CN105938013A - Spectrometer and correction method thereof - Google Patents

Abstract

The invention discloses a spectrometer and a correction method thereof. The spectrometer comprises an incident optical unit, an incident slit, a dispersion unit which is used for dispersing incident light and an array detector which is used for receiving dispersed light. The spectrometer also comprises a broadband detector which receives zero order light beams formed through diffraction of the dispersion unit. Besides, a color filter matched with a V(lambda) curve is arranged before the photoelectric conversion element of the broadband detector, and the linearity of the array detector is corrected by utilizing photometric information measured by the broadband detector so that the measured absolute radiant quantity is acquired, and the linearity and the measurement accuracy of the spectrometer can be further enhanced; furthermore, spectral mismatch correction can also be performed on the photometric information of the broadband detector by utilizing relative spectral power distribution measured by the array detector so that the photometric measurement precision can be enhanced.

Description

A kind of spectrogrph and bearing calibration thereof

[technical field]

The invention belongs to optical radiation measurement field, be specifically related to a kind of spectrogrph.

[background technology]

Existing rapid spectrometer (also referred to as array spectrogrph) is typically made up of beam incident optical unit, entrance slit, dispersion element, image-generating unit and detector array, structure is as shown in Figure 1, light beam enters inside spectrogrph through beam incident optical unit from entrance slit, through dispersion element, the dispersed light of different wave length is projected on the photosurface of detector array, to realize the measurement of spectral power distribution.

According to many seam diffraction effects, there is not dispersion, also referred to as zero level sub-beams in the zero level of grating time diffraction light, and the detector of spectrogrph is actual, and measure is the secondary diffraction spectrum of non-zero order of grating generation dispersion, generally one time diffraction light.In the prior art, one of zero level sub-beams often important sources being considered as stray light of spectrometer, it is constantly in the negative status of " elimination ", is not the most obtained effectively utilization.Common way has and coats black or the coating of antiradar reflectivity on the cavity of conventional optics cavity, or arranges delustring trap at zero level sub-beams convergence and absorbed by this segment beam.Meanwhile, rapid spectrometer is because have employed array type detector, such as CCD etc., although sensitivity is higher, should be not as good as silicon cell but the Larger Dynamic range of linearity is general.

[summary of the invention]

For above-mentioned technical problem, instant invention overcomes the technology " prejudice " to optical grating diffraction zero level sub-beams in prior art, turn waste into wealth, utilize the optical information comprised in zero level sub-beams that the detector array of spectrogrph is linearly corrected, improve the accuracy of measurement of spectrogrph further；Or supplement as detector array is measured wave band.

The present invention can be achieved through the following technical solutions: a kind of spectrogrph, including entrance slit, dispersion element, for receiving the detector array of dispersed light, it is characterized in that, also include the broadband detector for receiving the zero level sub-beams formed through dispersion element light splitting.

The operation principle of the present invention is as follows: tested light beam enters inside spectrogrph by beam incident optical unit from entrance slit, light splitting is carried out to by light-metering by dispersion element, and form the spectrum of the most at the same level time, the non-zero order time spectrum that wherein light splitting is formed is projected on the photo-electric conversion element of detector array, to realize the measurement of spectral power distribution, the zero level sub-beams that light splitting is formed then is received by broadband detector.

The present invention is the most perfect also by techniques below scheme:

As preferably, the photo-electric conversion element of described broadband detector is light cell or photodiode, and concrete can be silicon cell, silicon photoelectric diode or germanium battery, or the photo-electric conversion element of other materials.The luminosity probe being wherein made up of the photo-electric conversion element such as silicon cell or silicon photoelectric diode is fabulous linear owing to all having in large span dynamic range, is therefore widely used in the measurement of light quantity.

Further, light path before the photo-electric conversion element of described broadband detector is provided with light filter, makes the relative spectral response curve of the incident illumination (by light-metering) that broadband detector photo-electric conversion element incides beam incident optical unit match with special spectrum Efficiency Function.

