Compound prism design principles, III:linear-in-wavenumber and opticalcoherence tomography prisms
Nathan Hagen1,2 and Tomasz S. Tkaczyk1,*
1

Department of Bioengineering, Rice University, Houston Texas 77005, USA2e-mail:nhagen@optics.arizona.edu*Corresponding author: ttkaczyk@rice.edu
Received 26 May 2011; accepted 5 July 2011;
posted 12 July 2011 (Doc. ID 148165); published 30 August 2011

We extend the work of the first two papers in this series [Appl. Opt. 50, 4998–5011 (2011), Appl. Opt. 50,
5012–5022 (2011)] to design compound prisms for linear-in-wavenumber dispersion, especially for application in spectral domain optical coherence tomography (OCT). These dispersive prism designs are believed to be the first to meet the requirements of high resolution OCT systems in direct-view geometry,
where they can be used to shrink system size, to improve light throughput, to reduce stray light, and to
reduce errors resulting from interpolating between wavelength- and wavenumber-sampled domains. We
show prism designs that can be used for thermal sources or for wideband superluminescent diodes centered around wavelengths 850, 900, 1300, and 1375 nm. © 2011 Optical Society of America
OCIS codes: 230.5480, 080.2740, 260.2030, 300.6190, 110.4500.

1. Introduction

The first two papers of this series [1,2] (called Paper I
and Paper II below) introduced basic design principles for compound prisms, concentrating on compact and linear-in-wavelength dispersive prisms.
We showed that it is possible to achieve a degreeof dispersion previously only thought possible withgratings, 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 multiple
parameters to adjust for optimizing prism performance. Here we extend our design method to include
prisms whose dispersion is linear in the wavenumber
domain, and apply this method to the spectral ranges
and high dispersion required by spectral domain optical coherence tomography (OCT) systems, an application that has previously been considered as too
0003-6935/11/255023-08$15.00/0© 2011 Optical Society of America

demanding for using prisms as dispersive elements.
We show that not only can prisms replace gratings inthis application, but that they can be used in directview geometry and provide better dispersion linearity in the wavenumber domain than gratings can.
2. Linear-in-Wavenumber Dispersion

Spectral interferometric applications typically findit desirable to have a spectrum which is sampled uniformly in the wavenumber rather than the uniformity in wavelength common to grating-based
dispersive spectrometers, as this allows direct application of Fourier transform algorithms on the measured spectrum [3–5]. A spectrometer that samples
the spectrum uniformly in the wavenumber domain
requires a disperser whose ray deviation angle δvaries linearly with wavenumber σ, i.e., δðσÞ ∝ σ for
σ ¼ 1=λ. The refractive index of typical glass has anearly linear relationship with wavenumber, a characteristic where prisms have an advantage overgratings. Figure 1 shows the dispersion curve for
SF66 glass and compares its linearity in both thewavelength and wavenumber domains with that of1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5023

Snell’s law) at each interface. For a triplet compoundprism, the refraction equations are9
θ1 ¼ θ0 þ β1
θ03 ¼ arcsinðnn23 sin θ3 Þ >
>
>
>
>
1
0
0
>
θ1 ¼ arcsinðn1 sin θ1 Þ θ4 ¼ θ3 − α3=00
θ 2 ¼ θ 1 − α1
θ4 ¼ arcsinðn3 sin θ4 Þ ; ð1Þ
>>>θ02 ¼ arcsinðnn12 sin θ2 Þ θ5 ¼ θ04 þ γ 3>>>;
θ3 ¼ θ02 − α2Fig. 1. (Color online) Dispersion of SF66 glass is substantiallynonlinear when plotted against the wavelength (black solid curve),
but much less so when plotted against the wavenumber (black
dashed curve). The spectral dispersion of a grating, on the other
hand, is linear in the wavelength domain (red solid curve), but
substantially nonlinear in the wavenumber domain (red dashedcurve). The nonlinearity parameter NL used to label the grating
curves is defined in (2) and (3).

grating dispersion. Other glasses show a similar
shape. Since a prism’s angular dispersion is nonlinearly related to the glass dispersion, the linearityof nðσÞ is not by itself enough to provide linear-inwavenumber spectral dispersion.
In Paper I we presented a design procedure fordirect-vision doublet and double-Amici dispersive
prisms, and then adapted the procedure to triplet
and Janssen prisms in Paper II. There we showedthat the additional degrees of freedom available totriplet prisms above those of the better known
double-Amici prisms allows a designer to improve on
dispersion linearity while maintaining other prismcharacteristics. The designs presented in Paper II
achieved a limited success in modeling prisms withlinear dispersion in the wavelength domain. With
the switch to the wavenumber domain, achieving linearity is much easier, so that it becomes possible todesign prisms with negligible nonlinearity.
The nonlinear equation relating the deviation
angle δ to the prism design parameters can be derived by concatenating the refraction equation (using

