Op ik - In e na ional Jou nal o Ligh and Elec on Op ics 247 (2021) 167873
A ailable online 28 Augus 2021
0030-4026/© 2021 The Au ho (s). Published by Else ie GmbH. This is an open access a icle unde he CC BY-NC-ND license
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As igma ism in he basic O ne spec ome e
H´
ec o Gonz´
alez-Nú˜
nez, Ca los Mon e o-O ille, Raúl de la Fuen e
*
Depa amen o de Física Aplicada, Facul ade de ´
Op ica e Op ome ía, Uni e sidade de San iago de Compos ela, 15782 Galicia, Spain
ARTICLE INFO
Keywo ds:
O ne spec ome e
Rowland con igu a ion
Concen ic sys ems
As igma ism
Op ical sys em design
ABSTRACT
The in-plane con igu a ion o he basic O ne spec ome e is e ised. The loca ions o me idional
and sagi al images o he sli cen e a e analyzed when he sli is displaced in he axial di ec ion
om he usual con igu a ion, whe eas i s cen e is kep on he Rowland ci cle o he conca e
mi o . This ansla es he posi ion o he sagi al image plane and allows o he cancella ion o
as igma ism and me idional coma o wo wa eleng hs while hese abe a ions a e kep small
o e he whole spec al ange. This is accomplished wi hou spli ing he conca e mi o in o wo
di e en mi o s, simpli ying he design and i s p ac ical implemen a ion. A design example is
p esen ed wi h excellen op ical pe o mance.
1. In oduc ion
Almos i y yea s ago, H. O ne p oposed a wo mi o concen ic imaging sys em wi h uni magni ica ion which cancels all Seidel
abe a ions in a pa icula annula ield: he O ne elay o O ne imaging sys em [1]. La e , his de ice was con e ed o an imaging
spec ome e by pe o ming wo modi ica ions. Fi s , he con ex mi o was eplaced by a con ex g a ing o p o ide he equi ed ligh
dispe sion [2] and, second, as a consequence o symme y b eaking, he conca e mi o was spli in o wo mi o s o educe second
o de as igma ism [3]. Really, his spec ome e p o ides ze o as igma ism o a pa icula wa eleng h and o a pa icula objec
poin , whe e he cen e o he spec ome e sli is usually placed. In he i s con igu a ions o he O ne spec ome e , he so call
in-plane de ices, his poin was loca ed a he plane o symme y o he sys em. They we e analyzed using di e en heo e ical models
[4–7]. Howe e , new con igu a ions we e en isaged soon wi h he sli ou o his plane: he o -plane O ne con igu a ions [8–12].
They ha e he ad an age ha i is no necessa y o spli he conca e mi o , bu hey p esen a good op ical quali y only o a he sho
sli s. On he o he hand, di e en a ian s o he O ne imaging spec ome e we e analyzed o imp o e i s imaging quali y. Fo
example, depa u e om concen ici y [13], using non classical g a ings wi h modi ied g oo e pa e ns [14], adding new e ac ing
op ical elemen s [15,16] o using ee- o m op ics [17–19]. O he ecen s udies o he O ne imaging spec ome e can be ound in
e e ences [20–25], and an excellen e iew o imaging spec ome e s o emo e sensing including he O ne design is ound in [26].
In his pape , we e u n o he basic O ne dispe si e sys em, ha is, one conca e mi o and a con ex classical g a ing, in he in-
plane con igu a ion. We show ha i is possible o educe as igma ism in he en i e spec al ange, and canceling i o wo di e en
wa eleng hs. The key is o mo e he sli cen e om he mo e common con igu a ion in which i is placed in a plane o hogonal o he
op ical axis con aining he cu a u e cen e . This was ye conside ed in one o ou p e ious wo ks [27] bu in ha wo k we deal wi h a
h ee-componen sys em, and we only con empla ed s a iona y anas igma ic images a one wa eleng h.
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (R. de la Fuen e).