As preferably, described light filter make described broadband detector for being matched with human eye luminous efficiency function V (λ) curve or smooth straight line by the relative spectral sensitivity curve of light-metering by beam incident optical unit, or other specific spectral sensitivity response curves.When the photo-electric conversion element of broadband detector is silicon cell, the light filter matched with human eye luminous efficiency function V (λ) curve is set before silicon cell and just constitutes relative to by the luminosity probe of light-metering.Technology further above represents, before broadband detector, light filter is set, make by beam incident optical unit by light-metering after entrance slit, dispersion element, light filter, broadband detector matches or consistent with specific spectrum Power Function for the sensitive response curve of relative spectral of tested incident illumination, this spectrum Power Function can be V (λ) curve or smooth straight line, or other curves.When special applications, light filter can also be a band-pass filter, it it is possibly even a logical narrower band-pass filter of band, now, the actual measurement effect of broadband detector can be just for a certain wavelength near narrow-band spectrum scope, the most described light filter makes described broadband detector be band-pass response curve near a certain specific wavelength for the relative spectral sensitivity curve of tested incident illumination.

As a kind of technical scheme, described dispersion element can be integral type structure, i.e. has the ability of light splitting and imaging concurrently.Concrete described dispersion element can be concave grating or concave surface flat field grating.Concave surface flat field grating is to delineate a series of wire casing, the reflective diffraction gratings of composition on high reflecting metal concave surface, has preferable light splitting and light gathering.

As another kind of technical scheme, described dispersion element can be that fabricated structure, i.e. light splitting are realized by different optical elements respectively from imaging.Concrete dispersion element can include dispersion element and image-forming component, and wherein dispersion element can be plane grating, it is achieved be light splitting function；Image-forming component can be then concave mirror, it is achieved be imaging function.

Present invention also offers the bearing calibration of a kind of spectrogrph, the measurement wave band [λ of described detector array₁, λ₂] and the response wave band [λ of broadband detector_s1, λ_s2] match, that is to say the response wave band [λ of broadband detector_s1, λ_s2] scope measures wave band [λ at detector array₁, λ₂Within the scope of].Such as: the measurement wavelength band of detector array is [380nm, 780nm], the response wave band scope of broadband detector can be then at [380nm, 780nm] or at [450nm, 600nm] in, utilizing the bearing calibration that the present invention provides, the detector array spectrogrph linearity in [380nm, 780nm] can obtain correcting the most accurately by the signal that broadband detector records.

Concrete: to measure by the relative spectral power distributions information of light-metering and Radiation intensity information respectively by detector array and broadband detector, utilize broadband detector response value in its response range to correct linear in same wavelength band of detector array.Correction coefficient K is:

$K=\frac{I_s}{{\∫}_{{\mathrm{\λ}}_{s1}}^{{\mathrm{\λ}}_{s2}}P\left(\mathrm{\λ}\right)\·S\left(\mathrm{\λ}\right)d\mathrm{\λ}}---\left(1\right)$

Wherein, I_SFor broadband detector by measuring the response value that obtains of zero level sub-beams, P (λ) be detector array record by the relative spectral power distributions of light-metering, S (λ) is the absolute spectral sensitivity of broadband detector, λ_s1、λ_s2The integration start wavelength and the integration that are respectively broadband detector terminate wavelength, and unit is nm.

Utilizing above-mentioned correction coefficient, can obtain by the absolute light spectral power distributions of light-metering is K P (λ), then can be passed through formula by the luminosity value of light-meteringBeing calculated, wherein V (λ) is CIE standard human eye luminous efficiency function；K_mFor maximum spectral luminous efficacy, for vision improvement, K_mFor 683lm/W；λ₁、λ₂The integration start wavelength and the integration that are respectively detector array terminate wavelength, and unit is nm.