where β1 ¼ α1 þ 12 α2 and γ 3 ¼ α3 þ 12 α2 . All of thequantities used here are defined in Fig. 2. The angles
θi and θ0i are the angles of incidence and refraction atinterface i, and the ray deviation angle δ is defined asδ ¼ θ0 − θ5 . The quantities ni ðλÞ and αi are the glass
refractive indices and prism apex angles. The deviation angle is thus a nonlinear function of all of theseparameters, δðn1 ðλÞ; n2 ðλÞ; n3 ðλÞ; α1 ; α2 ; α3 ; θ0 Þ. For adouble-Amici prism, α3 ¼ α1 and n3 ðλÞ ¼ n1 ðλÞ.
Modifying the prism design algorithms from
Papers I and II for wavenumber-sampled data is simple: one need only sample the glass refractive indicesat uniformly spaced wavenumbers within the same
spectral range. Writing the dependence explicitly,
the nonlinearity penalty term in the merit function
for each setup becomesZ  2 d δ 
NLλ ¼  2 dλ;dλð2Þ

Z  2 d δ 
NLσ ¼  2 dσ;
dσ

ð3Þ

or

where the merit function is written as
M nl ¼ M 0 þ wnl NL;

ð4Þ

with NLλ or NLσ in the place of NL depending onwhether λ or σ space is uniformly sampled. The“base” merit function M 0 is

Fig. 2. Example triplet compound prism, where the second element is assumed to be oriented symmetrically with respect to the opticalaxis normal. The system shown here has prism apex angles ðα1 ; α2 ; α3 Þ ¼ ð−85°; 90°; −55°Þ, beam displacement Δy, and axial thicknesses t1 ,t2 , and t3 . For this example, the input ray has an angle θ0 ¼ 0°, such that δ ¼ 15°.
5024

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

M 0 ¼ ðδ − δ Þ2 þ ðΔ − Δ Þ2 þ Θ2 :

ð5ÞHere δ is the ray deviation angle for the central
wavenumber (i.e., δ ≡ δðσ Þ for σ ¼ ðσ min þ σ max Þ=2), Δ
is the spectral dispersion (Δ ≡ δðσ max Þ − δðσ min Þ), Θ
is a penalty term for large ray incidence angles(see Eq. 12 of Paper I), and wnl is the relative
weight applied to the nonlinearity penalty. In
order to clearly discriminate between the linear-inwavelength dispersion discussed in Paper I and thelinear-in-wavenumber dispersion discussed here, we
define the terms “λ-linearity” quantified by NLλ, and“σ-linearity” by NLσ, where the subscript indicates inwhich space the function is uniformly sampled.
Using the same system parameters as the λlinearized designs in Tables 1 and 2 of Paper I,
Table 1 below shows the results of designing prisms
for σ-linearized dispersion. All of the designs inTable 1 have been restricted to ray angles of incidence less than 65° at all prism interfaces. In the
doublet prisms of section (a), σ-linearity improvement is achieved by pairing two glasses with nearly
identical refractive indices but substantially different dispersions. This setup produces equal andopposite ray deviation at the first and last interfacesof the prism, but dispersion without deviation at
the central interface. The double-Amici and triplet
designs follow a different principle, pairing a very
low dispersion glass with an LAF or LAK material.
(Other glass combinations will work, but these produce the best dispersion linearity.) Note that in all of
the tabulated data below, we remove the prefixes on
the glass names (i.e., N-BK10 is written as BK10) in
order to keep the tables compact. In some cases, wehave also abbreviated names, such as LITHOSIL-Q/
LITHQ, LITHOTEC-CAF2/CAF2, CLEARTRAN/
CLEAR, and ZNS_BROAD/ZNSBR.
Table 1.

In addition to the nonlinearity value NL, two convenient measures of a prism design are the spectral
sampling ratio (SSR) and the beam compression
factor K. SSR is a measure introduced in Paper I(Eq. 1) to describe how much the spectral bands are
stretched by nonlinear dispersion, whereas K (introduced in Eq. 2 of Paper II) indicates the change inbeam width between the incident and exiting beams.
Typically, one wishes to have SSR ¼ 1 and K ≥ 0:9, as
these two values correspond to highly linear spectral
dispersion and to a beam experiencing little compression along the spectral spread axis.
While the designs of section (b) in Table 1 show larger nonlinearity values than those of section (a), thisis a result of a higher dispersion target. (A doubleAmici prism designed to the same target values asa doublet will almost always have better performance.
Exceptions occur when a highly asymmetric layout isneeded for good performance.) See Fig. 3 for the resulting dispersion curves of these designs. The prismsof section (c) have dispersion values somewhat lower
than the target as a result of the competing terms ofthe merit function. One can achieve a closer match to
the target by increasing the weight of the ðΔ − Δ Þ2term in M nl relative to wnl , although at some tradeoff in nonlinearity.
Comparing these results with those of the corresponding tables in Papers I and II, we find thatNLσ is much smaller than the NLλ values obtainedwith equivalent prisms:
Prism Type