Con en s lis s a ailable a ScienceDi ec
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Recei ed 10 Feb ua y 2021; Recei ed in e ised o m 2 July 2021; Accep ed 21 Augus 2021
Op ik 247 (2021) 167873
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2. The Rowland ci cle
Since he Rowland ci cle [28] is c ucial o unde s anding he p inciples o design p esen ed he e, i will be ins uc i e o e ise his
concep . The Rowland ci cle was i s de ined in ela ion o he heo y o abe a ions o conca e sphe ical di ac ion g a ings, bu i
can be applied as well o any kind o e lec i e o e ac i e sphe ical su ace, conca e o con ex (in his las case, objec and images a e
i ual). The Rowland ci cle is de ined as he ci cle angen o a sphe ical su ace a he poin o incidence o he p incipal ay and
ha ing i s diame e equal o he adius o he su ace (see Fig. 1). I has his main p ope y [29]: when an objec poin belongs o his
ci cle, i (s) me idional image(s) is (a e) coma- ee and belong(s) o his ci cle oo. F om he p ope ies o ci cles i ollows ha i h is he
adial coo dina e o an objec o image poin on he Rowland ci cle measu ed om he cen e o cu a u e o he sphe ical su ace ( he
so called o -cen e dis ance), i ul ills:
h=Rsin θ(1)
whe e R is he cu a u e adius and θ he angle o incidence, e lec ion, o e ac ion, as app op ia e. On he o he hand, he sagi al
image is ou o his ci cle, so images in he Rowland con igu a ions p esen a g ea amoun o as igma ism. This is o no conce n in non-
imaging spec ome e s, especially in he UV ange, which can be designed wi h only a conca e di ac ion g a ing and ha ing he
cen e o he sli on he Rowland ci cle.
Howe e , in imaging spec ome y, as igma ism mus be minimized and some elemen s ha e o be added o compensa e o he
as igma ism on he Rowland ci cle. In his case, you can ake ad an age o ano he p ope y o he Rowland ci cle ela ed o concen ic
sys ems. I an op ical sys em is composed only by sphe ical elemen s ha ing a common cen e o cu a u e, he comple e sys em owns
he same p ope ies o each single su ace in he Rowland disposi ion; ha means, i he objec belongs o he Rowland ci cle o he i s
op ical elemen , he me idional image alls on he Rowland ci cle o he las op ical elemen . This ollows because an image on he
Rowland ci cle o a gi en elemen also belongs o he Rowland ci cle o he nex elemen , as i is shown in Fig. 2. Al hough i has no
been demons a ed in gene al, he coma ee p ope y o he me idional Rowland image has been shown in some cases, namely, in he
O ne and Dyson spec ome e s [28,30]. In hese sys ems, choosing wisely he adius o each su ace, he posi ion o he sagi al image
can be made coinciden wi h he me idional one, a leas o a gi en objec poin o he sli and a gi en wa eleng h.
The possibili y o ge ing anas igma ic and me idional coma- ee imaging in spec ome e s disposed in a concen ic layou has
a ac ed much a en ion. Fu he mo e, concen ic spec ome e s ha e been shown o p esen e y low spa ial and spec al dis o ion,
so hey ha e been he p e e ed op ical con igu a ion in many spec oscopic imaging applica ions.
Fig. 1. Rowland ci cle in a sphe ical conca e di ac ion g a ing.
Fig. 2. Example o disposi ion o Rowland ci cles in a concen ic sys em composed by wo elemen s.
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3. Locus o me idional and sagi al images
In his sec ion, a basic O ne spec ome e wi h an objec poin loca ed on he Rowland ci cle o he conca e mi o is conside ed.
We will concen a e ou a en ion in he image plane, a e e lec ion a he g a ing. In he ligh o he abo e discussion, he spec al
me idional images belong o he co esponding Rowland ci cles o he conca e mi o a e he second e lec ion. The x, z coo dina es
o hese images (x′
m
, z′
m
) can be ela ed o he angle o di ac ion o he co esponding p incipal ay, θ’
2
. Acco ding o he geome y o
Fig. 3:
x′
m= − R2sin θ′
2cos γ′
m
z′
m= − R2sin θ′
2sin γ′
m
(2)
whe e γ′
m is he pola angle o he me idional image poin . and he in a iance o he o -cen e dis ance by e lec ion (o e ac ion) in a
concen ic sys em has been applied:
h′= − R1sin θ′
1= − R2sin θ′
2(3)
The nega i e signs in he abo e exp essions a e due o ou sign con en ion: posi i e angles ha e been aken o coun e clockwise
o a ion abou an axis o abou a no mal o a su ace, while he adii R
1
and R
2
we e aken posi i e. On he o he hand, no e ha he
angles a e ela ed by he ollowing equa ion: γ′
m=θ′
2−2θ′
1.
