Compound prism design principles, I
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 148098); published 30 August 2011

Prisms have been needlessly neglected as components used in modern optical design. In optical throughput, stray light, flexibility, and in their ability to be used in direct-view geometry, they excel over gratings.
Here we show that even their well-known weak dispersion relative to gratings has been overrated bydesigning doublet and double Amici direct-vision compound prisms that have 14° and 23° of dispersion
across the visible spectrum, equivalent to 800 and 1300 lines=mm gratings. By taking advantage of themultiple degrees of freedom available in a compound prism design, we also show prisms whose angulardispersion shows improved linearity in wavelength. In order to achieve these designs, we exploit thewell-behaved nature of prism design space to write customized algorithms that optimize directly inthe nonlinear design space. Using these algorithms, we showcase a number of prism designs that
illustrate a performance and flexibility that goes beyond what has often been considered possible withprisms. © 2011 Optical Society of America
OCIS codes: 230.5480, 080.2740, 260.2030, 300.6190.

1. Introduction

Prisms are commonly used components of spectrometers, and are often preferred over gratings when
the desired dispersion is weak. In this regime, they
possess better throughput than their grating counterparts, better stray light characteristics, are easier
to configure into a direct view geometry (in which thecentral wavelength passes through the system with
its angle unchanged), and are more customizable. We
concentrate below on these last two characteristics,
and argue that prisms have been unnecessarily neglected in modern spectrometry. With compoundprisms, the optical designer has excellent flexibility,
enabled by choosing the angles of the prism elements, the glasses that comprise them, and adding
additional elements into the design for optimal
performance.
The ability to configure prisms for direct viewgeometry is an advantage that many researchers
have used in their instruments, and which was first
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APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

realized by Amici in 1865 [1]. However, despite theirimportance for spectrometer design, the principles
of designing these direct view, or “nondeviating,”
prisms is not only scattered across a number of journal papers spanning 150 years, it is also incomplete.
Because of the scattered state of the historical literature, it can be difficult to give proper credit to prior
work, and so we start with a survey of past work
on direct-vision prisms going back to Amici’s initial
invention. Using the modern sign convention, we
present the nonlinear prism design equation andits linear counterpart derived from the small-angle
approximation. Regarding compound dispersive
prisms, the existing literature has focused exclusively on the latter, but we show that it is easy towork directly with the nonlinear equation for prism
dispersion. This provides for a rapid means of optimizing over all available glasses, and for customizing
a design for any given system requirements.
We next introduce the concept of dispersion linearity and show that one can improve the linearity of thespectrum by brute force optimization across all glass
combinations within a glass catalog. Because the design space is not of high dimension and is relatively

well behaved, nonlinear optimization algorithms canrapidly obtain solutions, making such a large search
practical. This is the first discussion, to our knowledge, of an end-to-end design for dispersion linearization. We find that while an optimization over glass
choice can substantially improve linearity, highly linear dispersion from prisms using only two elements
can only be achieved within short spectral ranges.
Three-element compound prisms such as the double
Amici design can achieve substantially linear-inwavelength dispersion, but we show that there isalways a trade-off in prism size.
A surprising result obtained with our design
algorithms is that, despite the weak dispersion allowed by two-element direct-vision prism geometry,
we show that it is possible to achieve >14° of dispersion across the visible spectrum in a two-element
direct-vision design. In a double Amici design, weshow that it is possible to achieve 23° of dispersion
across the visible spectrum, while maintaining
direct-vision geometry. The entire source code (written in Python) used for the designs and all algorithms used to produce the results in this paper canbe found on the authors’ website [2].
2. Historical Survey of Direct-Vision Dispersive
Prism Design

The historical literature on nondeviating prisms invariably miscites the original references, perhaps
due to their obscurity. But the recent opening up ofold documents to direct electronic access allows one
to trace developments, beginning with Amici’s initial
invention. Amici himself never published about his
nondeviating prism, but rather communicated theidea to his friend Donati, who was able to construct
the device and use it for observations of stellar spectra. These were published in 1862 [1,3] (an abstract
in English was also published in 1863, but mentions
only Donati’s observations on stellar spectral lines
and not Amici’s prism [4]). Ref. [1] is almost universally given with the wrong author—Amici instead of
Donati—and the incorrect publication date of 1860,
though the manuscript itself is signed by Donati witha date of 1860.
Upon hearing of Amici’s idea, the French astronomer P. J. C. Janssen (often referred to in the historical literature as Jules Janssen) contributed a paper
describing three modifications of the concept [5]. The
first describes splitting the double Amici prismin two at the plane of symmetry, allowing one tomanipulate the two sides independently. The second
modification takes advantage of this splitting tosilver the second element, so that the prism is usedin reflective Littrow geometry. The third design constructs a compound prism based on Amici’s principle
but composed of five elements instead of three in order to produce greater dispersion [6]. The first manuscript to show illustrations of Amici’s and Janssen’s
direct-view prisms is apparently Secchi’s treatise on
the stars [7].

In what was apparently an independent invention, in 1864, Browning in England came up with athree-element prism designed not for zero deviation
but rather for reduced deviation. This triplet setup
used two thin prisms of crown glass placed on bothsides of a single liquid prism of carbon disulfide, andwas intended to increase the number of prisms that
could be admitted in a circular arrangement common
at the time for increasing spectral dispersion. The
entire setup was manufactured on the instructionof Gassiot, and consisted of 11 prisms in all [8].
(The invention is also discussed by Browning himself
in Ref. [9].)In 1865, the English astronomer Hershel describeda method for constructing single-element directvision prisms based on the use of internal reflections[10]. (An illustration of the prism appears in [11]).
This design, however, seems to have been used only
rarely, possibly due to the low value of dispersion obtained. Emsmann also describes a single-element
direct-vision approach [12], and Fuchs later showed
how to achieve the same nondeviation effect usinga plane mirror external to the prism [13]. Liveing
and Dewar continued along these lines and developed a new nondeviating prism design using three
air-spaced prisms and two reflecting surfaces [14].
(See Ref. [15] for a modern paper that independently
developed this approach and gives alternative
geometries.)
Tait (1872) mentions the construction of a threeelement nondeviating prism in which all threeelements are composed of different materials—a generalization of the double Amici in which the outer elements are composed of the same material [16]. Thisis a subject we explore in Papers II and III of thisseries [17,18].
Thollon, in 1878, was apparently the first to give
explicit expressions for the design of double Amici
prisms, though this work was quickly followed by
the more detailed paper of Riccó [19–21]. Thollon
also goes on to show that one can reduce the anglesof the outer crown prism elements, and increase theangle of the inner flint prism, in order to increase thedispersion of the system as a whole [22].
In 1881, Zenger noted that a design need not use
prism materials whose refractive indices are substantially different [23]. Rather, one can constructa nondeviating prism in which the ray for the central
wavelength is entirely unrefracted: the prism elements have the same mean refractive index, and theexterior faces of the prism are orthogonal to the optical axis. Only the dispersion is different between
the three glass elements. This produces a weak dispersion, so that Wernicke proposed converting thetwo-element Zenger design into a double Amici configuration—a system in which the mean indices of all
three elements are equal [24]. Tyrrell and Conn provide a detailed analysis of this Wernicke-type prism
design [25], and also point out that the liquid prisms
are not much used because of the strong temperature
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