Owing to relative spectral sensitivity relatively its absolute spectral sensitivity of broadband detector is easier to know, knowing relative spectral sensitivity S (λ) of broadband detector_relTime, what available detector array recorded is corrected the spectrum mistake coupling of broadband detector by light-metering relative spectral power distributions P (λ), and correction coefficient K1 is:

$K1=\frac{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P\left(\mathrm{\λ}\right)\·Q\left(\mathrm{\λ}\right)\·d\mathrm{\λ}}{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P\left(\mathrm{\λ}\right)\·S{\left(\mathrm{\λ}\right)}_{rel}\·d\mathrm{\λ}}\·\frac{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P{\left(\mathrm{\λ}\right)}_S\·S{\left(\mathrm{\λ}\right)}_{rel}\·d\mathrm{\λ}}{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P{\left(\mathrm{\λ}\right)}_S\·Q\left(\mathrm{\λ}\right)\·d\mathrm{\λ}}---\left(2\right)$

The zero level sub-beams signal value I that correction coefficient K1 and broadband detector are directly recorded_TBe multiplied the signal value I after i.e. can being corrected_{T , C}。

Wherein, P (λ)_SFor the relative spectral power distributions to the standard source that broadband detector is calibrated；Q (λ) is the preferable spectral response functions of broadband detector, i.e. Q (λ) can be CIE standard human eye luminous efficiency function V (λ), it can also be smooth straight line, or band-pass response curve near a certain wavelength, or other specific spectral response curves；λ_s1、λ_s2Being respectively broadband detector integration start wavelength and integration terminates wavelength, unit is nm.

As a example by the probe of broadband detector is luminosity probe, measured by the relative spectral power distributions of light-metering and luminosity information respectively by detector array and broadband detector, and lose matching error according to the V (λ) of formula (3) effectively correct for photometric probe, thus it is greatly improved the certainty of measurement of shading value.Concrete updating formula is:

$I_{T,C}=I_T\·\frac{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P\left(\mathrm{\λ}\right)\·V\left(\mathrm{\λ}\right)\·d\mathrm{\λ}}{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P\left(\mathrm{\λ}\right)\·S{\left(\mathrm{\λ}\right)}_{rel}\·d\mathrm{\λ}}\·\frac{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P{\left(\mathrm{\λ}\right)}_S\·S{\left(\mathrm{\λ}\right)}_{rel}\·d\mathrm{\λ}}{\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P{\left(\mathrm{\λ}\right)}_S\·V\left(\mathrm{\λ}\right)\·d\mathrm{\λ}}---\left(3\right)$

Wherein, I_{T , C}For the shading value after correction, I_TFor the shading value of the zero level sub-beams that broadband detector directly records, P (λ) be detector array record by the relative spectral power distributions of light-metering, S (λ)_relFor the relative spectral sensitivity of broadband detector, P (λ)_SFor the relative spectral power distributions to the standard source that broadband detector is calibrated, V (λ) is CIE standard human eye luminous efficiency function.λ_s1、λ_s2Being respectively integration start wavelength and integration terminates wavelength, unit is nm, general λ_s1And λ_s2It is respectively 380nm and 780nm.

Through above-mentioned correction, broadband detector photometric measurement accuracy is greatly improved.Meanwhile, the measured value I after broadband detector correction is utilized_{T , C}The absolute spectrum radiation parameter I of detector array can be obtained according to the following formula_e(λ), it is shown below:

$I_e\left(\mathrm{\λ}\right)=\frac{I_{T,C}}{K_m\·\underset{{\mathrm{\λ}}_{s1}}{\overset{{\mathrm{\λ}}_{s2}}{\∫}}P\left(\mathrm{\λ}\right)\·V\left(\mathrm{\λ}\right)d\mathrm{\λ}}\·P\left(\mathrm{\λ}\right)---\left(4\right)$

Wherein, K_mFor maximum spectral luminous efficacy, for vision improvement, K_mFor 683lm/W；For scotopic vision, K_mFor 1725lm/W.

Absolute spectrum radiation parameter I according to detector array_e(λ), acquisition can be calculated and radiate value accurately.

Except above-mentioned utilize broadband detector and detector array measure wave band lap implement correction to reach to improve the scheme that spectrogrph is linear in addition to, present invention also offers a kind of measurement scheme, the wherein response wave band [λ of broadband detector_s1, λ_s2] at the measurement wave band [λ of detector array₁, λ₂Outside], i.e. λ_s1＞ λ₂Or λ_s2＜ λ₁；Or the response wave band [λ of broadband detector_s1, λ_s2] measure wave band [λ with detector array₁, λ₂] the most overlapping, i.e. λ_s1＜ λ₁＜ λ_s2＜ λ₂Or λ₁＜ λ_s1＜ λ₂＜ λ_s2.By the program, would know that the luminosity information outside detector array measurement wavelength band.Such as: measurement wave band [380nm in visible-range of detector array, 780nm], and broadband detector after being provided with band logical cut-off light filter its response wave band such as infrared part [1100,1200nm], so while the spectral power distribution in the range of obtaining visible light wave range, also would know that the luminosity information in the range of band logical cut-off light filter.