In Fig. 4 he locus o me idional image poin s co esponding o di e en di ac ion angles, ha is, di e en wa eleng hs, a e
ep esen ed o some pa icula adii a ios R
1
/R
2
. No e ha hese cu es also ep esen he possible loca ions o he objec poin
whene e i belongs o he Rowland ci cle o he i s mi o (in his case, by con en ion, only he le pa o he cu e, x′
m
<0, mus be
conside ed). In Fig. 4(a) he adius a io is less han wo. In his case, he main cha ac e is ics o hese cu es a e he ollowing:
i) They a e symme ic wi h espec o he z axis.
Fig. 3. Me idional image poin in he Rowland ci cle o he conca e mi o a e he las e lec ion.
Fig. 4. (a) Locus o me idional and sagi al images o R
1
/R
2
=1.93. The objec is loca ed a he beginning o he me idional cu e. (b) Same as 4a
bu R
1
/R
2
=2.03. These cu es a e ep esen a i e o he gene al cases R1/R2<2 and R1/R2≥2.
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ii) They c oss he x-axis a θ′
1=θ′
2/2= ±a ccos(0.5R1/R2), being posi i e be ween hese wo alues (excep o θ′
1 =θ′
2 =0whe e
hei alue is ze o).
iii) A hei maximums hey ul ill:
an γ′
m=2 an θ′
1− an θ′
2(4)
i ) The posi ion o he specula me idional image (ze o di ac ion o de ) is ob ained om he posi ion o he objec by specula
e lec ion wi h espec o he z-axis. Tha means x′
m
=-x, z′
m
=z, h´=h and θ′
2= − θ2, whe e x, z, h a e he x, z-coo dina es and o -
cen e dis ance o he objec poin , espec i ely, and θ
2
is he angle o incidence o he p incipal ay on he g a ing.
) Me idional images o posi i e di ac ion o de a e on he le o he specula image.
The shapes o he cu es a e e y di e en o R1/R2≥2 (see Fig. 4b). In ha case he cu es a e angen o he x-axis a θ′
1 =θ′
2 =
0 and dec ease symme ically as he angle inc eases.
No e ha he no malized posi ion o Rowland images only depends on he adius a io ( he adius R
2
ac s as a scale). I does no
depend on he objec posi ion, whene e i is in he Rowland ci cle a he i s e lec ion in he conca e mi o . The objec posi ion only
ixes he posi ion o he me idional specula image ( he image o ze o wa eleng h). On he o he hand, he ela ion be ween image
posi ion and wa eleng h λ is dic a ed by he g a ing equa ion:
sin θ2+sin θ′
2=mgλ (5)
whe e g is he g a ing g oo e densi y and m he di ac ion o de . Depending on he alue o g and he objec posi ion (which de ines
angle o incidence a he g a ing, θ
2
), he image o a gi en wa eleng h is placed in a pa icula poin o he me idional image cu e in
Fig. 4.
Wi h espec o he posi ions o sagi al images hey a e bound by he ollowing in a ian [16]:
sin θ2 an γ= − sin θ′
2 an γ′
s(6)
which ela es he z-coo dina es o sagi al images, z
s
´, and objec poin , z, as ollows:
z′
s= − R2sin θ′
2
sin γ′
s
cos(γ′
s−γ′
m)= − zcos(γ′
s)
cos(γ′
s−γ′
m)cos(γ)(7)
Fo ypical con igu a ions, he adius a io is close o wo and he objec is e y closed o he x-axis, so he pola angles γ and γ′
m a e
e y closed o ze o. The e o e, z’
s
≈-z, and he posi ion o sagi al images is cons an (see Fig. 4).
4. Anas igma ism
In his sec ion we analyze anas igma ic imaging in he basic O ne spec ome e wi h an objec poin in he Rowland ci cle o he
conca e mi o a he i s e lec ion. The condi ion o anas ima ism is:
z′
m=z′
s= − z(8)
Fo con igu a ions whe e R1/R2≥2 we ha e he beha io shown in Fig. 4b: all images a e as igma ic. Howe e , when R1/R2<2,
he e a e se e al anas igma ic images depending on he objec posi ion. I he objec is a z>0, he sagi al line is a z<0 and he e a e
Fig. 5. (a) Map o Rowland poin s. Cu es a e labeled by he alue o he z-coo dina e o he objec no malized o R
2.
. (b) no malized second o de
as igma ism e sus no malized image o -cen e dis ance o di e en adius a io R1/R2.