4999

dependence in the refractive index of optical liquids,
which makes calibration difficult.
Abbott and Fowle (1900) are the first researchersof which we are aware that explicitly attempt tolinearize a compound prism’s dispersion, and theyalso aim at doing this for a direct-vision system [26].
They mention using a double Amici prism designcomposed of a central 20° prism element of rocksalt between two elements of 5° 10 arc min flint
glass, but present only dispersion measurements
for a two-element system, illustrating the improvement in linearity empirically. More recently, thesame goal of a linearized nondeviating prism was
attempted by Ebizuka et al., using a three-element
prism of three different glasses [27]. Blechinger
et al. show several linearized prism designs for
deviating prisms [28], and Bittner et al. develop an
Offner-type imaging spectrometer using one or more
Féry prisms cemented together, showing that one canlinearize the dispersion by proper selection of theglasses and angles [29]. Refs. [26–29], however,
only briefly mention design issues, and do not givequantitative design methods. One is left with the impression that these authors constructed their designs using trial and error, aided by optical design
software.
E. and T. Dereniak, on the other hand, take advantage of theory developed for superachromat lenses[30,31] to give explicit expressions for the first-order
design of compound nondeviating prisms with linearized dispersion ([32]), allowing one to see the explicit
dependence on the mean index, Abbe V-number,and partial dispersion of the glasses used in thedesign. (See Section 4 below, and also Section 2 ofPaper II [17].) An early book that appears to containrelevant material, but which has proven inaccessible
to us, is Ref. [33].
A clever adaptation of the double Amici was developed by Wynne for atmospheric dispersion correction[34]. This takes advantage of the plane of symmetry
in the center of a double Amici system, allowing forthe physical separation of the two sides of the prism(as in Ref. [5]). By rotating both sides in equal andopposite azimuth angles, one has a nondeviating disperser with tunable dispersion.
Our own interest in the subject of nondeviating
prisms stems from their use in imaging spectrometers [35,36] and from our own research group’s
need to develop better performing designs.
3. Simple Prisms: First-Order Design Principles

The refraction of a ray through a simple prism (Fig. 1)
is given by applying Snell’s law at each interface
such that90
θ1 ¼ θ0 − β; = θ2 ¼ arcsinðn sin θ2 Þ; >; ð1Þ
θ01 ¼ arcsin n1 sin θ1 ; θ3 ¼ θ02 þ γ;>;
θ2 ¼ θ01 − α;where we assume that the optical axis is parallel tothe prism base, the external medium has refractive
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APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

Fig. 1. Ray trace through a simple prism. The ray is incident at
an angle θ0 from the optical axis (dashed); θ1 , θ01 , θ2 , and θ02 are theangles of incidence and refraction on the first and second prism
interfaces, and θ3 is the angle of the deviated ray with respect
to the axis. The prism shown here has an apex angle α ¼ 60° andis oriented such that the front and back faces of the prism haveangles β ¼ −15° and γ ¼ 45° with respect to the optical axis.
The prism has an index of refraction n ¼ 1:5, a height h, and anaxial thickness t. For the ray traced here, θ0 ¼ 30° and θ3 ¼ −7:4°,
such that δ ¼ 37:4°. Using the sign convention, θ1 and θ01 are positive quantities, while θ2 and θ02 are negative. The quantity Δy isthe beam displacement.

index 1, α ¼ γ − β, and the angles are defined asshown in Fig. 1. The angles θi follow the modern sign
convention [37], in which clockwise angles traced
from the surface normal to the ray have negative
sign, and counterclockwise angles have positive sign.
The deviation angle δ is simply
δ ¼ θ0 − θ3 ¼ θ1 − θ02 − α:Since the prism refractive indices vary with wavelength, so too does the deviation angle. The difference in deviation angle between the two extreme
wavelengths transmitted by the prism is thedispersion Δ:
Δ ¼ δðλmin Þ − δðλmax Þ ≡ δmax − δmin :In classical optical design, this is commonly written
as Δ ¼ δF − δC for the Fraunhofer F and C wavelengths (λF ¼ 486 cm, λC ¼ 656 cm). We use λ for thecentral wavelength, defined as λ ¼ 12 ðλmax þ λmin Þ,
rather than the conventional λd (note that the central
wavelength and the mean wavelength can be different when the dispersion is sampled nonuniformly, as
is typically the case in prism spectra). The spectral
dispersion is thus a function of the glass dispersionnðλÞ, the prism apex angle α, and the incident ray
angle θ0 .
The refraction Eqs. (1) represent the nonlinear
relationship of dispersion δðnðλÞ; α; θ0 Þ to each of thesystem variables. In order to obtain an estimate of
the appropriate design variables from the desired
dispersion, the historical approach has been to assume small prism angles α and small angles of incidence θ0 . Using these approximations reduces theexact form obtained by combining the refraction
equations,δ ¼ θ0 − γ






1
sinðθ0 − βÞ − α ;− arcsin n sin arcsinnto the simple linear form
δ ¼ ðn − 1Þα;ð2Þfrom which the prism apex angle α is trivially obtained from the desired dispersion and the glass refractive index. A nondeviating prism is defined as adisperser in which the deviation of the central wavelength is zero: δðλÞ ¼ 0. This can be achieved with theaddition of a second prism element.
Other useful quantities to know about a prism
are the minimum prism height hmin , the minimum
axial thickness tmin , minimum displacement Δymin ,and the exit beam width w0 for a given incident
beam width w. These are readily calculated fromgeometry astmin ¼ h0 ðtan γ − tan βÞ;Δymin ¼ h0sinðαÞ sinðθ01 þ βÞ;cosðβÞ cosðθ2 Þ

