 7. Dispersion of the sapphire–air–ZnS Mooney prism overthe visible and midwavelength IR bands. The prism prescription
and layout are shown in boxed insets.[7]together with a cross-dispersing element, would produce a system in which the spectrum travels threetimes across the length of the detector array. In thismode, such a disperser begins to resemble a lowresolution echelle system. Note that second-order dispersion is also readily achieved with a grating–prism
combination in which the two dispersions are made tocancel one another at two reference wavelengths.
The higher-order dispersion shown in the beamsteering prisms of Paper I is too weak to be usefulfor cross-dispersing spectrometry, and so it is interesting to see if we can take advantage of compoundprisms’ flexibility to intentionally increase the higher-order dispersion. Nelson et al. describe a methodof selecting glasses in order to produce a secondorder spectrum that involves selecting one glass thathas an absorption band just below the instrument’s
spectral range, and a second glass whose absorption
band is just above the spectral range [8]. The two
glasses have strong dispersion of opposite senses
at opposite ends of the spectrum, so the combination
can produce a strong higher-order spectrum. However, it can be difficult to find satisfactory glasses
for a given spectral range. Instead, our approachhere takes advantage of the smooth behavior of
the prism design space and uses the same nonlinear
algorithm to maximize the second-order component
Table 4.

Glass 1

ð4Þ

where wH ¼ 0:1 is the weight we use in our designs.
Table 4 shows ten prism designs optimized for second-order dispersion using two different spectral
ranges (400 nm ≤ λ ≤ 700 nm for section (a), 350 nm ≤λ ≤ 1000 nm for (b)). The resulting dispersions areshown in Fig. 8. (The reason that deviating prisms,δ ≠ 0, were chosen for the designs in this table is that
this greatly relaxes the design constraints, making iteasier to obtain larger values for the higher-order
spectra.) While this very wide spectral band is impractical for most commercial silicon-based detectors,
a detector having high quantum efficiency over thisentire range is possible with a deep-depletion silicondetector array.[9–11]
Each of the five designs of section (a) in the table use
LASF31A (one of the highest-index glasses available)
as the central element. In each design, the first element has a glass of similar dispersion to LASF31A
but a lower index, while the third element uses a
low-dispersion, low-index glass. In section (b), whichuses a much wider spectral range, all the designs except for 4b use low-index, low-dispersion glasses exclusively for all three elements. Design 4b uses two
high-index, high-dispersion glasses (SF6 and PBF2)
and produces a very different high-order spectrum—
one that has a large third-order component (seeFig. 8).
If used for spectrometry, the total travel of the dispersion spectrum is the main feature of interest. The
travel achieved by the dispersion in the designs ofTable 4 is only ∼0:2° for the visible spectrum designs,
but ∼1:8° (0:9° in one direction and 0:9° going back)for the extended spectrum designs. Design 4b isagain the exception here, as it achieves a total travelof about 2:8° (0:9° forward, 0:9° back, and 1:0° forward again).

Best-Performing Triplet Prisms Optimized for Second-Order Dispersion, for θ0  0, (a) the Schott Glass Catalog,
and (b) ZEMAX’s Infrared Glass Catalog

Glass 2

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)Δ1 (deg)

Δ2 ð×1000Þ

Δ3 ð×1000Þ

0.003
0.015
0.020
0.025
0.021

−0:451−0:426−0:424−0:408−0:408−0:054−0:386−0:483−0:565−0:312−0:167−0:194−0:219−0:227−0:07417.246
16.39416.00015.64415.642

2.290
2.682
2.577
4.190
1.578



1a
2a
3a
4a
5a

LASF43
KZFS5
BAF52
BAF52
BALF5

LASF31ALASF31ALASF31ALASF31ALASF31A

FK51A
PK51
BAK4
PSK3
BAF52

1b
2b
3b
4b
5b

K10
K10
F_SILICA
SF6
K10

KZFS2
BAF2
CAF2
PBF2
KZFS2

SRF2
SK11
BAK1
LAK21
CAF2

5020

merit func ¼ M 2 , λ range ¼ 400−700 nm, δ ¼ 45°
84.86−92:2184.33 44.997
0.099
92.65
−82:4678.47 44.998
0.099
95.73
−78:4172.93 44.997
0.099
94.66
−72:2666.79 44.996
0.097
98.82
−76:4171.45 44.998
0.098
merit func ¼ M 2 , λ range ¼ 350−1000 nm, δ ¼ 45°
73.26−96:8691.58 43.981
0.962−44:31121.78−19:04 44.370
0.896−44:94116.01−13:27 43.352
0.919−52:8396.02−86:01 44.759
0.967−72:4794.54−90:26 44.199
0.856

APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

Fig. 8. (Color online) Dispersion of prisms 1a–5a and 1b–5b of Table 4.