Paper, Table, Design

Δ

NLλ

NLσ

DoubletDoubletDouble-Amici
Double-Amici
Triplet
Triplet1,1,6a3,1,1a1,2,1a3,1,1b2,1,1b3,1,1c

1°
1°
4°
4°
4°
4°

0.323
1.770
1.762
-

0.069
0.479
0.411

Linear-in-Wavenumber Dispersive Prisms for Wideband Visible Wavelength Spectral Interferometry: the Best Performing Prisms are Listedin Order of Dispersion Linearity in the Wavenumber Domain and Optimized over the Schott Glass Catalog a

Prism Type

Glass 1Glass 2

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)

NLσ

SSR

K

1.000
0.990
1.000
1.000
1.000

0.069
0.071
0.071
0.072
0.076

1.47
1.38
1.43
1.50
1.41

0.49
0.84
0.40
0.47
0.72

4.000
4.000
4.000
4.000
4.000

0.479
0.506
0.513
0.524
0.528

1.89
1.97
1.99
1.98
2.02

1.00
1.00
1.00
1.00
1.00

3.957
3.901
3.997
3.901
3.906

0.411
0.452
0.478
0.487
0.494

1.79
1.84
1.89
1.95
1.96

0.59
0.95
1.00
0.95
0.95



1a.
2a.
3a.
4a.
5a.

DoubletDoubletDoubletDoubletDoublet

LAF34
LITHQ
LAK14
LASF44
FK5

1b.
2b.
3b.
4b.
5b.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici

CAF2
CAF2
CAF2
CAF2
CAF2

1c.
2c.
3c.
4c.
5c.

Triplet
Triplet
Triplet
Triplet
TripletLASF41
CAF2
CAF2
FK51A
CAF2

merit func: M 0 , λ range: 400–700 nm, Δ ¼ 1°
79.79
−101:860.000
93.15−99:570.001
77.86−106:750.000
76.73−101:280.000
87.23−100:72
0.000
Merit func: M 0 , λ range: 400–700 nm, Δ ¼ 4°
LAK33A
95.25−94:29
0.000
LAF34
92.27
−90:42
0.000
LAF35
92.32
−93:30
0.000
LAK10
93.73
−96:76
0.000
LAF21
89.69
−87:150.000
Merit func: M nl þ ðK − 1Þ2 , λ range: 400–700 nm, Δ ¼ 4°
SK11
PSK53A−59:61−10:08
81.20
0.016
LAK34
CAF2
95.55
−97:7097.39−0:019
LAK33A
CAF2
95.22−94:2895.22
0.000
LAK33ACAF2
93.00−97:6794.99−0:003
LAK33A
PK51
92.90−101:6795.91−0:004

SSK5
CAF2
BAK4
BAF10
CAF2

a
All of the designs are nondeviating (δ ¼ 0), assume axial illumination (θ0 ¼ 0), and are constrained to angles of interface <65°. The
dispersion gradient curves of all 10 prisms are illustrated in Fig. 3 together with the layouts of prisms 1a and 1b.

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

5025

Fig. 3. (Color online) Layouts and dispersion gradient, dδ=dσ, of the designs shown in Table 1 (only three representative layouts areshown). (a) Δ ¼ 1° doublets, (b) Δ ¼ 4° double-Amici prisms, (c) Δ ¼ 4° triplets. Ideal linear dispersion produces a horizontal line in theseplots, so that any variation from that line represents the nonlinear dispersion. Note that the spectra here are sampled uniformly inwavenumber; the corresponding wavelength labels are given on the upper x axis for reference.

3. Prism Designs for Spectral Domain OCT

One of the most important applications for σ-linear
dispersion is spectral domain OCT [5–8], which istypically performed in the near-IR or short-wave
IR (850, 1060, and 1300 nm are common choices
for the central wavelength), where penetration depth
into tissue is substantially greater than in the visible. However, the dispersion of commonly available
glasses is generally much smaller, and the spectral
ranges used in OCT (typically 20‒100 nm FWHM)
are also generally narrower than those used for general purpose spectrometry. This combination of narrow spectral range and greatly reduced dispersion
makes dispersive prism design a challenge. For example, a typical OCT setup may use a 75‒100 mm
focal length lens and a 1024-element detector arraywith 10 μm pixels, so that an angular dispersion of
Δ ¼ 5:9°‒7:8° is required. With diffraction gratings,
this can be achieved with a groove density of 500‒
3500 lines per mm(see Appendix A), depending on
the spectral width (20 or 100 nm FWHM) and focal
length used. Prisms generally require a wide spectral
range in order to achieve sufficient dispersion, and so
we aim for 6°‒8° dispersion across spectral ranges ofat least 100 nm—a setup which aims for a mean spectral sampling rate of Δλ ≈ 0:1 nm, and thus a maximum imaging depth of [9] zmax ¼ λ2 =ð4nΔλÞ ≈ 1:2 mmin biological tissue (n ≈ 1:45).
5026