Fig. 3. A ray trace through a doublet compound prism, where
the second element is assumed to be oriented symmetrically withrespect to the normal to the optical axis. The system shown here
has prism apex angles α1 ¼ 60° and α2 ¼ −40°, indices of refraction
n1 ¼ 1:5 and n2 ¼ 1:8, height h, axis-to-apex height h0 , beam displacement Δy, and axial thicknesses t1 and t2 . The input ray has
angle θ0 ¼ 10°, such that δ ¼ 0:4°.

in which a typical design uses α2 < 0, indicating thatthe prism is inverted relative to the first prism (as inFig. 3). If we further assume that the wavelength
dependence to the refractive index is approximately
linear, then the dispersion can be written as

hmin ¼ maxfh0 ; h0 þ Δyg;
w0 ¼ w cosðθ02 Þ= cosðθ1 Þ;
where h0 ¼ w cosðβÞ= cosðθ1 Þ, and they are illustrated
in Fig. 2.
4. Two-Element Prisms: Linear and Nonlinear Design
Approaches

A number of researchers [38–42] have constructed
direct-vision compound prisms with two elements(“doublets”) using linear design equations. With
the small-angle approximations, one can use linear
equations to design doublet prisms (as in Fig. 3) having both zero deviation and the desired dispersion[32]. Using the linearity property of these equations,the deviation of the compound element is simply thesum of the individual element deviations,δðλÞ ¼ δ1 ðλÞ þ δ2 ðλÞ ¼ ðn1 ðλÞ − 1Þα1 þ ðn2 ðλÞ − 1Þα2 ;ð3ÞFig. 2. Incident beam width w is transformed into an exiting
beam width w0 . The minimum prism height hmin and minimum
prism axial thickness tmin are constrained by the beam width,prism geometry and angle of incidence.

Δ¼

δ1 ðλÞ δ2 ðλÞ
þ
;
V1
V2

ð4Þ

where δi and V i are the dispersion and Abbe numberof element i within the compound prism, V i ¼
ðn − 1Þ=ðnF − nC Þ. If we define the shorthand notationδ ¼ δðλÞ and let δ ¼ 0, solving Eqs. (3) and (4) simultaneously gives−1δ1 ¼ −δ2 ¼ −Δ 1 − 1
;
V2 V1

ð5Þ

from which one can obtain the element apex angles
α1 and α2 from the mean refractive indices of theglasses chosen:
α1 ¼



Δ
1
1 −1
−
;
1 − 1 V 1 V 2
n

ð6Þ

α2 ¼



Δ
1
1 −1
−
:
2 − 1 V 2 V 1nð7ÞMost practical prisms do not satisfy the smallvalue approximations closely, so that the typical design procedure is to use the linear Eqs. (3) and (4) toobtain a preliminary solution, after which commercial optical design software is used to perform thefinal nonlinear optimization. An alternative to thelinear design equations is to use the optimization
code directly with the nonlinear equationδðn1 ðλÞ; n2 ðλÞ; α1 ; α2 ; θ0 Þ;
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5001

readily obtained by concatenating
doublet’s refraction equations,

the

prism

9
>
>
>
>
1
0
0
=
θ1 ¼ arcsin n1 sin θ1 ; θ3 ¼ arcsinðn2 sin θ3 Þ; >;100
θ2 ¼ θ1 − α1;>> θ 4 ¼ θ 3 þ 2 α2 ;>>>;
θ02 ¼ arcsin nn12 sin θ2 ;
θ1 ¼ θ0 − β1;

θ3 ¼ θ02 − α2 ;ð8Þwhere β1 ¼ −α1 þ 12 α2 , δ ¼ θ0 − θ4 , and Δ ¼ δðλmax Þ−δðλmin Þ. (See Fig. 3 for an illustration of these quantities.) The convention, followed in all of our two- andthree-element prism designs, is to have the second
prism placed such that its plane of symmetry is normal to the optical axis. All other prism elements havetheir positions defined relative to this second element. Angles βi and γ i are defined such that αi ¼ γ i −βi for prism element i, and β and γ define the angles ofthe prism’s front and rear faces relative to the opticalaxis normal. (Thus, for our convention, the second
element always has β2 ¼ −γ 2 .)In order to gain more flexibility over the design
procedure, we use custom written algorithms incorporating nonlinear optimization code [43]. This allows us to illustrate the design space of each prismand analyze its behavior.
From the two quantities δ and Δ, one can definea merit function M whose optimization code can beused to perform the search for a solution. A common
choice for M is the sum of square errors in the variables, which in this case gives
M ¼ ðδ − δ Þ2 þ ðΔ − Δ Þ2 ;where δ and Δ are the design target values and δ
and Δ are values during a given step of the optimization. Figure 4 illustrates the design space for the ray
deviation and the spectral dispersion of a BK7–SF6
doublet operating over the visible spectral range(400–700 nm). For the example chosen, the linear
equations provide a design of ðα1 ; α2 Þ ¼ ð35°; −22°Þ
at the intersection of the δ ¼ 0 and Δ ¼ 1° lines
shown. The nonlinear approach finds the optimumat ðα1 ; α2 Þ ¼ ð33°; −31°Þ, given by the intersection of
the dashed curves shown. In this simple case, the linear equations provide a good starting point for theexact design.
Looking at Fig. 4(a), we can see that the ðα1 ; α2 Þ
design space for the deviation δ is well behaved, withthe linear equations somewhat underestimating
the magnitude for α2 needed to achieve direct-vision(δ ¼ 0). Figure 4(b), likewise, shows that the design
space for the dispersion (Δ ¼ 1°) is also well behaved,with the linear equations again underestimating themagnitude of α2 over much of its range. One feature
to note, however, is the presence of a maximum between the two 0.400 contour lines at the bottom of
the figure. This indicates that a secondary minimumof the dispersion function Δðn1 ðλÞ; n2 ðλÞ; α1 ; α2 ; θ0 Þ is5002

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

Fig. 4. (Color online) Doublet prism design space, showing
contour plots of a, the central deviation δðα1 ; α2 Þ, and b, the difference of the dispersion Δðα1 ; α2 Þ from the target value Δ ¼ 1° for
a 400–700 nm spectral range and glass choices of BK7 and SF6.
The dashed blue lines indicate the ðα1 ; α2 Þ values required to meeteither the δ ¼ 0 design target (for direct-vision) or the Δ ¼ 1°
target, while the red solid lines indicate prism angles satisfying
linear Eq. (5). A design satisfies both targets where the two curves
meet, and the gray background indicates the valid design space,outside of which total internal reflections occur at one of the prism
interfaces.

