Converting Between λ and ν

For non-spectral quantities, such as absorbance (dI/I), converting datasets is as simple as inverting the x-values. For spectral quantities, such as irradiance, conservation of energy must be taken into account:

\[f(\lambda)d\lambda = f(\nu)(-d\nu)\]

where f is any radiometric function, e.g. spectral radiance. The sign difference is due to the direction of increasing wavelength being opposite to the direction of increasing wavenumber. By differentiation,

\[f(\lambda) = f(\nu)\frac{-d\nu}{d\lambda} = f(\nu)\frac{d}{d\lambda}\left(\frac{-1}{\lambda}\right) = \frac{f(\nu)}{\lambda^2} = f(\nu)\nu^2\]

Similarly,

\[f(\nu) = f(\lambda)\frac{d\lambda}{-d\nu} = f(\lambda)\frac{d}{d\nu}\left(\frac{-1}{\nu}\right) = \frac{f(\lambda)}{\nu^2} = f(\lambda)\lambda^2\]

In the code, below, solar irradiance is calculated separately using the wavenumber blackbody function and wavelength blackbody function. Each result is converted.

'''
Python 3.x
'''

import numpy as np
import matplotlib.pyplot as plt

c = 2.998e8 # speed of light, m/s       
h = 6.626e-34 # Planck constant, J*s
kB = 1.381e-23 # Boltzmann constant, J/K

def wn2wl(x, y0, spectral=True):
    y = y0*x**2 if spectral else y0
    return 1/x, y

def wl2wn(x, y0, spectral=True):
    return wn2wl(x, y0, spectral)

def blackbody_wl(wl, T):
    return 2*h*c**2/wl**5/(np.exp(h*c/wl/kB/T)-1) # W/sr/m^2/m

def blackbody_wn(wn, T):
    return 2*h*c**2*wn**3/(np.exp(h*c*wn/kB/T)-1) # W/sr/m^2/m^-1

T = 5778 # blackbody temperature of the Sun, Kelvin
sun_distance = 92.935e6 # mi
sun_diameter = 865370 # mi
theta = np.arctan(sun_diameter/2/sun_distance) # Sun half-angle
Omega = np.pi*np.sin(theta)**2 # projected solid angle

wl_bb = np.linspace(200e-9, 2500e-9, 50) # wavelengths, m
wn_bb = 1/wl_bb # wavenumbers, m^-1
E_wl_bb = Omega*blackbody_wl(wl_bb, T) # solar irradiance, W/m^2/m
E_wn_bb = Omega*blackbody_wn(wn_bb, T) # W/m^2/m^-1
wn_converted, E_wn_converted = wl2wn(wl_bb, E_wl_bb, spectral=True)
wl_converted, E_wl_converted = wn2wl(wn_bb, E_wn_bb, spectral=True)

plt.figure(num='Conversions'); plt.clf()
plt.subplot(211)
plt.plot(wl_bb*1e9, E_wl_bb*1e6*1e-4*1e-9, label='Ω*BB(λ)')
plt.plot(wl_converted*1e9, E_wl_converted*1e6*1e-4*1e-9, 
         'x', label='Ω*BB(ν), Converted')
plt.xlabel('nm'); 
plt.ylabel('μW/cm$^2$/nm'); 
plt.grid(); plt.legend()
plt.title('Solar Irradiance at Top of Atmosphere')

plt.subplot(212)
plt.plot(wn_bb*1e-2, E_wn_bb*1e6*1e-4*1e2, label='Ω*BB(ν)')
plt.plot(wn_converted*1e-2, E_wn_converted*1e6*1e-4*1e2, 
         'x', label='Ω*BB(λ), Converted')
plt.xlabel('ν, cm$^{-1}$'); 
plt.ylabel('μW/cm$^2$/cm$^{-1}$'); 
plt.grid(); plt.legend()
plt.title('Solar Irradiance at Top of Atmosphere')
plt.tight_layout()