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

If the 6°‒10° dispersion used for the prism design
is insufficient for a given OCT instrument, anothermeans of achieving sufficient sampling is to increase
the focal length of the spectrometer’s reimaging lens.
Since most spectral domain OCT systems operatewith diffraction-limited imaging, simply increasing
the focal length causes a corresponding increase in
the size of the focal spot, so that one must also increase the size of the beam by using a larger aperture
objective lens (i.e., a lens with a longer focal length
but the same NA). The trade-off is that a higher quality objective lens is needed.
Table 2 shows a number of prisms designed for σlinearized dispersion with parameters appropriate
for use in retinal OCT systems, with spectral ranges
centered at 850 nm. Designs in sections (a) and (b) of
the table are for a 6° and 8° dispersion angle, while
the designs in (c) and (d) use wider spectral ranges
and are for 10° dispersion. Until now, the algorithm
has constrained the ray angles of incidence in theprisms to θ ≤ 65°. For the double-Amici designs used
here, however, the dispersion targets used in Table 2
are no longer obtainable with this constraint whenusing the ZEMAX Infrared catalog glasses. Thus,
the double-Amici designs in Table 2 use θlim ¼ 70°
instead in order to allow sufficient dispersion.
Although the triplet prisms, on the other hand, have

Table 2. OCT Prism Designs: the Best Performing Three-Element Prisms Listed in Order of Dispersion σ-Linearity, for δ  0 and θ0  0, and optimized(for sections (a)–(c)) over the ZEMAX Infrared Glass Catalog. Section (d) Optimizes over the Combined Schott Catalog and Infrared Catalog a

Prism Type

Glass 1Glass 2

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)NLσ (×104 )

SSR

K

5.889
5.999
6.000
5.789
6.000
5.985
5.971
5.907
5.945
6.000

0.031
0.032
0.034
0.034
0.034
0.031
0.031
0.031
0.032
0.032

1.01
1.02
1.02
1.03
1.02
1.02
1.01
1.01
1.01
1.02

1.00
1.00
1.00
1.00
1.00
0.99
0.97
0.91
0.95
1.00

8.000
7.998
8.000
7.795
7.999
7.931
8.000
8.001
7.929
8.000

0.068
0.073
0.083
0.107
0.112
0.068
0.068
0.069
0.069
0.069

1.03
1.03
1.03
1.05
1.06
1.02
1.02
1.03
1.03
1.03

1.00
1.00
1.00
1.00
1.00
0.92
1.00
1.00
0.92
1.00

9.999
10.001
9.999
9.999
10.000
9.909
9.931
9.916
9.960
9.914

0.218
0.218
0.220
0.246
0.262
0.184
0.200
0.201
0.208
0.215

1.07
1.08
1.08
1.10
1.08
1.09
1.07
1.10
1.06
1.10

0.97
0.97
0.97
0.97
0.97
0.84
0.88
0.85
0.94
0.84

9.998
10.00010.00110.000
9.872
9.868
9.857
9.845
9.854
9.840

1.682
1.770
1.825
1.844
1.850
0.726
0.853
0.869
0.874
0.878

2.16
2.33
2.38
2.40
2.29
1.43
1.55
1.39
1.58
1.60

1.00
1.00
1.00
1.00
1.00
0.53
0.57
0.52
0.57
0.52

þ ðK − 1Þ2 , λ range: 800–900 nm, Δ ¼ 6°

1a.
2a.
3a.
4a.
5a.
6a.
7a.
8a.
9a.
10a.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
Triplet
Triplet
Triplet
Triplet
Triplet

ZBLA
BAF2
KCL
F_SILICA
ZBLAN
CAF2
CAF2
IRG9
SRF2
BAF2

1b.
2b.
3b.
4b.
5b.
6b.
7b.
8b.
9b.
10b.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
Triplet
Triplet
Triplet
Triplet
Triplet

IRGN6
GEO2
IRG7
IRGN6
IRG15
BAF2
BEO
IRG9
IRG9
CALCITE

1c.
2c.
3c.
4c.
5c.
6c.
7c.
8c.
9c.
10c.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
Triplet
Triplet
Triplet
Triplet
Triplet

CAF2
CAF2
SRF2
KCL
BAF2
CAF2
LIF
BAF2
CAF2
CAF2

1d.
2d.
3d.
4d.
5d.
6d.
7d.
8d.
9d.
10d.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
triplet
Triplet
Triplet
Triplet
Triplet