forming in the bottom right corner. This featureturns out to be a common one, even in the design
space of prisms with three or more elements, andcan cause an optimization routine to get stuck inside
the secondary minimum. Thus, for doublets we findthat it is generally a good idea to overestimate theapex angle of the higher index element (α2 in thiscase) when providing a starting guess in doublet
designs.
Also, we can observe that since the second element
in a doublet compound prism tends to “achromatize”
the first (i.e., the dispersions of the two elements
tend to cancel rather than to sum constructively), theachievable dispersion is typically weak. Using NLAK34 and N-SF66 as an example choice of glasses(this glass pair produces a design with small angles
αi ), Fig. 5 shows curves for the α1 and α2 design
points as a function of the dispersion target Δ forthe visible spectral range (400–700 nm). Beyond
about Δ ¼ 14°, the angle of incidence θ1 becomes

150
100
50
0
-50
-100
0

2

4

6

8

10

12

14

16

Fig. 5. (Color online) Design parameters of a nondeviating NLAK34/N-SF66 prism doublet as a function of the target dispersion. The α1 and α2 curves proceed in an approximately parabolic
curves until about 14:6°, beyond which the system can no longer
achieve the target values. In this case, the limit is reached as thefront face of the prism approaches the grazing angle to the incident
beam (i.e., θ1 approaches 90°).

extreme, and the design can no longer simultaneously maintain the direct-vision (δ ¼ 0) anddispersion targets. This is the design limit for a nondeviating prism doublet using a standard glass set.
Considerably larger dispersion than this can beachieved by extending the spectral range, thoughwith a reduced number of glasses available due toabsorption effects. (Note that fully depleted silicondetector arrays allow working at high quantum efficiency over a 300–1000 nm range [44,45].)
5. Optimizing over Glass Combination

Unless the types of glasses to be used in the prismare greatly constrained, it is also useful to optimize
over available glass types, as it provides additional
design flexibility. One can choose a glass pair that
Table 1.

not only satisfy the design targets δ and Δ , but that
also, for example, provide the most linear dispersion
or the most compact layout among all glass pairs considered. In order to implement glass optimization,
we have written code that reads the parameters of
all glasses listed in a standard catalog. (This script
is made available on the authors’ website [2] for
all interested readers.) For the greatest accessibility,
we have chosen the catalog format provided by thecommercial optical design software ZEMAX [46], andperformed a brute force search over all glass combinations within the full set or any subset of catalogs.
The results of all glass combinations that can satisfy
the design targets are then tabulated and sorted in
the form of Table 1.
If we choose to optimize over only the Schott glass
catalog [47], then the number of different glasses
available is N ¼ 110. The algorithm provided in ourcode optimizes over any catalog, or any set of catalogs, provided with ZEMAX, but the Schott catalog
is the one most widely used, and so is the one wechoose here. After removing redundant and historical glasses, the Schott glass catalog is left with
N ¼ 100. The total number T 2 of unique glass pairs
within this list is therefore T 2 ¼ NðN − 1Þ ¼ 9900.
(Because of nonlinearities in the design space, the ordering of the glasses does matter.) Although not all
combinations are capable of satisfying both design
targets simultaneously, the resulting table of designs
produced by the algorithm is generally quite long,and so the list is then sorted by merit function value
to show the best performing designs (see the authors’
website [2] for example tables of designs). Assumingthat the algorithm has successfully located the global
minimum in the design space for each glass combination, the design whose merit function is lowest within
the resulting list is the overall optimum for the given
glass catalog (or set of catalogs).

Best Performing Doublet Prisms for Design Targets δ  0 and Δ  1° (400 nm ≤ λ ≤ 700 nm, θ0  0)aGlass 1Glass 2α1 (deg)

α2 (deg)δ (deg)Δ (deg)NL (×104 )

SSR

1a
2a
3a
4a
5a

N-PK51
N-BAF4
N-BAF52
N-KZFS4
N-KZFS11

N-BK10N-KZFS11
N-KZFS4
N-BAF52
N-BAF4−104:55−114:72−114:61110.7399.32

124.14104.49112.39−112:78−108:82

0.000
0.000
0.000
0.000
0.000

1.000
1.000
1.000
1.000
1.000

0.182
0.193
0.195
0.205
0.223

2.99
2.71
2.49
2.53
2.40

6a
7a
8a
9a
10a

N-LAK33A
N-LASF31ALAFN7N-LASF44
N-LAF34

N-BALF5
N-BAF4
N-BASF2
N-BAF10
N-SSK5

65.07
56.17
82.49
76.74
79.81

−103:62−91:94−98:03−101:15−101:73−0:001
0.000
0.000
0.000
0.000

1.000
1.000
1.000
1.000
1.000

0.323
0.324
0.327
0.328
0.335

4.70
5.22
4.72
4.56
4.45

1b
2b
3b
4b
5b

N-SF66
N-SF66
N-LAK34
N-LAK33A
N-SF66

N-LAK34
N-LAK33A
N-SF66
N-SF66
N-LAK14

13.62
13.97
−17:49−17:33
13.47−17:36−17:21
13.74
14.08−17:98

0.000
0.000
0.000
0.000
0.000

1.000
1.000
1.000
1.000
1.000

0.742
0.748
0.742
0.749
0.740

11.25
11.34
11.25
11.34
11.24

a
Rows 1a–10a list the designs in order of dispersion linearity (1a–5a are unconstrained, while 6a–10a are constrained to incidence angles<65°); 1b–5b list the designs ordered by thickness. Prisms 1a and 1b are illustrated in Fig. 7.

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

5003

Fig. 6. Typical angular dispersion curve for a prism disperser (left axis) and the corresponding dispersion gradient (right axis). A detector
array placed at the focal plane of a lens will provide a uniformly sampled spectrum if the dispersion gradient is constant across the
spectrum. For the case shown here, the sampling will be 10× finer in the blue end of the spectrum than in the red.