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wo anas igma ic images ( emembe ha we conside he objec a x<0): one on i s le and he o he on i s igh . The e o e, hey
co espond o di ac ion o de s o di e en sign. O he wise, i z<0, he e a e up o ou anas igma ic images (see Fig. 4a). Howe e ,
he spec a canno expand o co e all hese images because he di ac ion g a ing gene a es igne ing. Thus, we a e le o conside
only wo anas igma ic images, ei he o he le o o he igh o he z-axis. In he pa icula case in which he sagi al line ouches he
me idional cu e a i s maximum he image is anas igma ic and s a iona y, ha is, he de i a i e o as igma ism wi h espec o
wa eleng h also anishes. I he objec is mo ed away om he z axis, u he anas igma ic images will no be ound. No e ha o he
ypical con igu a ion in which he objec is loca ed on he x-axis, he e a e h ee anas igma ic images. Howe e , one o hem o e laps
he objec , o he is i s specula image and he las one is on he op ical axis.
We can ge a mo e gene al ision looking a Fig. 5(a) whe e i is plo ed a map o poin s in he Rowland ci cle as a unc ion o adius
a io and o -cen e dis ance. Each cu e co esponds o a gi en alue o he z-coo dina e o he objec . The con inuous cu es a e o
z>0 and he dashes cu es o z<0. The do ed-dashed line co esponds o z=0. Conside ing ha he objec is on a cu e wi h z=z
0
, and
acco ding o Eq. (8), anas igma ic images i in he cu e z=-z
0
. D awing a ho izon al line a a gi en adius a io, he in e sec ion o his
line wi h hose cu es p o ides he o -cen e dis ance o he objec and i s anas igma ic images co esponding o his adius a io.
As i was men ioned abo e, he e a e no anas igma ic images o objec s in he Rowland ci cle i R1/R2≥2, since in his case he e
a e only cu es wi h nega i e z. I he objec is loca ed a a poin wi h z>0, he e a e wo possible posi ions in he Rowland ci cle,
being, in p inciple, he posi ion close o he cu a u e cen e o he su aces p e e able since i co esponds o a mo e pa axial egime.
In his case, he image cu e con ains a single anas igma ic image a each side o he z axis. The spec a can be expanded owa ds he
cu a u e cen e which co esponds also o a mo e pa axial egion. On he o he hand, placing he objec a a poin wi h z<0, allows o
he use o he +1 di ac ion o de , enabling he educ ion o he as igma ism o e he whole spec al ange by using he wo co -
esponding anas igma ic images. Besides, you mus be ca e ul o loca e he spec a nea he s a iona y anas igma ic poin o limi he
amoun o anas igma ism o e he whole spec a. This is an ad an age o e a con igu a ion in which he objec has posi i e z-coo -
dina e. Indeed, in Fig. 5(b), he as igma ism, calcula ed in a good app oxima ion as z′
m
-z′
s
, is plo ed as a unc ion o h′ o se e al
adius a ios R
1
/R
2
. I is clea om his igu e ha o a gi en spec al ange (Δλ o Δh’) he e is an op imum adius a io which
minimizes he as igma ism o e he whole spec al ange. This will be conside ed in he nex sec ion.
5. Simula ions
Some simula ions using Ma lab [31] and Oslo design so wa e [32] we e pe o med o show ha basic O ne spec ome e s o high
op ical quali y can be designed p o ided ha second o de as igma ism o he images o he sli cen e is educed. We de ised a design
p ocedu e in Ma lab which comp ises he ollowing s eps:
i) he speci ica ions o he spec ome e o be designed a e conside ed, namely, he spec al band (λ
-
, λ
+
), he leng hs o bo h sli
and spec al image, he di ac ion o de and he -numbe ( /#).
ii) An ini ial alue o he g a ing densi y g is assumed.
iii) The cu a u e adius o he g a ing is calcula ed by [16]:
R2=hspec
|m|gΔλ =50mm (9)
whe e h
spec
is he leng h o he spec al image, o he conside ed spec al band, Δλ =λ+−λ−.
i ) Vigne ing is a oided, app oxima ely, by imposing he ollowing condi ion: sin θ′
2 = ∓1/( /#)whe e θ′
2 is he angle o
di ac ion o he p incipal ay o he wa eleng h, λ
, which is di ac ed closes o he di ac ion g a ing, and he minus sign
applies i ligh is di ac ed o he le side o he z-axis.