MGF2
MGF2
MGF2
MGF2
MGF2
GEO2
F5
F5
F5
LASF44a

Merit func: M nl
ZNSE
103.26−66:400.018
KRS5
100.03−62:310.000
KRS5
101.02−63:740.001
KRS5100.80−61:370.039
KRS5
100.59−63:11
0.001
KRS5
IRG9
100.67−61:3599.93 −0:003
KRS5
ZBLAN
100.61−61:22100.20 −0:004
ZNSE
IRG15
102.58−65:16103.23
0.003
KRS5
F_SILICA
100.25−60:50100.74 −0:003
KRS5
BAF2
100.03−62:32100.030.000
Merit func: M nl þ ðK − 1Þ2 , λ range: 800–900 nm, Δ ¼ 8°
KRS5103.23−72:34
0.002
KRS5
103.59−73:75−0:002
KRS5
104.33−70:860.001
ZNSE
106.35−72:360.049
KRS5
104.27−68:74−0:000ZNSE
BEO
106.30−72:59103.83 −0:001
KRS5
IRG9
100.56−73:13104.030.000
KRS5
CALCITE
105.00−71:31102.37 −0:001
ZNSE
MGO
106.89−73:78104.32
0.005
KRS5
IRG15
102.63−72:97104.480.000
Merit func: M nl þ ðK − 1Þ2 , λ range: 750–1050 nm, Δ ¼ 10°
SRTIO3102.58−65:260.001
SRTIO3102.59−65:270.000
SRTIO3102.58−65:59−0:002SRTIO3103.50−70:090.002
SRTIO3101.81−68:23−0:001
CLEAR
SRF2
103.86−67:73106.39 −0:010
SRTIO3
CAF2
101.66−63:32102.86 −0:006
ZNSBR
LIF
103.94−67:89107.87 −0:008
SRTIO3LIF
101.79−63:58103.71 −0:007
CLEAR
KCL
105.06−70:13106.67 −0:008
Merit func: M nl þ ðK − 1Þ2 , λ range: 400–1050 nm, Δ ¼ 10°
LAK33A109.63−94:860.000
LAF34
107.90−92:32
0.002
LAF21
106.30−90:13−0:001
LAF35
108.26−95:03−0:003LAK10
108.92−97:53−0:001
SAPPHIRE MGF2
79.45
−18:89−71:98
0.014
LAK9
MGF2
83.83
−27:67−66:93
0.019
ALON
IRG9
67.62
4.75
−82:34
0.031
LAK14
MGF2
83.45
−26:90−67:300.022
BAF2
SAPPHIRE
76.56−13:12−71:23 −0:073

All designs have been constrained to angles of incidence <70°.

no difficulty achieving the design targets under a 65°
constraint, the triplets in Table 2 use θlim ¼ 70° foreasy comparison with the double-Amici designs.
The results of Table 2 show that the prisms in the
800–900 nm spectral range can achieve a spectral
sampling rate dδ=dσ which varies by no more than
2% across the entire spectral range. The prisms in
sections (c) and (d), however, have a more restrictive
design space and thus have much higher dispersion
nonlinearity.
In the 800–900 nm spectral range, all of the designsof Table 2 use either KRS5 (thallium bromoiodide) or ZNSE (zinc-selenide) glass, the two mostdispersive amorphous materials of the Infraredcatalog in this spectral range. The design targets
for the 750–1050 nm spectral range, despite having
a higher dispersion angle, are easier for prisms toachieve and so we see that SRTIO3 and CLEARTRAN
become the preferred materials in section (c) due totheir lower dispersion nonlinearity than KRS5 and
ZNSE. In general, we can also see that due to the restriction on beam compression, triplet prisms provide
only a marginal improvement over double-Amici designs for the requirements used here.
In the widest spectral range (400–1050 nm), thetriplet designs provide a large improvement in nonlinearity over the double-Amici designs, though at
the cost of compressing the beam in the dispersion
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5027

axis by 50%. As one increases the weight of the beamcompression penalty relative to the other terms ofthe merit function, the resulting K value of the best
designs comes closer to 1, but the NL and SSR values
also come closer to the values of the double-Amici
designs.
There exist some extraordinarily dispersive materials in the 800–900 nm spectral range: the chalcogenides proustite (silver sulfarsenide, AgAsS3 ) andsilver gallium selenide (AgGaSe2 ) have about twice
the dispersion of KRS5. Both of these crystals havean absorption band in the red end of the visible spectrum, and thus appear a deep red color to the eye.
Unfortunately, these crystals are strongly birefringent, making their implementation in spectrometers
problematic. There are amorphous materials ofsimilar composition [10], but these are not widely
available.
The types of light sources, which can be used forthese spectral ranges for OCT, would include
superluminescent diodes (e.g., Superlum T870-HP),
thermal sources [11–13] (e.g., a Tungsten-Halogen
bulb, or Xenon lamp), or tunable Ti:sapphire lasers.
In the 800–900 nm spectral ranges, near-IR
Table 3.