6. Dispersion Linearity

Nonlinearity in the spectral dispersion (see Fig. 6) isoften a nuisance to users of spectrometers, especiallyfor spectrometers incorporating prism dispersing elements. We can attempt to minimize the dispersion
nonlinearity (NL) by incorporating a penalty term
into our design merit function. From Eq. (8), we know
the ray deviation as a function of wavelength, δðλÞ.
The gradient of the deviation angle, dδ=dλ, gives
the spectral dispersion per unit wavelength. The difference of the gradient from a constant, or equivalently the area under the second derivative, givesa measure of the dispersion nonlinearity, defined as
Z
NL ¼

j

d2 δ
jdλ:
dλ2

ð9Þ

Thus, the new merit function becomes
M ¼ ðδ − δ Þ2 þ ðΔ − Δ Þ2 þ wnl NL;

ð10Þ

where wnl is a weight factor for weighing the relative
importance of the NL term versus the remaining
terms of the merit function. In practice, the δ andΔ design constraints fix the values of α1 and α2 fora given glass choice, so that the first two terms of
Eq. (10) become zero. In order to optimize for dispersion linearity, therefore, the only degree of freedom
left is glass choice. Thus, the algorithm performs abrute force search over all glass combinations within
the glass catalog in order to locate designs for which
NL is smallest.
Another measure of the linearity of the spectrum is
to take the ratio of the dispersion gradient maximum
and minimum. This provides a measure of the maximum amount that the spectral bands get stretchedby the nonlinear dispersion. This “spectral sampling
ratio” (SSR) is defined as
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APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

 
 
dδ 
dδ 


SSR ¼ max   = min   :
dλ
dλ

ð11Þ

Thus, for example, an SSR of 5 would mean that if
the short wavelength end of the spectrum has a spectral bin that is 1 nm wide, the long wavelength end of
the spectrum will have a spectral bin 5 nm wide.
Section (a) of Table 1 shows the best performing
prisms from optimizing a δ ¼ 0, Δ ¼ 1° design over
the Schott glass catalog. Only those prisms with thesmallest NL are shown, with the first five designs(1a–5a) having angles of incidence unconstrained,
and the second five (6a–10a) having angles of incidence constrained to <65°. We can see that in eachcase the two glasses chosen in Section (a) of the tabletend to be close to one another on the Abbe diagram

(i.e., they tend to have similar refractive indices nand similar linear dispersions V). (Note that the fiveglass combinations within each section of the tabletend to be quite similar. The complete list of designs
shows many different configurations, but these do
not appear in the tables shown here due to space
restrictions.)
Among the glass combinations in Table 1, we cansee that the highly linear dispersion designs, which
can have interface angles up to 89°, use a pair ofglasses that lie very near one another on the glass
chart, so that the glass–glass interface in the centerof the prism induces very little beam deflection.
When designing prisms for dispersion linearity, one
finds that even for small amounts of dispersion (<1°),
the angles of incidence on the prism interfaces canbe substantial, indicating that these designs attempt
to balance the dispersion equation’s nonlinearity
in incidence angle against the refractive index
nonlinearity.
Constraining solutions to use angles of incidence less than a given value θlimit (65° for theprisms of Table 1) can be accomplished by adding ahalf-quadratic cost function to the merit function. For
the I interfaces of a given prism (I ¼ 3 for a doublet),
we calculateI X0;θi < θlimit
:
ð12Þ
Θ¼
ðθlimit − θi Þ2 ; θi > θlimit
i¼1

Thus, the merit function is unaffected when a prism’s
interface angles remain under the limiting value, but
increases quadratically as the design goes beyond
θlimit . The choice of 65° that we use here, rather thansome other angle, is somewhat arbitrary, but has
been selected as being small enough to be practicalbut large enough to allow sufficient design flexibility.
Once we restrict the angles of the interface to <65°,
we see the glass combinations in Table 1 change, sothat now we have pairs of glasses with almost identical linear dispersion values (the Abbe V-number)but substantially different mean refractive indices.
One finds that, in general, direct-vision doublet
prism designs have poorer dispersion linearity thantheir counterpart (non-direct-vision) singlet designs.
Using the Schott glass catalog, the more linear singlet designs for Δ ¼ 1° have 0:09 ≤ NL ≤ 0:12, whichlies below the doublet results shown in Table 1.
The bottom section of the Table 1 (designs 1b–5b)shows the thinnest designs. From this list, one cannote that the best performing doublets use highindex glasses for both elements, and that among
the high-index glasses, the linear dispersions of thetwo glasses chosen (i.e., V 1 and V 2 ) differ by as muchas the catalog allows while keeping the mean refractive index relatively close. From the tabulated designs, one can note that the ordering of the glasses
alters the performance somewhat. That is, if one flips
the prism upside down and illuminates from the left
instead of the right, the resulting dispersion is close
to but not exactly the same as that of the original
prism.
Note that direct-vision prisms with small dispersion values (<1°) are most commonly found inimaging spectrometry, where the number of spectral
bands resolved is typically much smaller than in nonimaging spectrometry, and where the direct-visiongeometry can provide substantial reduction in system size.
7. Doublet Prism Thickness

Another commonly desired characteristic is for a
prism design to be as thin as possible. The overall
prism thickness t can be defined by the sum of axial
thicknesses as t ¼ t1 þ t2, or by taking the z distancefrom the prism extreme corners (i.e., front bottomedge to rear top edge, for the prism shown in Fig. 3).
For an arbitrary prism height h, the former value canbe obtained by the formulas
t1 ¼ h0 ðtan γ 1 − tan β1 Þ;t2 ¼ 2ðh − h0 Þ tan β2 ;where h0 ¼ w cosðβ1 Þ= cosðθ1 Þ. Here h0 is the heightfrom the optical axis to the first element apex andh is the full height of the element, from apex to base(see Fig. 3). All elements of the compound prism areassumed to share the same height h.
If an approximate relative measure of thickness t
is all one needs, then one can also
P simply minimize
the sum of apex angles, i.e., t ≈ h Ii¼1 jα1 j, in which his the prism height and I the total number of prismelements. This approximate measure is what is used
to sort the results shown in Table 1.
Taking a look at Fig. 7, we can see that for design
1a, the beam input is small relative to the prismheight, indicating that for most systems this design
will require large prisms. Here the prism height isthree times that of the input beam width (i.e.,h ≈ 3 w). This h ≫ w property is characteristic of
all of the linearized designs 1a–5a in Table 1.
8. Achromatic Beam Deviation

So far, we have considered only direct-view dispersive prisms. One important prism application
actually requires ray deviation without dispersion:
beam steering [48]. With a small modification to ourbasic merit function, we can easily adopt the design
algorithms to optimize for beam steering applications. While designing beam steering prisms, we
run into the phenomenon of higher-order dispersion.
Investigating this effect forces us once again toresort to asymmetric triplet prisms to achieve full
performance.
Beam steering uses a set of rotating prism elements placed in front of an imaging system to steer
the field of view within a much larger “field of regard”(see Fig. 8). In this application, dispersion in the deflected beam is unwanted, so it becomes necessary toachromatize with the proper selection of glasses andprism angles.
Although it is a well-known technique, publishedresearch on achromatic beam deviation is small enough to allow a nearly comprehensive citation list.
The earliest publication of which we are aware isRef. [33], an extremely scarce book available onlyin Russian. In the archive literature, one finds a
handful of papers [49–55]. Only Ref. [52], however,presents a method for optimal glass selection, so thatthe reader is often left to wonder whether the results
obtained are the best achievable. This is easily done,
however, with the algorithms we have been working
with so far.