In sum, the present invention, by the utilization to dispersion zero level sub-beams, can reach to correct the purpose that spectrogrph detector array is linear, reduce the level of spectrogrph internal stray light the most to a certain extent.

[accompanying drawing explanation]

Accompanying drawing 1 is the basic structure of existing spectrogrph.

Accompanying drawing 2 is the structural representation of embodiment 1；

Accompanying drawing 3 is the structural representation of embodiment 2；

Accompanying drawing 4 is the result schematic diagram of embodiment 3；

1-dispersion element；2-detector array；3-broadband detector；1-1 dispersion element；1-2 image-forming component；

[detailed description of the invention]

Embodiment 1

As shown in Figure 2, present embodiment discloses a kind of spectrometer architecture, including entrance slit, dispersion element 1, detector array 2 and broadband detector 3, wherein dispersion element 1 is concave surface flat field grating, detector array 2 is ccd array detector, broadband detector 3 is silicon cell, and is provided with the light filter that the zero level sub-beams generation relative spectral sensitivity curve making silicon cell form light splitting matches with CIE standard human eye luminous efficiency function V (λ) before the photosurface of silicon cell.Beam incident optical unit is optical fiber or fibre bundle.

The operation principle of this spectrogrph is as follows: received by optical fiber or fibre bundle beam incident optical unit by light-metering, enter into inside spectrogrph by entrance slit, and it is incident to the surface of dispersion element 1, and then dispersion element 1 is carried out light splitting to incidence by light-metering, owing to concave surface flat field grating collection light splitting and imaging function are with integrally, the spectrum of the most at the same level time therefore will be formed in its image planes；The non-zero order time spectrum that wherein light splitting is formed is received by detector array 2 and measures, and the zero-order beam that light splitting is formed is received measurement by broadband detector array 3.

If absolute spectral sensitivity S of broadband detector 3 (λ) it is known that, the photometric signal I recorded by light-metering relative spectral power distributions P (λ) and broadband detector 3 that can be recorded by detector array 2_sDrawing the absolute spectral response of detector array 2, and then the measurement linearity of correction detector array 2, its correction coefficient isK P (λ) is then absolute spectral response.

Additionally, the photometric signal I that broadband detector 3 can be recorded by light-metering relative spectral power distributions P (λ) recorded by detector array 2_TIt is corrected, and then the V (λ) reducing broadband detector 3 loses matching error, obtains accurate light quantity.Concrete updating formula is:

$I_{T,C}=I_T\·\frac{\underset{{\mathrm{\λ}}_1}{\overset{{\mathrm{\λ}}_2}{\∫}}P\left(\mathrm{\λ}\right)\·V\left(\mathrm{\λ}\right)\·d\mathrm{\λ}}{\underset{{\mathrm{\λ}}_1}{\overset{{\mathrm{\λ}}_2}{\∫}}P\left(\mathrm{\λ}\right)\·S{\left(\mathrm{\λ}\right)}_{rel}\·d\mathrm{\λ}}\·\frac{\underset{{\mathrm{\λ}}_1}{\overset{{\mathrm{\λ}}_2}{\∫}}P{\left(\mathrm{\λ}\right)}_S\·S{\left(\mathrm{\λ}\right)}_{rel}\·d\mathrm{\λ}}{\underset{{\mathrm{\λ}}_1}{\overset{{\mathrm{\λ}}_2}{\∫}}P{\left(\mathrm{\λ}\right)}_S\·V\left(\mathrm{\λ}\right)\·d\mathrm{\λ}}---\left(1\right)$

Wherein, I_{T , C}For the shading value after correction, I_TFor the shading value of the zero level sub-beams that broadband detector 3 directly records, P (λ) be detector array 2 record by the relative spectral power distributions of light-metering, S (λ)_relFor the relative spectral sensitivity of broadband detector 3, P (λ)_SFor the relative spectral power distributions to the standard source that broadband detector 3 is calibrated, V (λ) is CIE standard human eye luminous efficiency function.λ₁、λ₂Being respectively integration start wavelength and integration terminates wavelength, unit is nm, λ₁And λ₂It is respectively 380nm and 780nm.