optimized silicon detector arrays are ideal, whereas
for the 400–1050 nm spectral range a deep-depletion
silicon CCD [14] is an ideal detector. For the
750–1050 nm range, either a deep-depletion Si CCD,
or a Ge, InSb, or HgCdTe detector array is needed.
4. OCT Prisms for Longer Wavelengths

To show that prisms can be used for the longer
wavelength spectral ranges analyzed by spectral
domain OCT, Table 3 presents a number of prism designs tailored to the spectral ranges of three commercially available wideband superluminescent diodes.
Achieving 6° of angular dispersion for wavelengths
beyond 1 μm generally requires a spectral band
which is at least 150 nm wide, and this has defined
the commercial sources we show here for our example designs. Thus, for example, the spectral band ofsection (a), 1:175–1:425 μm, can be used with theThorlabs LS2000B superluminescent diode, the1:21–1:41 μm band of section (b) with InPhenix’s
IPSDD1308, and the 1:25–1:50 μm range of section(c) with Superlum’s Q1350-HP.
The results of using the double-Amici and triplet
prism design algorithms for these spectral ranges

OCT Prism Designs: the Best Performing Double-Amici Prisms Listed in Order of Dispersion σ-Linearity, for δ  0 andθ0  0 Optimized over the ZEMAX Infrared Glass Catalog a

Prism Type

Glass 1Glass 2

Glass 3α1 (deg)

α2 (deg)

α3 (deg)δ (deg)Δ (deg)NLσ (×104 )

SSR

K

0.139
0.157
0.170
0.172
0.190
6.029
5.928
5.946
5.915
5.908

0.570
0.587
0.601
0.604
0.624
0.034
0.052
0.055
0.098
0.126

1.14
1.16
1.17
1.17
1.19
1.03
1.05
1.03
1.10
1.13

1.00
1.00
1.00
1.00
1.00
0.69
0.70
0.68
0.87
0.94

6.001
6.001
5.871
6.000
5.824
5.904
5.908
6.081
6.000
6.001

0.074
0.076
0.160
0.175
0.198
0.040
0.040
0.044
0.045
0.048

1.06
1.07
1.16
1.18
1.21
1.02
1.03
1.03
1.03
1.03

1.00
1.00
1.00
1.00
1.00
0.91
0.91
0.80
1.00
1.00

6.001
6.000
6.001
6.000
5.886
5.976
5.987
6.061
6.011
6.000

0.124
0.129
0.210
0.221
0.304
0.024
0.049
0.050
0.067
0.072

1.12
1.13
1.22
1.23
1.33
1.02
1.04
1.02
1.05
1.07

1.00
1.00
1.00
1.00
1.00
0.53
0.85
0.58
0.87
1.00

þ ðK − 1Þ2 , λ range: 1:175–1:425 μm, Δ ¼ 6°

1a.
2a.
3a.
4a.
5a.
6a.
7a.
8a.
9a.
10a.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
Triplet
Triplet
Triplettriplettriplet

KCL
BAF2
KBR
NACL
KBR
IRG9
IRG9
F_SILICA
NACL
KBR

1b.
2b.
3b.
4b.
5b.
6b.
7b.
8b.
9b.
10b.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
Triplet
Triplet
Triplet
Triplettriplet

NACL
KBR
CSBR
CSBR
GEO2
KBR
KBR
KCL
CSBR
CSBR

1c.
2c.
3c.
4c.
5c.
6b.
7b.
8b.
9b.
10b.

d-Amici
d-Amici
d-Amici
d-Amici
d-Amici
Triplet
Triplet
Triplet
Triplet
Triplet

KBR
NACL
CSBR
CSBR
GEO2
SRF2
KBR
IRG9
KCL
CSBR

a

Merit func: M nl
CDSE
100.72−62:36−0:001
CDSE
100.51−61:470.001
AMTIR1101.94−65:810.000
AMTIR1101.69−64:880.000
CDSE
100.10−66:81
0.000
KRS5
KCL
103.98−67:96108.82 −0:066
KRS5
BAF2
103.71−67:41108.51 −0:001
SRTIO3IRG11
110.54−81:09111.38 −0:005
AMTIR1 MGF2
99.61
−59:23104.38
0.096
KRS5
CSBR
107.02−78:11105.890.006
Merit func: M nl þ ðK − 1Þ2 , λ range: 1:21–1:41 μm, Δ ¼ 6°
CDSE103.83−67:820.001
CDSE
104.00−68:730.000
AMTIR1102.80−75:81
0.015
CDSE
101.22−77:400.002
AMTIR1105.61−71:01
0.033
CDSE
CAF2
102.21−64:42105.49 −0:006
CDSE
SRF2
102.29−64:58105.45 −0:007
CDSE
IRG9
101.87−63:74104.97
0.050
CDSE
MGF2
102.02−67:61107.09 −0:000
AMTIR1 SRF2
103.29−68:18107.140.001
Merit func: M nl þ ðK − 1Þ2 , λ range: 1:25–1:50 μm, Δ ¼ 6°
CDSE103.26−68:540.000
CDSE
103.12−67:650.001
AMTIR1102.39−75:790.000
CDSE
100.35−77:190.002
AMTIR1105.64−71:15
0.016
KRS5
ZBLA
104.03−68:07110.40
0.061
CDSE
MGF2
101.21−62:42106.41 −0:078
SRTIO3
BEO
108.40−83:94111.38 −0:019
CDSE
KCL
101.80−63:60103.79 −0:014
AMTIR1 MGF2
103.07−66:29108.010.000