Fig. 7. Two prism designs from Table 1, design 1a (for improved
dispersion linearity) and 1b (for a compact system). Both designs
have δ ¼ 0 and Δ ¼ 1°. The rays are drawn for the central
wavelength, λ. Note that both prisms show some negative beamdisplacement.
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5005

Fig. 8. As the prism is rotated, the deviation angle δ steers the field of view (with half-angle θ0 ) through a wider “field of regard”(with half-angle δ þ θ0 ).

For achromatic beam deviation designs, we need
only select a target central ray deviation δ and dispersion value Δ (presumably zero in this case) as
the inputs to our algorithm and run the code to provide a set of designs. However, for this application it
is common to choose a slightly different form for thesecond term to limit the total change in deviation
angle at all points across the spectral range. This
leads to a new merit functionthat the Θ term in Eq. (13) has a strong effect on
the chromaticity.
The most compact designs in designs 1a–5a of
Table 2 use a combination of the highest-dispersion
glasses (SF68 and SF66) together with a glass thathas one of the lowest ratios of partial dispersion
to refractive index (LAK34, LAK14, LAK33A, and
PSK53A).
9. Double Amici Prisms

M chr ¼ ðδ − δ Þ2 þ C þ Θ;

ð13Þ

where
C ¼ j maxfδðλÞg − minfδðλÞgjis what we will call the “chromaticity.” With this
merit function, we optimize over all glass pair combinations within the catalog to obtain a design withminimum chromaticity.
Table 2 shows a list of doublet designs for achromatic 30° deviation across most of the visible
spectrum, optimized across the Schott glass catalog(see Fig. 9 for chromaticity curves corresponding todesigns 1b–5b). The results show that 30° deviation
can be achieved with less than 20 mdeg of dispersion
across the 450–650 nm spectral range. Note that thechromaticity generally decreases with higher angles
of incidence θi allowed on the prism interfaces, so
Table 2.

The double Amici prism design is a three-element
system (a “triplet”) in which the first and third elements share the same glass and the same apex
angles (see Fig. 10). The design layout is thus symmetric about the plane passing through the center of
its second element. In general, if we allow air to be
used as a “glass,” then this layout can also model apair of air-spaced singlet prisms. To allow this in
our optimization code, we have added air as a glass
to the standard catalog in each of our designs.
The symmetry property of the double Amici meansthat the linear design equations are almost identicalto those of the doublet prism, with only the additionof a factor of 2 in front of the first term in each:δ ¼ 2δ1 þ δ2 ¼ 2ðn1 − 1Þα1 þ ðn2 − 1Þα2 ;
Δ¼2

δ1
δ
þ 2:
V1 V2

Achromatic Beam Deviation: the Best Performing Doublet Prisms (for θ0  0), Listed in Order of (a) Compactness or (b) Dispersion
Chromaticity, Where Chromaticity C is Defined by Eq. (13)aGlass 1α1 (deg)

Glass 2α2 (deg)δ (deg)

C (×100)

30.00030.00030.00030.00030.00055.55457.25045.75255.69455.11830.00030.00030.00030.00030.000

0.210
0.248
0.251
0.253
0.259

, λ range ¼ 450–650 nm, δ ¼ 30°, Δ ¼ 0

a

1a
2a
3a
4a
5a

SF68
SF68
SF68
SF68
SF66

1b
2b
3b
4b
5b

SK5
SK57
SK11
LAK21
SK16

merit func.: M chr
LAK34−13:7451.61
LAK33A−14:72
51.23
PSK53A−10:73
55.42
LAK14−13:2453.08
LAK34−14:8851.46
merit func.: M chr , λ range ¼ 450–650 nm, δ ¼ 30°, Δ ¼ 0
CAF2−39:8993.20
CAF2−38:66
91.88
CAF2−41:70
92.76
CAF2−35:14
92.39
CAF2−36:6892.51

The dispersions of prisms 1d–5d are illustrated in Fig. 9.

5006

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

Fig. 9. (Color online) Dispersion, in millidegrees, of the achromatized beam deviation designs 1b–5b from Table 2.

Similar to Eq. (5), we can derive the expressions forthe prism angles using the linear equations, giving


Δ
1
1 −1
α1 ¼
−
;
2ð
n1 − 1Þ V 1 V 2


Δ
1
1 −1
−
:
α2 ¼
2 − 1 V 2 V 1n
The nonlinear equation for the deviation angle
δ requires including an additional two terms in therefraction equations, so that9>
θ03 ¼ arcsinðnn21 sin θ3 Þ; >
θ1 ¼ θ0 − β1 ;
>
>


>
>
1
0
0
>
>
θ1 ¼ arcsin n1 sin θ1 ; θ4 ¼ θ3 − α1 ;=00
θ2 ¼ θ1 − α1;θ¼arcsinðn
sin
θ
Þ;
1
4
4

>
>
>
>
θ02 ¼ arcsin nn12 sin θ2 ; θ5 ¼ θ04 − β1
>
>
>
>
0
;
θ 3 ¼ θ 2 − α2 ;ð14Þcan be concatenated to form δðn1 ðλÞ; n2 ðλÞ; α1 ; α2 ; θ0 Þ.
Here β1 ¼ −α1 þ 12 α2 and δ ¼ θ0 − θ5 .
Figure 11 shows the ðα1 ; α2 Þ design space for a
fused silica/N-LASF31A/fused silica double Amici
prism used in the 400–700 nm spectral range. We

can see that the design space for the central deviation is still well behaved, and that the design space
for the dispersion has begun to diverge significantly
from linearity. And we also find that for the design
targets of δ ¼ 0 and Δ ¼ 2°, the linear equations
provide a design ðα1 ; α2 Þ ¼ ð95:4°; −98:6°Þ, whichlies outside the valid design space, where total internal reflection occurs at an interface between oneof the prism elements. The nonlinear algorithm
indicates an optimum solution located at ðα1 ; α2 Þ ¼
ð64:9°; −63:6°Þ. Additionally, the minimum for jΔ −Δ j occurring in the bottom right corner of the design
space has deepened, making it easier for an optimization algorithm to get trapped there.
Note that the double Amici prism, despite having
an element more than the doublet, does not actually
have additional degrees of freedom for design. Thus,
after achieving the design targets δ and Δ , the onlymeans of optimizing auxiliary design variables suchas dispersion linearity or total thickness is to adjust
the glass types. Accordingly, as with the doublet design, we perform a complete search over all glass
pairs available within the glass catalog.
Table 3 shows the best designs resulting from ouralgorithm, as determined by either dispersion linearity or system thinness. Because the “achromatizing”
effect of the glasses in a double Amici prism is lesssevere than in a simple direct-vision doublet, we findFig. 10. Raytrace through a double Amici prism: the first and third elements are identical, and the second element is oriented symmetrically with respect to the normal to the optical axis. The system shown here has prism apex angles α1 ¼ 70° and α2 ¼ −60°, indices ofrefraction n1 ¼ 1:5 and n2 ¼ 1:8, height h, and axial thicknesses t1 and t2 . The input ray has angle θ0 ¼ 15°, such that δ ¼ 30°. For thissetup, the beam displacement Δy is nearly zero.
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5007