The shading value that broadband detector 3 records is after above-mentioned correction, and accuracy of measurement is greatly improved.Meanwhile, the measured value I after broadband detector 3 correction it is also with_{T , C}The absolute spectrum radiation parameter of acquisition detector array 2 correspondence according to the following formula:

$I_e\left(\mathrm{\λ}\right)=\frac{I_{T,C}}{K_m\·\underset{{\mathrm{\λ}}_1}{\overset{{\mathrm{\λ}}_2}{\∫}}P\left(\mathrm{\λ}\right)\·V\left(\mathrm{\λ}\right)d\mathrm{\λ}}\·P\left(\mathrm{\λ}\right)---\left(2\right)$

Wherein, K_mFor maximum spectral luminous efficacy, for vision improvement, K_mFor 683lm/W；For scotopic vision, K_mFor 1725lm/W.

Further, according to the absolute spectrum radiation parameter I of detector array 2_e(λ), acquisition can be calculated and radiate value accurately.

The structure of this spectrogrph overcomes those skilled in the art's technology " prejudice " to zero level time diffraction spectrum, turn waste into wealth, the superior Larger Dynamic scope utilizing the optical information in zero level sub-beams and silicon cell (or silicon photoelectric diode) is linear, to being linearly corrected of spectrogrph detector array, further increase the accuracy of measurement of spectrogrph.

Embodiment 2

As shown in Figure 3, present embodiment discloses a kind of spectrometer architecture, including entrance slit, collimating mirror, dispersion element 1, detector array 2 and broadband detector 3, wherein, dispersion element 1 includes dispersion element 1-1 and image-forming component 1-2, and dispersion element 1-1 is plane grating, and image-forming component 1-2 is concave mirror；Detector array 2 is ccd array detector, broadband detector 3 is silicon photo diode, and is provided with the light filter that the zero level sub-beams generation relative spectral sensitivity curve making silicon photo diode form light splitting matches with CIE light efficiency visibility function V (λ) before the photosurface of silicon photo diode.

The operation principle of this spectrogrph is as follows: entered into inside spectrogrph by light-metering by entrance spectrometer slit, and formation collimated light beam is incident upon at dispersion element 1-1 after collimating mirror, dispersion element 1-1 carries out light splitting to incident tested light, and be incident upon at image-forming component 1-2 with zero level sub-beams by the non-zero order time spectrum of light splitting formation, and then on the focal plane of image-forming component 1-2, form the spectrum of the most at the same level time；Wherein received by light-metering non-zero order time spectrum by detector array 2 and measure, received measurement by the zero-order beam of light-metering by broadband detector array 3.

Following mutual trimming process and computational methods are similar to Example 1.

Embodiment 3

As shown in Figure 4, present embodiment discloses a kind of spectrometer architecture, including entrance slit, collimating mirror, dispersion element 1, detector array 2 and broadband detector 3, wherein, dispersion element 1 includes dispersion element 1-1 and image-forming component 1-2, and dispersion element 1-1 is plane grating, and image-forming component 1-2 is concave mirror；Detector array 2 is ccd array detector, broadband detector 3 is silicon photo diode, and is provided with the light filter that the zero level sub-beams generation relative spectral sensitivity curve making silicon photo diode form light splitting matches with CIE light efficiency visibility function V (λ) before the photosurface of silicon photo diode.

The operation principle of this spectrogrph is as follows: entered into inside spectrogrph by light-metering by entrance spectrometer slit, and formation collimated light beam is incident upon at dispersion element 1-1 after collimating mirror, dispersion element 1-1 carries out light splitting to incident tested light, and the non-zero order time spectrum of light splitting formation is incident upon at image-forming component 1-2, and then the spectrum being arranged in order by wavelength order occurs on the focal plane of image-forming component 1-2, and received by detector array 2 and measure；The zero level sub-beams wherein formed via dispersion element 1-1 light splitting is not incident upon on image-forming component 1-2, but directly by the received measurement of broadband detector array 3.