) The g a ing equa ion is used o calcula e he angle o incidence a he g a ing a λ
: sin θ2 =mgλ ±1/( /#),
i) A alue o he adius a ioR1/R2is es ablished. Eqs. (2) and (3) a e applied in he objec domain (angles wi hou single quo a ion
ma k) o calcula e de coo dina es o he objec . Then, hese equa ions a e used o calcula e he z-coo dina e o he me idional
image o e e y wa eleng h in o de o compu e he second o de longi udinal as igma ism as zm−zs≅zm+z.
ii) S ep i) is pe o med i e a i ely o a se o adius a ios and he as igma ism is minimized in he ull spec al ange by
minimizing he ollowing sum, om λ
-
o λ
+
: ∑(zm−zs)2. This gi es an op imum adius a io.
iii) The s eps ii) o ii) a e i e a ed un il zm−zs|λ−=zm−zs|λ+, ha is, g is chosen so ha he second o de as igma ism is mini-
mized a he edges o he spec al band.
Table 1
Design pa ame e s.
Sli leng h (mm) 9.5
Spec al band (nm) 900–1700
-numbe 2.75
Di ac ion o de +1
G a ing densi y (g /mm) 175
G a ing adius (mm) 50
Radius a io 1.885
Coo dina es o sli cen e (x
0
,z
0
) (mm) (−33.98, −0.33)
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A e comple ing hese s eps, he e may s ill be some igne ing ( his can be checked wi h he Oslo design so wa e), so we can s a
he design p ocedu e om a sligh ly smalle -numbe and inally se he -numbe o he one gi en in he speci ica ions.
In o de o check his design p ocedu e, we designed an O ne spec ome e in he SWIR (900–1700 nm), he sensi i e spec al
ange o a common InGaAs senso . We conside ed a ypical senso size 9.6×7.68 mm
2
wi h 15 µm squa e pixels and a -numbe , /
#=2.75. The senso is loca ed a he sagi al image plane: he spec um, wi h spa ial ex ension h
spec
=7 mm, lies along he sho e
senso dimension; and we choose a sli leng h o 9.5 mm. Then, he design p ocedu e desc ibed abo e was applied. The spec ome e
speci ica ions a e shown in ows 1–4 in Table 1 while ows 5–8 display he esul s o he design. All o hese pa ame e s a e calcula ed
Fig. 6. (a) Spo diag ams o 0, 0.7 and ull ield, and selec ed wa eleng hs he Ai y disk is also plo ed. (b) The PSF o he same ields and
wa eleng hs. Numbe s in he uppe le co ne o each plo co espond o he S ehl a io. Each plo is enclosed in a squa e o 20 µm o side.
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Op ik 247 (2021) 167873
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conside ing λ
=λ
+
=1700 nm, he wa eleng h ha di ac s closes o he g a ing. Finally, he whole spec ome e has a size o
35×100×95 mm.
In Fig. 6(a) we ep esen di e en spo diag ams calcula ed wi h Oslo design so wa e a e da a en y om Table 1, o 0, 0.7 and
ull ield (4.75 mm) a selec ed wa eleng hs. These a e he cen e and ex eme wa eleng hs, he anas igma ic wa eleng hs (1040 and
1497 nm). I is seen ha a 0 ield, as igma ism is he ele an abe a ion o all he spec al images and he deg ee o as igma ism is
simila in all he cases. This means ha highe o de as igma isms a e signi ican , once second o de as igma ism has been minimized.
A 0.7 and ull ield, e e y spo diag am has an S shape, g ea e o la ge ields and mo e elonga ed o smalle wa eleng hs. In Fig. 6
(b) he co esponding poin sp ead unc ions (PSF) a e ep esen ed. Because o he e ec o di ac ion, he spa ial esolu ion ( e ical
dimension) dec eases wi h espec o he alues expec ed om he spo diag ams, being he PSF shape ellip ical a 0 ield poin s. No e
ha excep he las one, all PSFs a e con ained in he 15-mic on came a pixels. This co esponds o a spec al esolu ion (ho izon al
dimension) lowe han 0.9 nm. As i is ypical in he in-plane O ne con igu a ion, spa ial esolu ion is be e han spec al esolu ion.
No e also ha he S ehl a ios a 0 and 0.7 ields achie e ou s anding alues.