that substantially greater dispersion is possible.
Comparing the designs in Tables 1 and 3, we findthat the double Amici designs can achieve muchlarger dispersion (23:5° instead of 14° in the case ofdesign 1b in Table 1 versus design 1c in Table 3).
Dispersion of 23:5° across the visible spectrum(400–700 nm) is roughly equivalent to that of a
1300 lines=mm diffraction grating, while still achieving direct-view geometry (zero mean beam deviation). The double Amici design angles as a functionof dispersion are illustrated in Fig. 12 for NLAK34 and N-SF66 glasses. Note that the doublet
and double Amici designs of Figs. 5 and 12 use thesame glasses and the same spectral range. If we
restrict the prism design space to systems having
maximum angles of incidence of 65°, then we obtain
the designs shown.
Comparing the double Amici prisms of Table 1’s
section (a) with their doublet counterparts, we cansee a modest improvement in the linearity, and very
similar glass combinations. Comparing the prisms
with the most linear dispersion (in Table 1 section(b)) with the most compact designs (Table 1 section(c)), we find that there is not a lot of flexibility in thisdesign space. If we allow angles of incidence >65°,
then the linearity can improve dramatically, but
the cost is a prism that is quite large in comparison
to the transmitted beam size. At smaller amounts ofdispersion, the trade-off is less severe, and substantial improvements in dispersion linearity can beachieved.
The double Amici glass choices in Table 3 differ
from the doublets of Table 1 in glass composition:
the highly linear double Amici designs use a very low
dispersion glass combined with an LAK/LAF glass,while the thinnest designs use a pair of highly dispersive materials. Figure 13 illustrates the layouts anddispersions for prism designs 1b and 1c from Table 3,

Fig. 11. (Color online) Double Amici prism design space, showing
contour plots of (a) the deviation δ and (b) the dispersion Δ for a
400–700 nm spectral range, and for a fused silica/N-LASF31A/
fused silica glass triplet, with (a) δ ¼ 0 and (b) Δ ¼ 2°. As in
Fig. 4, the dashed blue line indicates ðα1 ; α2 Þ designs meeting target values, the red solid lines indicate designs using linear Eq. (5),and the gray background indicates the valid design space.

Table 3.

Best Performing Double Amici Prisms for Design Targets δ  0 and Δ  4° (400 nm ≤ λ ≤ 700 nm, θ0  0)aGlass 1Glass 2α1 (deg)

α2 (deg)δ (deg)Δ (deg)NL (×104 )

SSR

0.903
1.000
1.000
0.929
0.951

0.239
0.269
0.302
0.336
0.339

4.31
4.54
4.79
5.55
4.04

4.000
4.000
4.000
4.000
4.000

1.770
1.814
1.830
1.859
1.860

5.74
5.97
6.03
6.13
6.21

4.000
4.000
4.000
4.000
4.000

2.995
2.937
3.019
2.962
2.984

11.35
11.09
11.45
11.18
11.33



1a
2a
3a
4a
5a

N-BAF4
N-BAF4
N-BAF52
N-BALF5
LITH-CAF2

1b
2b
3b
4b
5b

LITH-CAF2
LITH-CAF2
LITH-CAF2
LITH-CAF2
LITH-CAF2

1c
2c
3c
4c
5c

N-LAK34
N-SF66
N-LAK33A
N-SF66
N-LAK14λ range: 400–700 nm, Δ ¼ 1°, sorted by linearity
P-LASF47−82:04106.980.006
N-LASF31A−81:8697.68
0.000
N-LASF41−82:24103.91
0.000
N-LAF34−85:66104.260.002
N-FK5−74:33121.630.002
λ range: 400–700 nm, Δ ¼ 4°, sorted by linearity
N-LAK33A−95:0494.37
0.000
N-LAF34−92:0590.50
0.000
N-LAF35−92:0893.38
0.000
N-LAF21−89:4887.24
0.001
N-LASF44−88:1784.87
0.000
λ range: 400–700 nm, Δ ¼ 4°, sorted by thickness
N-SF66−32:5750.29
0.000
N-LAK3425.09−65:700.000
N-SF66−32:2351.55
0.000
N-LAK33A
25.77−64:97
0.000
N-SF66−33:68
49.57
0.000

a
Sections (a) and (b) of the table list the designs in order of dispersion linearity; section (c) lists the designs ordered by thickness. All of
the designs are constrained to angles of incidence <65°. The layout and dispersions for prisms 1a and 1b are illustrated in Fig. 13.

5008

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

Fig. 12. (Color online) (a) Design parameters of a nondeviating
N-LAK34/N-SF66 double Amici prism as a function of the target
dispersion. The kink in the curves occurs at Δ ¼ 23:5°, beyond
which the system can no longer achieve the target values. The
vertical bar at the right side of the plot indicates the design
configuration whose layout is shown in (b).

showing again that trying to achieve dispersion linearity involves a trade-off in prism compactness.
For prisms with more modest amounts of dispersion(<1° for dispersion across the full visible spectral
range, for example), the linearity achievable isgreatly increased. Thus, the fact that sections (b) and(c) of Table 3 have similar values of NL (within a factor of 2) is a result of the confined design space available for 4° dispersion across this spectral range.
Among the designs shown in Table 3, we can seethat the dispersion linearity is substantially worse
than for the doublet designs—a result due entirely tothe higher spectral dispersion requirements in the

double Amici designs. If we compare double Amici
prisms with doublets having the same dispersion,
however, we find what one would intuitively expect:a double Amici’s dispersion designs can be mademore linear than an equivalent doublet’s can.
One feature that a double Amici design can achievethat is difficult to accomplish with a doublet is zero
beam displacement (i.e., Δy ¼ 0). In some optical systems, and especially in prism arrays, it may be problematic to allow substantial beam displacement,
and this can be an important design constraint. ThisΔy ¼ 0 bound creates a nonlinear constraint on the 1 , and n
 2 . Also, since
relationship between θ0 , α1 , α2 , n
the beam deviation is a function of wavelength, thebeam displacement Δy will vary across the spectrum,
so that zero displacement is typically achieved for
only one wavelength and only one angle of incidence.
In typical double Amici prisms, however, the chromatic transverse displacement is quite small for onaxis illumination (θ0 ¼ 0).
If we assume θ0 ¼ 0, then zero displacement will
occur when the ray inside the second element is parallel to the optical axis (i.e., θ02 ¼ −β2 ). This simplifies
the equation of constraint (for Δy ¼ 0) relating α1and α2 to