All designs have been constrained to angles of incidence < 70°.

5028

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

are shown in Table 3. We can see that the dispersion nonlinearity for the double-Amici prisms is
substantially higher than it was for the 800–900 nm
spectral range used for the retinal OCT prisms. If,
however, we compare the triplet designs here withthose of Table 2, we find only slight differences—atthese longer wavelengths the triplets can provide asubstantial improvement in dispersion σ-linearity
over their double-Amici counterparts. If one wishes
to use these prisms to eliminate the interpolation
step typically used in OCT reconstruction algorithms, then the latter can provide data with minimal artifacts.
A remarkable feature of the designs in Table 3 isthat all of them share an almost identical layout, regardless of the spectral ranges used. In each of thethree example spectral ranges, one can find designs
in which the spectral sampling rate varies by no
more than 3% across the full spectral range, albeit
with some trade-off in beam compression.
CDSE and AMTIR1 are used for the central element of most of the designs in Table 3, whereas there
is considerable variety in the materials used for theouter elements. For the three spectral ranges used
here, CDSE and AMTIR1 are the most dispersivematerials in the Schott and Infrared catalogs used.
5. Engineering Issues for Prism Selection

In addition to factors such as dispersion nonlinearityand prism thickness, there are important engineering issues which may decide whether a given prism
design is optimal for a system. These can include theprice of obtaining optical quality material (ZnSe, forexample, is typically more than 10 times the price ofcommon borosilicate glasses), and properties such as
hygroscopy, water solubility, birefringence, thermal
stability, toxicity, and corrosiveness.
Hygroscopy describes a material’s tendency to takeup water from its environment, causing a change inthe material optical properties. This can occur either
through direct contact with water (as is common in
biological imaging) or through exposure to humid
air, protection against which requires a sealed environment. Since dispersive prisms are rarely used inwater contact, water solubility is less important thanhygroscopy per se, though the two properties are closely linked. The materials used in the prism designsof Tables 2 and 3, which have significant hygroscopy,
are BAF2, CSBR, KBR, KCL, LIF, MGO, and NACL.
Prism birefringence is another property which cancause difficulty to a spectrometer, due to the difference in dispersion values between the ordinary andextraordinary beams transmitted by the material.
While one can compensate for this to some extent
by cutting each single prism element as two symmetric pieces from orthogonal crystal orientations,an easier solution is of course to avoid birefringent
materials in the design. Birefringent materials
used in the prism designs of Tables 2 and 3 include
BEO (0.0156), CALCITE (−0:172), CDSE (0.02), andMGF2 (0.0118), where the parenthetical values givethe amount of birefringence Δn at λ ¼ 1 μm.
Among IR-transmitting glasses, toxicity is anotherimportant material characteristic, and is a property
that can also influence cost and material availability.
Toxic materials used in the prism designs of Tables 2
and 3 include CDSE, CSBR, KBR, KCL, and KRS5.
In general, robust coatings of such materials canmake them safe for general laboratory use, but their
toxicity prior to coating restricts the manufacture ofprisms made from them to specialized optics companies.
6. Conclusion