Following mutual trimming process and computational methods are similar to Example 1.

Claims (10)

1. a spectrogrph, including entrance slit, dispersion element (1), is used for receiving the detector array (2) of dispersed light, and its feature exists In, also including broadband detector (3), described broadband detector (3) receives the zero level time formed through dispersion element (1) light splitting Light beam.

2. spectrogrph as claimed in claim 1, it is characterised in that the photo-electric conversion element of described broadband detector (3) is photoelectricity Pond or photodiode.

3. spectrogrph as claimed in claim 1 or 2, it is characterised in that before the photo-electric conversion element of described broadband detector (3) It is provided with the filter making broadband detector (3) that the relative spectral sensitivity curve of incident illumination to be matched with special spectrum Efficiency Function Color chips.

4. spectrogrph as claimed in claim 3, it is characterised in that described light filter makes the relative spectral of broadband detector (3) Sensitivity curve matches with human eye luminous efficiency function V (λ) curve or smooth straight line；Or described light filter makes institute The relative spectral sensitivity curve stating broadband detector (3) is certain band-pass response curve.

5. spectrogrph as claimed in claim 1, it is characterised in that described dispersion element (1) is concave surface flat field grating.

6. spectrogrph as claimed in claim 1, it is characterised in that described dispersion element (1) includes dispersion element (1-1) and becomes pixel Part (1-2)；Described dispersion element (1-1) is plane grating, and described image-forming component (1-2) is concave mirror.

7. a kind of spectrogrph bearing calibration in claim 1, it is characterised in that the measurement wave band of described detector array (2) [λ₁,λ₂] and the response wave band [λ of broadband detector (3)_s1,λ_s2] match, the response wave band of broadband detector (3) [λ_s1,λ_s2] scope detector array (2) measure wave band [λ₁,λ₂Within the scope of].

8. spectrogrph bearing calibration as claimed in claim 7, it is characterised in that described detector array (2) and broadband detector (3) measure respectively by the relative spectral power distributions information of light-metering non-zero order time spectrum and believed by the luminosity of light-metering zero level sub-beams Breath, utilizes correction coefficientObtain by the absolute light spectral power distributions of light-metering with the product of P (λ)；

Wherein, P (λ) be detector array (2) record by the relative spectral power distributions of light-metering, I_sFor broadband detector (3) response value by light-metering zero level sub-beams recorded；S (λ) is that the absolute spectral response of broadband detector (3) is sensitive Degree, λ_s1、λ_s2The integration start wavelength and the integration that are respectively broadband detector (3) terminate wavelength, and unit is nm.

9. spectrogrph bearing calibration as claimed in claim 7, it is characterised in that utilize that detector array (2) records by light-metering phase To the measured value of broadband detector (3), spectral power distribution P (λ) being carried out spectrum and loses matching and correlation, correction coefficient isBy correction coefficient K1 and broadband detector (3) directly measured value I_T Be multiplied the signal value I after i.e. can being corrected_T,C。

Wherein, S (λ)_relIt is the relative spectral sensitivity of broadband detector (3), P (λ)_SFor broadband detector (3) is entered The relative spectral power distributions of the standard source of row calibration, Q (λ) is the preferable spectral response functions of broadband detector (3). λ_s1、λ_s2Being respectively broadband detector (3) integration start wavelength and integration terminates wavelength, unit is nm.

10. a kind of spectrometer measurement method in claim 1, it is characterised in that the response wave band of described broadband detector (3) [λ_s1,λ_s2] measure wave band [λ at detector array (2)₁,λ₂Outside], i.e. λ_s1>λ₂Or λ_s2<λ₁；Or broadband detector (3) Response wave band [λ_s1,λ_s2] measure wave band [λ with detector array (2)₁,λ₂] the most overlapping, i.e. λ_s1<λ₁<λ_s2<λ₂Or λ₁<λ_s1<λ₂<λ_s2。