6. Conclusion

Compound prisms composed of three independent
elements provide remarkable power for tailoring dispersive properties. With this flexibility, however, alsocomes a need to constrain the design space in order toprevent unphysical or impractical solutions. The
nonlinearity operand NL, beam compression operand K, thinness operands A and B, chromaticity
operand C, angle-of-incidence operand Θ, andsecond-order operand Δ2 presented here provide a
set of tools for designers to use for greatly improving
on such characteristics as dispersion linearity andhigher-order spectrum while minimizing negative effects on beam compression and chromaticity. The algorithm combines these with a brute force searchover glass combinations, producing a long list of valid
designs to choose from, among which one can thenselect a given prism based on alternative criteriasuch as cost, glass availability, a balance among
the merit criteria used, etc. This allows a general
method for designing linear-in-wavelength prisms,
and we have shown that this can reduce the variationin spectral sampling rate across the spectrum to as
low as 13% across the visible spectral range (for unconstrained triplets) or a factor of 2 (for Janssen
prisms).
We have also shown how to use a penalty on beamcompression to trade off unwanted changes in the
beam width for a reduction in the dispersion linearity achieved. For systems in which minimization ofbeam displacement is also important, the symmetricdouble Amici and Janssen prism designs provide a
simple means of achieving zero displacement. Not
only can this general approach help in spectrometer
design, but the same algorithms can also be applied
to prism design in other applications such as beamsteering and prism beam expanders (not discussedhere). We have shown that the approach is even flexible enough to produce two-dimensional dispersion
design (achieved by cross dispersing the high-order
spectra shown in Section 5).
While all of the achievements of our customized
design algorithm are possible using standard opticaldesign software, our algorithm provides an easierinterface for manipulating low-level features of thesoftware, such as the core optimization routine andthe various merit function operands, and allows useof a wide suite of numerical and text-manipulation
functions available in a widely used high-level
language.
Though we have mentioned some methods for reducing the calculation time of the design algorithms,
the code generally requires hours in order to produce
a full set of triplet or Janssen prism designs optimized over a given glass catalog (such as the Schottcatalog). Since all of the code used is written in
Python, one can expect an order of magnitude reduction in calculation time if the nonlinear optimization
code is rewritten directly in C or Fortran.
An obvious extension of the algorithms we haveshown here is to use four-glass asymmetric prisms(“quadruplets”) in order to take advantage of theeven greater degrees of freedom available. Doing thisblindly, however, produces an algorithm that requires weeks of computation in order to produce afull set of designs to analyze. Rather than use a brute
force search through all glass combinations, as wehave used up to this point, it becomes necessary to
select the glasses carefully and perform optimization
over a small subset of the full glass catalog. We are
currently developing methods for this approach.
We encourage interested readers to download andmodify the design code we have written for this work,
available at the authors’ website [12].
Appendix A: Summary of Merit Function Operands

We have used various merit functions in order totailor compound prisms to different tasks. Here we
provide a table that summarizes these operands, providing an easy means for designers to tailor theirown prisms. The base merit function used throughout this series is
M 0 ¼ ðδ − δ Þ2 þ ðΔ − Δ Þ2 þ Θ:With the addition of a third independent prism element, we gain the ability to further modify the merit
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5021

function without sacrificing the two target parameters in M 0 . These additional penalty terms, alongwith the relative weights used in our designs, are
Merit Function

First Term

Default Weight

Added Term

Linearity
Beam Compr.
Chromaticity
Thinness
Higher-orderM0M0ðδ −δ Þ2
M0
M0

25
0.00251
0:001Δ
0.1

NLðk − 1Þ2
CþΘ
AþB
1=NL

M ¼ ðδ − δ Þ2 þ ðΔ − Δ Þ2 þ Θ2 þ 0:01ðK − 1Þ2

where the individual operands are defined as
A¼

N
Xþ 0:1 NL:
α2n ;

ðA1Þ

n

B¼

I 
X
ðα

low − jαi jÞ

0

i¼1

2

∶αi < αlow;
∶αi > αlow

C ¼ j maxfδðλÞg − minfδðλÞgj;K¼

NY
j cos θn j

NL ¼

Θ¼

I 
X
0
i¼1

This work was partially supported by National Institutes of Health (NIH) grants RO1-CA124319 andR21-EB009186.
ðA2Þ
References

ðA3Þ

;

ðA4Þ

Z  2 d δ dλ2 dλ;

ðA5Þ

n¼1

j cos θ0n jðθlimit − θi Þ2∶θi < θlimit;∶θi > θlimit

ðA6Þ

where N is the number of prism elements and I is thenumber of refractive interfaces. The A operand is
added to the merit function to generate thin prism
designs. B is used to prevent individual prism elements from becoming too thin to be practical. C is
the “chromaticity” term used for achromatic beamsteering prisms. K is added to prevent the designfrom inducing excessive beam compression. NL is
the term used to improve dispersion linearity. Finally, Θ is used to constrain the design space to practicalelements, at which the interface angles are limited tovalues less than θlimit (for which we have used 65°

5022

throughout both papers). The weights associatedwith the operands are the ones we have used forthe tabulated designs and have been determined empirically. The weight for the “thinness” merit function
is determined in proportion to the target dispersion
because higher dispersion tends to produce larger
prism angles, and thus it becomes necessary to proportionally weight the two. As an example, the exact
merit function used for Table 2, with the weightterms explicitly included, is

APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

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2. P. J. C. Janssen, “Note sur trois spectroscopes [Note on three
spectroscopes],” C.R. Hebd. Seances Acad. Sci. 55, 576–578(1862), in French.
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50, 5023–5030 (2011).
4. E. L. Dereniak and T. D. Dereniak, Geometric and Trigonometric Optics (Cambridge University, 2008), pp. 347–350.
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51, 41–46 (1984).
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Wei, “Fully depleted, back-illuminated charge-coupled devices
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