The OCT prism designs provided in Tables 2 and 3
show that it is possible to provide practical directvision OCT prisms in which the angular dispersion
is sufficiently linear-in-wavenumber to minimize
or eliminate the commonly used interpolation step(from wavelength to wavenumber sampling) prior to
fast Fourier transform computation [4,5]. Moreover,the use of prism dispersers in place of gratings allow
system designers to reduce stray light, to improve optical throughput, and to use the convenient directview geometry. Despite these advantages, somedifficulties remain for using compound prisms in
OCT. Although one can achieve substantial dispersion with these elements, their dispersive power remains weaker than their grating counterparts, sothat a relatively wide spectral range is needed in order to achieve sufficient sampling of the spectrum.
One can, in general, compensate for this by using
longer focal lengths for both the objective lens andfor the spectrometer’s reimaging lens while maintaining the objective lens’ NA. The main trade-off
in this approach is a reduction in the depth of focus,
which causes signal loss for measuring reflections
from surfaces deep within the sample.
The algorithms used for designing the OCT prisms
use the techniques discussed extensively in Paper I
and II, in which a nonlinear optimization routine isused directly on the prism refraction Eq. (1) to iteratively locate a solution. In order to restrict the algorithm to practical prism designs, several modifiers
are added to the optimizer merit function such asa penalty on beam compression and a penalty on dispersion nonlinearity, in addition to the standard merit function terms (5). While all of the achievementsof the customized design algorithms we use here
are possible using standard optical design software,
our algorithms provide an easier interface for manipulating the low-level features of the software, suchas the optimization algorithm, the various featuresof the merit function, such as the penalty terms,
and also allow use of a wide suite of numerical- andtext-manipulation functions available in a high-leveland powerful language. We encourage interested
readers to download and modify the design code wehave written for this work, available at the authors’
website [15]. The entire suite of functions is written
in Python using standard modules.
1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS

5029

Appendix A: Deriving the Grating Groove Densityfrom Angular Dispersion

The basic diffraction grating equation in air is
G sin δ ¼ mλ, where G is the spacing between
grooves, δ is the ray deviation angle, m is the diffraction order, and λ is the wavelength. Taking the derivative of this equation and assuming small angles
gives the dispersion equation Λ ≈ mdδ=dλ, whereΛ ¼ 1=G is the grating groove density (typically
expressed in lines per millimeter). Using the nomenclature of Section 2, and assuming the first diffraction order, we can write this as Λ ≈ Δ=λrange , where
λrange ¼ λmax − λmin . In a spectrometer, the linear
diffraction length at the detector is given as
tanðΔ=2Þ ¼ Nx=ð2f Þ, where N is the number ofdetector pixels, x is the pixel pitch, and f is the spectrometer’s reimaging lens focal length. Again, assuming small angles, we can combine the two
equations to obtainΛ≈Nx:f λrange

This is the equation used to obtain the diffraction
grating examples of Section 3, where we assumed
that the detector would see a full spectral rangeof twice the FWHM of the source spectral width(i.e., λrange ¼ 2 FWHM).
This work was partially supported by the National
Institutes of Health (NIH) grants RO1-CA124319
and R21-EB009186.
References
1. N. Hagen and T. S. Tkaczyk, “Compound prism designprinciples, I,” Appl. Opt. 50, 4998–5011 (2011).

5030

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

2. N. Hagen and T. S. Tkaczyk, “Compound prism designprinciples, II: triplet and Janssen prisms,” Appl. Opt. 50,
5012–5022 (2011).
3. K. Oka and T. Kato, “Spectroscopic polarimetry with achanneled spectrum,” Opt. Lett. 24, 1475–1477 (1999).
4. C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral
resolution and sampling issues in Fourier-transform spectralinterferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
5. Z. Hu and A. M. Rollins, “Fourier domain optical coherence
tomography with a linear-in-wavenumber spectrometer,”
Opt. Lett. 32, 3525–3527 (2007).
6. C. M. Eigenwillig, B. R. Biedermann, G. Palte, and R. Huber,“K-space linear Fourier domain mode locked laser and applications for optical coherence tomography,” Opt. Express 16,
8916–8937 (2008).
7. V. M. Gelikonov, G. V. Gelikonov, and P. A. Shilyagin,“Linear-wavenumber spectrometer for high-speed spectraldomain optical coherence tomography,” Opt. Spectrosc. 106,459–465 (2009).
8. M. Jeon, J. Kim, U. Jung, C. Lee, W. Jung, and S. A. Boppart,“Full-range k-domain linearization in spectral-domain opticalcoherence tomography,” Appl. Opt. 50, 1158–1163 (2011).
9. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser,“Optical coherence tomography—principles and applications,”
Rep. Prog. Phys. 66, 239–303 (2003).
10. M. A. Popescu, Non-Crystalline Chalcogenides (Springer,
2002).
11. A. F. Fercher, C. K. Hitzenberger, M. Sticker, E. MorenoBarriuso, R. Leitgeb, W. Drexler, and H. Sattmann, “A thermal
light source technique for optical coherence tomography,” Opt.
Commun. 185, 57–64 (2000).
12. L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light fullfield optical coherence tomography,” Opt. Lett. 27, 530–532(2002).
13. B. Laude, A. D. Martino, B. Drévillon, L. Benattar, and
L. Schwartz, “Full-field optical coherence tomography withthermal light,” Appl. Opt. 41, 6637–6645 (2002).
14. S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and
M. Wei, “Fully depleted, back-illuminated charge-coupleddevices fabricated on high-resistivity silicon,” IEEE Trans.
Electron Devices 50, 225–238 (2003).
15. URL: http://www.owlnet.rice.edu/~tt3/.