In Fig. 7a we ep esen he spec al cu es o second o de longi udinal as igma ism calcula ed o he sli cen e by Oslo and Ma lab
so wa e om he da a o he design made by he la e . We also ep esen he co esponding as igma ism cu es o he images o he
0.7 and ull ield poin s on he sli . I is seen ha o any poin on he sli and wa eleng h he longi udinal second o de as igma ism is
less han 0.09 mm. Fu he mo e, he cu es a e e y simila excep o a e ical ansla ion. In Fig. 7b we ep esen he spec al cu es
o 80% ensqua ed ene gy o he same sli poin s. All he cu es a e con ained in a squa e o side less han 14 µm, which is smalle han
he size o a pixel senso . These cu es include he con ibu ion o as igma ism and highe o de abe a ions. As commen ed in Re . [4],
main con ibu ions come om hi d o de sagi al coma and ou h o de sphe ical abe a ion. I is seen ha o he wo bounda ies o
he spec um, abe a ions inc ease as we mo e away he sli cen e . Howe e , o he cen e o he spec um, abe a ions a e g ea e o
he sli cen e , and hey a e he g ea es ones. This is consis en wi h he g ea leng h o he as igma ic lines shown in Fig. 6. I is wo h
no ing ha he da a in Table 1 co esponds o he inal design, wi hou any u he op imiza ion. Howe e , we also checked i an
op imiza ion would imp o e no iceably his inal design. So, by using he Oslo design so wa e and i s op imiza ion ou ines, we
allowed he decen e ing o he elemen s in he XZ plane, he il ing o he image plane a ound he y-axis, and he cu a u e o he i s
mi o , o a y eely. The esul was an op imized sys em ou o concen ici y whose wo s image poin had an RMS spo adius o
5.2 µm; a alue only a li le be e han he one (5.6 µm) o ou inal design. This is a e y small imp o emen ha does no compensa e
o he incon eniences o aking he sys em ou o concen ici y; some hing especially ele an du ing alignmen asks in an expe i-
men al se up.
On he o he hand, we also compa ed ou design wi h he classical O ne imaging spec ome e , ha is, he con igu a ion whe e
bo h sli and image plane a e in a plane pe pendicula o he z-axis which con ains he cen e o cu a u e o he sys em. Since in his
con igu a ion and a e ze o he ollowing ela ionship is ul illed: R
1
=2R
2
cos θ
1
, being θ
1
he minimum angle which a oids igne ing.
By conside ing hese cons ains we calcula ed a new adius a io o 1.859 and new coo dina es o he sli cen e x
0
=− 34.25 mm and
z
0
=0 mm. Once in oduced hese alues in he Oslo so wa e, we ob ained an RMS spo adius o 40.39 µm o he bes image poin o
his classical sys em, and he S ehl a ios each e y low alues (less han 0.07). Tha co esponds o a e y low-quali y design. O
cou se, e y good designs can be achie ed b eaking he layou symme y by spli ing he conca e mi o in o wo mi o s o di e en
adii, bu his would be con a y o he pu pose o he p esen wo k: o ob ain an O ne design o excellen quali y wi h a e y simple
con igu a ion o only a conca e mi o and a con ex di ac ion g a ing.
Fig. 7. (a) Second o de as igma ism calcula ed wi h Oslo So wa e o 0 (blue poin s), 0.7 (pu ple poin s) and ull ield ( ed poin s). (b) 80%
ensqua ed ene gy e sus wa eleng h o 0, 0.7 and ull ield. The black cu e in (a) co espond o he esul ob ained h ough he design p ocedu e
implemen ed in Ma lab and he black line in (b) co espond o he di ac ion limi . (Fo in e p e a ion o he e e ences o colo in his igu e legend,
he eade is e e ed o he web e sion o his a icle.).
H. Gonz´
alez-Nú˜
nez e al.
Op ik 247 (2021) 167873
8
6. Conclusions
The locus o me idional and sagi al images o a poin objec in a basic O ne spec ome e has been analyzed. The me idional
cu e is symme ic wi h espec o he z axis and has a ze o alue a he cen e o cu a u e o he sys em and, o R1/R2<2, wo o he
symme ic poin s. Fu he mo e, he cu e is bounded a z>0. On he o he hand, he sagi al cu e is, o all he pu poses, a line
pe pendicula o he z axis. This allows o changing he as igma ism by only ansla ing he posi ion o he objec along he z-axis and
placing he imaging senso in he sagi al image plane. This is ue o small z-coo dina es o he objec , which is he case in mos
p ac ical spec ome e s. Anas igma ic images a up o ou wa eleng hs a e ound i he adius a io R
1
/R
2
is less han wo. Howe e ,
only wo images a e a ailable p o ided ha igne ing is a oided a he cen al pa o he spec um. A simple design has been
p esen ed showing he quali y o his basic O ne spec ome e .
Funding
Xun a de Galicia, Conselle ia de Educacion, Uni e sidades e FP, Spain, G an GRC numbe ED431C2018/11 and ED431E2018/08.
Decla a ion o Compe ing In e es
The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
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