1
n
α2
1
:
¼ arcsin
sin α1 þ arcsin
sin β1122nnUnder the small-angle approximations, this can befurther approximated by the simple linear equation
 2 ¼ 2n1 − 1:nThus, in order to achieve zero beam displacement,
we can expect the interior element of a double
Amici prism to use very high-index glasses, and theexterior elements to use very low-index glasses, inorder to satisfy this constraint. Calcium fluoride (i.e., ¼ 1:436) and N-LAK33A (LITHOTEC-CAF2, nn¼
1:762)—two prominent glasses in Table 3—do not

Fig. 13. (Color online) Layout, dispersion, and dispersion gradient for prism designs 1b and 1c from Table 3. The rays drawn in the prism
layouts are for the central wavelength, λ. The dashed line in (a) is the best-fit linear portion of 1b’s dispersion curve. The gradient curves in(b) indicate that the spectral sampling rate in the blue end of the spectrum will be 6× (design 1b) or 10× (design 1c) that in the red end due
to dispersion NL.
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5009

quite satisfy this relation, and so allow some beamdisplacement. A glass pair that gives very low displa ¼ 1:462)
cement is fused silica (i.e., LITHOSIL-Q, n
and P-SF67 (n ¼ 1:933).
10.

Conclusion

Historically, the first spectrometers used prisms as
dispersing elements, a characteristic that continued
until the development of advanced ruling enginesand high-resolution grating spectrometers in thetwentieth century. With these new grating dispersers, the use of prisms declined sharply, a trend
driven in part by problems with volume absorption within the prism glass and the lack of good antireflection coatings on the prism interfaces. Gratings
acquired a reputation not only of having much stronger dispersion, but also of having greater throughput. But the modern development of high-purity
glasses and excellent thin film coatings has largely
eliminated prisms’ difficulties with absorption andpartial reflections. In contrast, gratings still struggle
with the loss and stray light incurred due to lightdiffracted into unwanted diffraction orders. For themodern optical designer, prisms can thus provide
higher throughput as well as decreased stray light.
There remains a strong perception among designers that prisms are incapable of providing
dispersion comparable to gratings. The results ofSection 9, however, show that with a proper choiceof glasses, compound prisms can provide comparable
dispersion to that of many gratings, even while maintaining a direct-vision geometry. There also remains
a perception, derived from well-known theory [56]that prisms will always have a lower resolving power
than their grating counterparts, a result based on theassumption of using singlet prisms in their minimum
deviation geometry. We have shown, however, thatcompound prisms can achieve substantially higher
dispersion than singlets can, while also maintaining
the input beam width, such that the resolving power
will actually surpass that of gratings with equivalentdispersion.
Finally, compound prisms provide a number ofdegrees of freedom with which to customize the dispersion characteristics, allowing one to tune thedisperser to any special system requirements. Thisis a flexibility that is largely unavailable to standard
gratings, though it can be realized by combining
gratings with singlet prisms [57]. Thus, while highquality compound prisms may be more expensive
than comparable gratings, their throughput is oftenmuch higher, their design flexibility is greater, andtheir direct-vision capability can be a substantial advantage when system compactness is needed.
In order to take advantage of these features, however, an optical designer needs to have a better understanding of the design space than that provided
by the conventional linear design as in Eqs. (2)–(7).
Commercial optical design software provides all of
the necessary functionality for this work, but can
be clumsy to customize for the special requirements
5010

APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011of compound prism design. For example, although
the prism apex angle α design parameter is a continuous variable, and thus easy for optimization
routines to use, the glass refractive index and dispersion nðλÞ is a discrete quantity constrained by available glass choices. Optimizing over glasses thus
requires a search over available glass combinations.
And due to the nonlinear design space, tailoring themerit function is a necessity for obtaining a good setof designs from which to choose.
In the third paper of this series [18], we go on to
use the methods described here to design linear-inwavenumber prisms, a property that turns out to bemuch easier to achieve than linearity in wavelength.
Dispersion linearity in wavenumber allows us to design high-dispersion direct-vision prisms specificallyfor the demanding application of optical coherence
tomography, which also requires using alternative
glasses for the near-infrared spectral range.
Among the designs shown in Sections 4 and 9
above, one of the undesirable properties of prism
spectra is their dispersion nonlinearity. Although
one can select the optimal glass combination, thishas limited ability to improve on the dispersion linearity. In order to achieve a large improvement in linearity, it is necessary to add more degrees of freedom
into the design, such as using asymmetric threeelement prisms (“triplets”) or symmetric fiveelement prisms (Janssen prisms). These compoundprisms have sufficient degrees of freedom to provide
impressive performance, in both dispersion linearityand dispersive power, while maintaining direct-viewgeometry.
We encourage interested readers to download andmodify the design code we have written for this work,
available at the authors’ website [2].
This work was partially supported by National
Institutes of Health (NIH) grants RO1-CA124319
and R21-EB009186.
References and Note
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in stellar spectra],” Annali del Reale Museo di Fisica e Storia
Naturale di Firenze 1, 1–20 (1866), in Italian.
2. http://www.owlnet.rice.edu/~tt3/.
3. G. B. Donati, “Intorno alle strie degli spettri stellari [On lines
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11. H. Schellen, Spectrum Analysis in its Application to Terrestrial
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Bodies (Appleton, 1872), p.82, translated by J. Lassell and
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prisma [A direct vision spectroscope with a single prism],”Ann. Phys. 150, 636–640 (1873), in German.
13. F. Fuchs, “Vorschläge zur construction einiger optischer
vorichtungen [Proposals for construction of some optical devices],” Z. Instrumentenkd. 1, 326–329 (1881), in German.
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vision spectroscope,” Proc. R. Soc. London 41, 449–452 (1886).
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vision dispersion prism systems,” Appl. Opt. 9, 859–862(1970).
16. P. G. Tait, “On anomalous spectra, and on a simple directvision spectroscope,” Proc. R. Soc. Edinburgh 7, 410–414(1872).
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